From d9a1ab8a1765acf6bcd517259cdf48b4f9b5bd6f Mon Sep 17 00:00:00 2001
From: "Documenter.jl" <documenter@juliadocs.github.io>
Date: Mon, 13 Nov 2023 14:17:29 +0000
Subject: [PATCH] build based on 39af259

---
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diff --git a/previews/PR345/.documenter-siteinfo.json b/previews/PR345/.documenter-siteinfo.json
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diff --git a/previews/PR345/assets/custom.css b/previews/PR345/assets/custom.css
new file mode 100644
index 000000000..e3610172e
--- /dev/null
+++ b/previews/PR345/assets/custom.css
@@ -0,0 +1,4 @@
+#documenter .docs-sidebar .docs-logo > img,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img {
+    max-height: 8rem;
+}
\ No newline at end of file
diff --git a/previews/PR345/assets/documenter.js b/previews/PR345/assets/documenter.js
new file mode 100644
index 000000000..f5311607b
--- /dev/null
+++ b/previews/PR345/assets/documenter.js
@@ -0,0 +1,889 @@
+// Generated by Documenter.jl
+requirejs.config({
+  paths: {
+    'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min',
+    'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min',
+    'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min',
+    'minisearch': 'https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min',
+    'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min',
+    'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min',
+    'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min',
+    'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min',
+    'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min',
+    'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min',
+  },
+  shim: {
+  "highlight-julia": {
+    "deps": [
+      "highlight"
+    ]
+  },
+  "katex-auto-render": {
+    "deps": [
+      "katex"
+    ]
+  },
+  "headroom-jquery": {
+    "deps": [
+      "jquery",
+      "headroom"
+    ]
+  },
+  "highlight-julia-repl": {
+    "deps": [
+      "highlight"
+    ]
+  }
+}
+});
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) {
+$(document).ready(function() {
+  renderMathInElement(
+    document.body,
+    {
+  "delimiters": [
+    {
+      "left": "$",
+      "right": "$",
+      "display": false
+    },
+    {
+      "left": "$$",
+      "right": "$$",
+      "display": true
+    },
+    {
+      "left": "\\[",
+      "right": "\\]",
+      "display": true
+    }
+  ]
+}
+
+  );
+})
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) {
+$(document).ready(function() {
+    hljs.highlightAll();
+})
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+let timer = 0;
+var isExpanded = true;
+
+$(document).on("click", ".docstring header", function () {
+  let articleToggleTitle = "Expand docstring";
+
+  debounce(() => {
+    if ($(this).siblings("section").is(":visible")) {
+      $(this)
+        .find(".docstring-article-toggle-button")
+        .removeClass("fa-chevron-down")
+        .addClass("fa-chevron-right");
+    } else {
+      $(this)
+        .find(".docstring-article-toggle-button")
+        .removeClass("fa-chevron-right")
+        .addClass("fa-chevron-down");
+
+      articleToggleTitle = "Collapse docstring";
+    }
+
+    $(this)
+      .find(".docstring-article-toggle-button")
+      .prop("title", articleToggleTitle);
+    $(this).siblings("section").slideToggle();
+  });
+});
+
+$(document).on("click", ".docs-article-toggle-button", function () {
+  let articleToggleTitle = "Expand docstring";
+  let navArticleToggleTitle = "Expand all docstrings";
+
+  debounce(() => {
+    if (isExpanded) {
+      $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down");
+      $(".docstring-article-toggle-button")
+        .removeClass("fa-chevron-down")
+        .addClass("fa-chevron-right");
+
+      isExpanded = false;
+
+      $(".docstring section").slideUp();
+    } else {
+      $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up");
+      $(".docstring-article-toggle-button")
+        .removeClass("fa-chevron-right")
+        .addClass("fa-chevron-down");
+
+      isExpanded = true;
+      articleToggleTitle = "Collapse docstring";
+      navArticleToggleTitle = "Collapse all docstrings";
+
+      $(".docstring section").slideDown();
+    }
+
+    $(this).prop("title", navArticleToggleTitle);
+    $(".docstring-article-toggle-button").prop("title", articleToggleTitle);
+  });
+});
+
+function debounce(callback, timeout = 300) {
+  if (Date.now() - timer > timeout) {
+    callback();
+  }
+
+  clearTimeout(timer);
+
+  timer = Date.now();
+}
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require([], function() {
+function addCopyButtonCallbacks() {
+  for (const el of document.getElementsByTagName("pre")) {
+    const button = document.createElement("button");
+    button.classList.add("copy-button", "fa-solid", "fa-copy");
+    button.setAttribute("aria-label", "Copy this code block");
+    button.setAttribute("title", "Copy");
+
+    el.appendChild(button);
+
+    const success = function () {
+      button.classList.add("success", "fa-check");
+      button.classList.remove("fa-copy");
+    };
+
+    const failure = function () {
+      button.classList.add("error", "fa-xmark");
+      button.classList.remove("fa-copy");
+    };
+
+    button.addEventListener("click", function () {
+      copyToClipboard(el.innerText).then(success, failure);
+
+      setTimeout(function () {
+        button.classList.add("fa-copy");
+        button.classList.remove("success", "fa-check", "fa-xmark");
+      }, 5000);
+    });
+  }
+}
+
+function copyToClipboard(text) {
+  // clipboard API is only available in secure contexts
+  if (window.navigator && window.navigator.clipboard) {
+    return window.navigator.clipboard.writeText(text);
+  } else {
+    return new Promise(function (resolve, reject) {
+      try {
+        const el = document.createElement("textarea");
+        el.textContent = text;
+        el.style.position = "fixed";
+        el.style.opacity = 0;
+        document.body.appendChild(el);
+        el.select();
+        document.execCommand("copy");
+
+        resolve();
+      } catch (err) {
+        reject(err);
+      } finally {
+        document.body.removeChild(el);
+      }
+    });
+  }
+}
+
+if (document.readyState === "loading") {
+  document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks);
+} else {
+  addCopyButtonCallbacks();
+}
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) {
+
+// Manages the top navigation bar (hides it when the user starts scrolling down on the
+// mobile).
+window.Headroom = Headroom; // work around buggy module loading?
+$(document).ready(function () {
+  $("#documenter .docs-navbar").headroom({
+    tolerance: { up: 10, down: 10 },
+  });
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'minisearch'], function($, minisearch) {
+
+// In general, most search related things will have "search" as a prefix.
+// To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc
+
+let results = [];
+let timer = undefined;
+
+let data = documenterSearchIndex["docs"].map((x, key) => {
+  x["id"] = key; // minisearch requires a unique for each object
+  return x;
+});
+
+// list below is the lunr 2.1.3 list minus the intersect with names(Base)
+// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with)
+// ideally we'd just filter the original list but it's not available as a variable
+const stopWords = new Set([
+  "a",
+  "able",
+  "about",
+  "across",
+  "after",
+  "almost",
+  "also",
+  "am",
+  "among",
+  "an",
+  "and",
+  "are",
+  "as",
+  "at",
+  "be",
+  "because",
+  "been",
+  "but",
+  "by",
+  "can",
+  "cannot",
+  "could",
+  "dear",
+  "did",
+  "does",
+  "either",
+  "ever",
+  "every",
+  "from",
+  "got",
+  "had",
+  "has",
+  "have",
+  "he",
+  "her",
+  "hers",
+  "him",
+  "his",
+  "how",
+  "however",
+  "i",
+  "if",
+  "into",
+  "it",
+  "its",
+  "just",
+  "least",
+  "like",
+  "likely",
+  "may",
+  "me",
+  "might",
+  "most",
+  "must",
+  "my",
+  "neither",
+  "no",
+  "nor",
+  "not",
+  "of",
+  "off",
+  "often",
+  "on",
+  "or",
+  "other",
+  "our",
+  "own",
+  "rather",
+  "said",
+  "say",
+  "says",
+  "she",
+  "should",
+  "since",
+  "so",
+  "some",
+  "than",
+  "that",
+  "the",
+  "their",
+  "them",
+  "then",
+  "there",
+  "these",
+  "they",
+  "this",
+  "tis",
+  "to",
+  "too",
+  "twas",
+  "us",
+  "wants",
+  "was",
+  "we",
+  "were",
+  "what",
+  "when",
+  "who",
+  "whom",
+  "why",
+  "will",
+  "would",
+  "yet",
+  "you",
+  "your",
+]);
+
+let index = new minisearch({
+  fields: ["title", "text"], // fields to index for full-text search
+  storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results
+  processTerm: (term) => {
+    let word = stopWords.has(term) ? null : term;
+    if (word) {
+      // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names
+      word = word
+        .replace(/^[^a-zA-Z0-9@!]+/, "")
+        .replace(/[^a-zA-Z0-9@!]+$/, "");
+    }
+
+    return word ?? null;
+  },
+  // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not find anything if searching for "add!", only for the entire qualification
+  tokenize: (string) => string.split(/[\s\-\.]+/),
+  // options which will be applied during the search
+  searchOptions: {
+    boost: { title: 100 },
+    fuzzy: 2,
+    processTerm: (term) => {
+      let word = stopWords.has(term) ? null : term;
+      if (word) {
+        word = word
+          .replace(/^[^a-zA-Z0-9@!]+/, "")
+          .replace(/[^a-zA-Z0-9@!]+$/, "");
+      }
+
+      return word ?? null;
+    },
+    tokenize: (string) => string.split(/[\s\-\.]+/),
+  },
+});
+
+index.addAll(data);
+
+let filters = [...new Set(data.map((x) => x.category))];
+var modal_filters = make_modal_body_filters(filters);
+var filter_results = [];
+
+$(document).on("keyup", ".documenter-search-input", function (event) {
+  // Adding a debounce to prevent disruptions from super-speed typing!
+  debounce(() => update_search(filter_results), 300);
+});
+
+$(document).on("click", ".search-filter", function () {
+  if ($(this).hasClass("search-filter-selected")) {
+    $(this).removeClass("search-filter-selected");
+  } else {
+    $(this).addClass("search-filter-selected");
+  }
+
+  // Adding a debounce to prevent disruptions from crazy clicking!
+  debounce(() => get_filters(), 300);
+});
+
+/**
+ * A debounce function, takes a function and an optional timeout in milliseconds
+ *
+ * @function callback
+ * @param {number} timeout
+ */
+function debounce(callback, timeout = 300) {
+  clearTimeout(timer);
+  timer = setTimeout(callback, timeout);
+}
+
+/**
+ * Make/Update the search component
+ *
+ * @param {string[]} selected_filters
+ */
+function update_search(selected_filters = []) {
+  let initial_search_body = `
+      <div class="has-text-centered my-5 py-5">Type something to get started!</div>
+    `;
+
+  let querystring = $(".documenter-search-input").val();
+
+  if (querystring.trim()) {
+    results = index.search(querystring, {
+      filter: (result) => {
+        // Filtering results
+        if (selected_filters.length === 0) {
+          return result.score >= 1;
+        } else {
+          return (
+            result.score >= 1 && selected_filters.includes(result.category)
+          );
+        }
+      },
+    });
+
+    let search_result_container = ``;
+    let search_divider = `<div class="search-divider w-100"></div>`;
+
+    if (results.length) {
+      let links = [];
+      let count = 0;
+      let search_results = "";
+
+      results.forEach(function (result) {
+        if (result.location) {
+          // Checking for duplication of results for the same page
+          if (!links.includes(result.location)) {
+            search_results += make_search_result(result, querystring);
+            count++;
+          }
+
+          links.push(result.location);
+        }
+      });
+
+      let result_count = `<div class="is-size-6">${count} result(s)</div>`;
+
+      search_result_container = `
+            <div class="is-flex is-flex-direction-column gap-2 is-align-items-flex-start">
+                ${modal_filters}
+                ${search_divider}
+                ${result_count}
+                <div class="is-clipped w-100 is-flex is-flex-direction-column gap-2 is-align-items-flex-start has-text-justified mt-1">
+                  ${search_results}
+                </div>
+            </div>
+        `;
+    } else {
+      search_result_container = `
+           <div class="is-flex is-flex-direction-column gap-2 is-align-items-flex-start">
+               ${modal_filters}
+               ${search_divider}
+               <div class="is-size-6">0 result(s)</div>
+            </div>
+            <div class="has-text-centered my-5 py-5">No result found!</div>
+       `;
+    }
+
+    if ($(".search-modal-card-body").hasClass("is-justify-content-center")) {
+      $(".search-modal-card-body").removeClass("is-justify-content-center");
+    }
+
+    $(".search-modal-card-body").html(search_result_container);
+  } else {
+    filter_results = [];
+    modal_filters = make_modal_body_filters(filters, filter_results);
+
+    if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) {
+      $(".search-modal-card-body").addClass("is-justify-content-center");
+    }
+
+    $(".search-modal-card-body").html(initial_search_body);
+  }
+}
+
+/**
+ * Make the modal filter html
+ *
+ * @param {string[]} filters
+ * @param {string[]} selected_filters
+ * @returns string
+ */
+function make_modal_body_filters(filters, selected_filters = []) {
+  let str = ``;
+
+  filters.forEach((val) => {
+    if (selected_filters.includes(val)) {
+      str += `<a href="javascript:;" class="search-filter search-filter-selected"><span>${val}</span></a>`;
+    } else {
+      str += `<a href="javascript:;" class="search-filter"><span>${val}</span></a>`;
+    }
+  });
+
+  let filter_html = `
+        <div class="is-flex gap-2 is-flex-wrap-wrap is-justify-content-flex-start is-align-items-center search-filters">
+            <span class="is-size-6">Filters:</span>
+            ${str}
+        </div>
+    `;
+
+  return filter_html;
+}
+
+/**
+ * Make the result component given a minisearch result data object and the value of the search input as queryString.
+ * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult
+ *
+ * @param {object} result
+ * @param {string} querystring
+ * @returns string
+ */
+function make_search_result(result, querystring) {
+  let search_divider = `<div class="search-divider w-100"></div>`;
+  let display_link =
+    result.location.slice(Math.max(0), Math.min(50, result.location.length)) +
+    (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div
+
+  if (result.page !== "") {
+    display_link += ` (${result.page})`;
+  }
+
+  let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text);
+  let text =
+    textindex !== null
+      ? result.text.slice(
+          Math.max(textindex.index - 100, 0),
+          Math.min(
+            textindex.index + querystring.length + 100,
+            result.text.length
+          )
+        )
+      : ""; // cut-off text before and after from the match
+
+  let display_result = text.length
+    ? "..." +
+      text.replace(
+        new RegExp(`\\b${querystring}\\b`, "i"), // For first occurrence
+        '<span class="search-result-highlight p-1">$&</span>'
+      ) +
+      "..."
+    : ""; // highlights the match
+
+  let in_code = false;
+  if (!["page", "section"].includes(result.category.toLowerCase())) {
+    in_code = true;
+  }
+
+  // We encode the full url to escape some special characters which can lead to broken links
+  let result_div = `
+      <a href="${encodeURI(
+        documenterBaseURL + "/" + result.location
+      )}" class="search-result-link w-100 is-flex is-flex-direction-column gap-2 px-4 py-2">
+        <div class="w-100 is-flex is-flex-wrap-wrap is-justify-content-space-between is-align-items-flex-start">
+          <div class="search-result-title has-text-weight-bold ${
+            in_code ? "search-result-code-title" : ""
+          }">${result.title}</div>
+          <div class="property-search-result-badge">${result.category}</div>
+        </div>
+        <p>
+          ${display_result}
+        </p>
+        <div
+          class="has-text-left"
+          style="font-size: smaller;"
+          title="${result.location}"
+        >
+          <i class="fas fa-link"></i> ${display_link}
+        </div>
+      </a>
+      ${search_divider}
+    `;
+
+  return result_div;
+}
+
+/**
+ * Get selected filters, remake the filter html and lastly update the search modal
+ */
+function get_filters() {
+  let ele = $(".search-filters .search-filter-selected").get();
+  filter_results = ele.map((x) => $(x).text().toLowerCase());
+  modal_filters = make_modal_body_filters(filters, filter_results);
+  update_search(filter_results);
+}
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// Modal settings dialog
+$(document).ready(function () {
+  var settings = $("#documenter-settings");
+  $("#documenter-settings-button").click(function () {
+    settings.toggleClass("is-active");
+  });
+  // Close the dialog if X is clicked
+  $("#documenter-settings button.delete").click(function () {
+    settings.removeClass("is-active");
+  });
+  // Close dialog if ESC is pressed
+  $(document).keyup(function (e) {
+    if (e.keyCode == 27) settings.removeClass("is-active");
+  });
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+let search_modal_header = `
+  <header class="modal-card-head gap-2 is-align-items-center is-justify-content-space-between w-100 px-3">
+    <div class="field mb-0 w-100">
+      <p class="control has-icons-right">
+        <input class="input documenter-search-input" type="text" placeholder="Search" />
+        <span class="icon is-small is-right has-text-primary-dark">
+          <i class="fas fa-magnifying-glass"></i>
+        </span>
+      </p>
+    </div>
+    <div class="icon is-size-4 is-clickable close-search-modal">
+      <i class="fas fa-times"></i>
+    </div>
+  </header>
+`;
+
+let initial_search_body = `
+  <div class="has-text-centered my-5 py-5">Type something to get started!</div>
+`;
+
+let search_modal_footer = `
+  <footer class="modal-card-foot">
+    <span>
+      <kbd class="search-modal-key-hints">Ctrl</kbd> +
+      <kbd class="search-modal-key-hints">/</kbd> to search
+    </span>
+    <span class="ml-3"> <kbd class="search-modal-key-hints">esc</kbd> to close </span>
+  </footer>
+`;
+
+$(document.body).append(
+  `
+    <div class="modal" id="search-modal">
+      <div class="modal-background"></div>
+      <div class="modal-card search-min-width-50 search-min-height-100 is-justify-content-center">
+        ${search_modal_header}
+        <section class="modal-card-body is-flex is-flex-direction-column is-justify-content-center gap-4 search-modal-card-body">
+          ${initial_search_body}
+        </section>
+        ${search_modal_footer}
+      </div>
+    </div>
+  `
+);
+
+document.querySelector(".docs-search-query").addEventListener("click", () => {
+  openModal();
+});
+
+document.querySelector(".close-search-modal").addEventListener("click", () => {
+  closeModal();
+});
+
+$(document).on("click", ".search-result-link", function () {
+  closeModal();
+});
+
+document.addEventListener("keydown", (event) => {
+  if ((event.ctrlKey || event.metaKey) && event.key === "/") {
+    openModal();
+  } else if (event.key === "Escape") {
+    closeModal();
+  }
+
+  return false;
+});
+
+// Functions to open and close a modal
+function openModal() {
+  let searchModal = document.querySelector("#search-modal");
+
+  searchModal.classList.add("is-active");
+  document.querySelector(".documenter-search-input").focus();
+}
+
+function closeModal() {
+  let searchModal = document.querySelector("#search-modal");
+  let initial_search_body = `
+    <div class="has-text-centered my-5 py-5">Type something to get started!</div>
+  `;
+
+  searchModal.classList.remove("is-active");
+  document.querySelector(".documenter-search-input").blur();
+
+  if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) {
+    $(".search-modal-card-body").addClass("is-justify-content-center");
+  }
+
+  $(".documenter-search-input").val("");
+  $(".search-modal-card-body").html(initial_search_body);
+}
+
+document
+  .querySelector("#search-modal .modal-background")
+  .addEventListener("click", () => {
+    closeModal();
+  });
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// Manages the showing and hiding of the sidebar.
+$(document).ready(function () {
+  var sidebar = $("#documenter > .docs-sidebar");
+  var sidebar_button = $("#documenter-sidebar-button");
+  sidebar_button.click(function (ev) {
+    ev.preventDefault();
+    sidebar.toggleClass("visible");
+    if (sidebar.hasClass("visible")) {
+      // Makes sure that the current menu item is visible in the sidebar.
+      $("#documenter .docs-menu a.is-active").focus();
+    }
+  });
+  $("#documenter > .docs-main").bind("click", function (ev) {
+    if ($(ev.target).is(sidebar_button)) {
+      return;
+    }
+    if (sidebar.hasClass("visible")) {
+      sidebar.removeClass("visible");
+    }
+  });
+});
+
+// Resizes the package name / sitename in the sidebar if it is too wide.
+// Inspired by: https://github.com/davatron5000/FitText.js
+$(document).ready(function () {
+  e = $("#documenter .docs-autofit");
+  function resize() {
+    var L = parseInt(e.css("max-width"), 10);
+    var L0 = e.width();
+    if (L0 > L) {
+      var h0 = parseInt(e.css("font-size"), 10);
+      e.css("font-size", (L * h0) / L0);
+      // TODO: make sure it survives resizes?
+    }
+  }
+  // call once and then register events
+  resize();
+  $(window).resize(resize);
+  $(window).on("orientationchange", resize);
+});
+
+// Scroll the navigation bar to the currently selected menu item
+$(document).ready(function () {
+  var sidebar = $("#documenter .docs-menu").get(0);
+  var active = $("#documenter .docs-menu .is-active").get(0);
+  if (typeof active !== "undefined") {
+    sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15;
+  }
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// Theme picker setup
+$(document).ready(function () {
+  // onchange callback
+  $("#documenter-themepicker").change(function themepick_callback(ev) {
+    var themename = $("#documenter-themepicker option:selected").attr("value");
+    if (themename === "auto") {
+      // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light');
+      window.localStorage.removeItem("documenter-theme");
+    } else {
+      // set_theme(themename);
+      window.localStorage.setItem("documenter-theme", themename);
+    }
+    // We re-use the global function from themeswap.js to actually do the swapping.
+    set_theme_from_local_storage();
+  });
+
+  // Make sure that the themepicker displays the correct theme when the theme is retrieved
+  // from localStorage
+  if (typeof window.localStorage !== "undefined") {
+    var theme = window.localStorage.getItem("documenter-theme");
+    if (theme !== null) {
+      $("#documenter-themepicker option").each(function (i, e) {
+        e.selected = e.value === theme;
+      });
+    }
+  }
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// update the version selector with info from the siteinfo.js and ../versions.js files
+$(document).ready(function () {
+  // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the
+  // siteinfo.js file, we just return immediately and not display the version selector.
+  if (
+    typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" &&
+    DOCUMENTER_VERSION_SELECTOR_DISABLED
+  ) {
+    return;
+  }
+
+  var version_selector = $("#documenter .docs-version-selector");
+  var version_selector_select = $("#documenter .docs-version-selector select");
+
+  version_selector_select.change(function (x) {
+    target_href = version_selector_select
+      .children("option:selected")
+      .get(0).value;
+    window.location.href = target_href;
+  });
+
+  // add the current version to the selector based on siteinfo.js, but only if the selector is empty
+  if (
+    typeof DOCUMENTER_CURRENT_VERSION !== "undefined" &&
+    $("#version-selector > option").length == 0
+  ) {
+    var option = $(
+      "<option value='#' selected='selected'>" +
+        DOCUMENTER_CURRENT_VERSION +
+        "</option>"
+    );
+    version_selector_select.append(option);
+  }
+
+  if (typeof DOC_VERSIONS !== "undefined") {
+    var existing_versions = version_selector_select.children("option");
+    var existing_versions_texts = existing_versions.map(function (i, x) {
+      return x.text;
+    });
+    DOC_VERSIONS.forEach(function (each) {
+      var version_url = documenterBaseURL + "/../" + each + "/";
+      var existing_id = $.inArray(each, existing_versions_texts);
+      // if not already in the version selector, add it as a new option,
+      // otherwise update the old option with the URL and enable it
+      if (existing_id == -1) {
+        var option = $(
+          "<option value='" + version_url + "'>" + each + "</option>"
+        );
+        version_selector_select.append(option);
+      } else {
+        var option = existing_versions[existing_id];
+        option.value = version_url;
+        option.disabled = false;
+      }
+    });
+  }
+
+  // only show the version selector if the selector has been populated
+  if (version_selector_select.children("option").length > 0) {
+    version_selector.toggleClass("visible");
+  }
+});
+
+})
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kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark 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kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content 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kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-focused.docs-sourcelink{border-color:transparent;color:#fff}html.theme--documenter-dark 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h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content 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html.theme--documenter-dark .navbar-dropdown,html.theme--documenter-dark .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--documenter-dark .navbar-dropdown.is-right{left:auto;right:0}html.theme--documenter-dark .navbar-divider{display:block}html.theme--documenter-dark .navbar>.container .navbar-brand,html.theme--documenter-dark .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--documenter-dark .navbar>.container .navbar-menu,html.theme--documenter-dark .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--documenter-dark .navbar.is-fixed-bottom-desktop,html.theme--documenter-dark .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--documenter-dark 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a.navbar-item.is-active:not(:focus):not(:hover),html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--documenter-dark .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--documenter-dark .pagination{font-size:1rem;margin:-.25rem}html.theme--documenter-dark .pagination.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--documenter-dark .pagination.is-medium{font-size:1.25rem}html.theme--documenter-dark .pagination.is-large{font-size:1.5rem}html.theme--documenter-dark .pagination.is-rounded .pagination-previous,html.theme--documenter-dark #documenter .docs-sidebar 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diff --git a/previews/PR345/assets/themes/documenter-light.css b/previews/PR345/assets/themes/documenter-light.css
new file mode 100644
index 000000000..1262ec506
--- /dev/null
+++ b/previews/PR345/assets/themes/documenter-light.css
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+  Theme: Default
+  Description: Original highlight.js style
+  Author: (c) Ivan Sagalaev <maniac@softwaremaniacs.org>
+  Maintainer: @highlightjs/core-team
+  Website: https://highlightjs.org/
+  License: see project LICENSE
+  Touched: 2021
+*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem}
diff --git a/previews/PR345/assets/themeswap.js b/previews/PR345/assets/themeswap.js
new file mode 100644
index 000000000..9f5eebe6a
--- /dev/null
+++ b/previews/PR345/assets/themeswap.js
@@ -0,0 +1,84 @@
+// Small function to quickly swap out themes. Gets put into the <head> tag..
+function set_theme_from_local_storage() {
+  // Initialize the theme to null, which means default
+  var theme = null;
+  // If the browser supports the localstorage and is not disabled then try to get the
+  // documenter theme
+  if (window.localStorage != null) {
+    // Get the user-picked theme from localStorage. May be `null`, which means the default
+    // theme.
+    theme = window.localStorage.getItem("documenter-theme");
+  }
+  // Check if the users preference is for dark color scheme
+  var darkPreference =
+    window.matchMedia("(prefers-color-scheme: dark)").matches === true;
+  // Initialize a few variables for the loop:
+  //
+  //  - active: will contain the index of the theme that should be active. Note that there
+  //    is no guarantee that localStorage contains sane values. If `active` stays `null`
+  //    we either could not find the theme or it is the default (primary) theme anyway.
+  //    Either way, we then need to stick to the primary theme.
+  //
+  //  - disabled: style sheets that should be disabled (i.e. all the theme style sheets
+  //    that are not the currently active theme)
+  var active = null;
+  var disabled = [];
+  var primaryLightTheme = null;
+  var primaryDarkTheme = null;
+  for (var i = 0; i < document.styleSheets.length; i++) {
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+    // The <link> tag of each style sheet is expected to have a data-theme-name attribute
+    // which must contain the name of the theme. The names in localStorage much match this.
+    var themename = ss.ownerNode.getAttribute("data-theme-name");
+    // attribute not set => non-theme stylesheet => ignore
+    if (themename === null) continue;
+    // To distinguish the default (primary) theme, it needs to have the data-theme-primary
+    // attribute set.
+    if (ss.ownerNode.getAttribute("data-theme-primary") !== null) {
+      primaryLightTheme = themename;
+    }
+    // Check if the theme is primary dark theme so that we could store its name in darkTheme
+    if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) {
+      primaryDarkTheme = themename;
+    }
+    // If we find a matching theme (and it's not the default), we'll set active to non-null
+    if (themename === theme) active = i;
+    // Store the style sheets of inactive themes so that we could disable them
+    if (themename !== theme) disabled.push(ss);
+  }
+  var activeTheme = null;
+  if (active !== null) {
+    // If we did find an active theme, we'll (1) add the theme--$(theme) class to <html>
+    document.getElementsByTagName("html")[0].className = "theme--" + theme;
+    activeTheme = theme;
+  } else {
+    // If we did _not_ find an active theme, then we need to fall back to the primary theme
+    // which can either be dark or light, depending on the user's OS preference.
+    var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme;
+    // In case it somehow happens that the relevant primary theme was not found in the
+    // preceding loop, we abort without doing anything.
+    if (activeTheme === null) {
+      console.error("Unable to determine primary theme.");
+      return;
+    }
+    // When switching to the primary light theme, then we must not have a class name
+    // for the <html> tag. That's only for non-primary or the primary dark theme.
+    if (darkPreference) {
+      document.getElementsByTagName("html")[0].className =
+        "theme--" + activeTheme;
+    } else {
+      document.getElementsByTagName("html")[0].className = "";
+    }
+  }
+  for (var i = 0; i < document.styleSheets.length; i++) {
+    var ss = document.styleSheets[i];
+    // The <link> tag of each style sheet is expected to have a data-theme-name attribute
+    // which must contain the name of the theme. The names in localStorage much match this.
+    var themename = ss.ownerNode.getAttribute("data-theme-name");
+    // attribute not set => non-theme stylesheet => ignore
+    if (themename === null) continue;
+    // we'll disable all the stylesheets, except for the active one
+    ss.disabled = !(themename == activeTheme);
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diff --git a/previews/PR345/assets/warner.js b/previews/PR345/assets/warner.js
new file mode 100644
index 000000000..3f6f5d008
--- /dev/null
+++ b/previews/PR345/assets/warner.js
@@ -0,0 +1,52 @@
+function maybeAddWarning() {
+  // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE
+  // in siteinfo.js.
+  // If either of these are undefined something went horribly wrong, so we abort.
+  if (
+    window.DOCUMENTER_NEWEST === undefined ||
+    window.DOCUMENTER_CURRENT_VERSION === undefined ||
+    window.DOCUMENTER_STABLE === undefined
+  ) {
+    return;
+  }
+
+  // Current version is not a version number, so we can't tell if it's the newest version. Abort.
+  if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) {
+    return;
+  }
+
+  // Current version is newest version, so no need to add a warning.
+  if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) {
+    return;
+  }
+
+  // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs.
+  if (document.body.querySelector('meta[name="robots"]') === null) {
+    const meta = document.createElement("meta");
+    meta.name = "robots";
+    meta.content = "noindex";
+
+    document.getElementsByTagName("head")[0].appendChild(meta);
+  }
+
+  const div = document.createElement("div");
+  div.classList.add("outdated-warning-overlay");
+  const closer = document.createElement("button");
+  closer.classList.add("outdated-warning-closer", "delete");
+  closer.addEventListener("click", function () {
+    document.body.removeChild(div);
+  });
+  const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE;
+  div.innerHTML =
+    'This documentation is not for the latest stable release, but for either the development version or an older release.<br><a href="' +
+    href +
+    '">Click here to go to the documentation for the latest stable release.</a>';
+  div.appendChild(closer);
+  document.body.appendChild(div);
+}
+
+if (document.readyState === "loading") {
+  document.addEventListener("DOMContentLoaded", maybeAddWarning);
+} else {
+  maybeAddWarning();
+}
diff --git a/previews/PR345/basicbsplineexporter/index.html b/previews/PR345/basicbsplineexporter/index.html
new file mode 100644
index 000000000..f6f9fef3f
--- /dev/null
+++ b/previews/PR345/basicbsplineexporter/index.html
@@ -0,0 +1,15 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>BasicBSplineExporter.jl · BasicBSpline.jl</title><meta name="title" content="BasicBSplineExporter.jl · BasicBSpline.jl"/><meta property="og:title" content="BasicBSplineExporter.jl · BasicBSpline.jl"/><meta property="twitter:title" content="BasicBSplineExporter.jl · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/basicbsplineexporter/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/basicbsplineexporter/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/basicbsplineexporter/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/><link href="../assets/custom.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="BasicBSpline.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="BasicBSpline.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">BasicBSpline.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li><a class="tocitem" href="../math-knotvector/">Knot vector</a></li><li><a class="tocitem" href="../math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="../math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li><a class="tocitem" href="../math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li><a class="tocitem" href="../math-fitting/">Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li class="is-active"><a class="tocitem" href>BasicBSplineExporter.jl</a><ul class="internal"><li><a class="tocitem" href="#Installation"><span>Installation</span></a></li><li><a class="tocitem" href="#First-example"><span>First example</span></a></li><li><a class="tocitem" href="#Other-examples"><span>Other examples</span></a></li></ul></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Visualization</a></li><li class="is-active"><a href>BasicBSplineExporter.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>BasicBSplineExporter.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/basicbsplineexporter.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="BasicBSplineExporter.jl"><a class="docs-heading-anchor" href="#BasicBSplineExporter.jl">BasicBSplineExporter.jl</a><a id="BasicBSplineExporter.jl-1"></a><a class="docs-heading-anchor-permalink" href="#BasicBSplineExporter.jl" title="Permalink"></a></h1><p><a href="https://github.com/hyrodium/BasicBSplineExporter.jl">BasicBSplineExporter.jl</a> supports export <code>BasicBSpline.BSplineManifold{Dim,Deg,&lt;:StaticVector}</code> to:</p><ul><li>PNG image (<code>.png</code>)</li><li>SVG image (<code>.png</code>)</li><li>POV-Ray mesh (<code>.inc</code>)</li></ul><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><pre><code class="language-julia hljs">] add https://github.com/hyrodium/BasicBSplineExporter.jl</code></pre><h2 id="First-example"><a class="docs-heading-anchor" href="#First-example">First example</a><a id="First-example-1"></a><a class="docs-heading-anchor-permalink" href="#First-example" title="Permalink"></a></h2><pre><code class="language-julia hljs">using BasicBSpline
+using BasicBSplineExporter
+using StaticArrays
+
+p = 2 # degree of polynomial
+k1 = KnotVector(1:8) # knot vector
+k2 = KnotVector(rand(7))+(p+1)*KnotVector([1])
+P1 = BSplineSpace{p}(k1) # B-spline space
+P2 = BSplineSpace{p}(k2)
+n1 = dim(P1) # dimension of B-spline space
+n2 = dim(P2)
+a = [SVector(2i-6.5+rand(),1.5j-6.5+rand()) for i in 1:dim(P1), j in 1:dim(P2)] # random generated control points
+M = BSplineManifold(a,(P1,P2)) # Define B-spline manifold
+save_png(&quot;BasicBSplineExporter_2dim.png&quot;, M) # save image</code></pre><p><img src="../BasicBSplineExporter_2dim.png" alt/></p><h2 id="Other-examples"><a class="docs-heading-anchor" href="#Other-examples">Other examples</a><a id="Other-examples-1"></a><a class="docs-heading-anchor-permalink" href="#Other-examples" title="Permalink"></a></h2><p>Here are some images rendared with POV-Ray.</p><p><img src="../img/pov_1d3d.png" alt/> <img src="../img/pov_2d3d.png" alt/> <img src="../img/pov_3d3d.png" alt/></p><p>See <a href="https://github.com/hyrodium/BasicBSplineExporter.jl/tree/main/test"><code>BasicBSplineExporter.jl/test</code></a> for more examples.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../plotlyjs/">« PlotlyJS.jl</a><a class="docs-footer-nextpage" href="../geometricmodeling/">Geometric modeling »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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new file mode 100644
index 000000000..af45641d4
--- /dev/null
+++ b/previews/PR345/bsplinebasisall-1.html
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+<script src="https://cdn.plot.ly/plotly-2.6.3.min.js"></script>    <div id="ecfc78d8-974d-4741-95cf-22e6cf1cb7e7" style="width:600px;height:400px;"></div>
+    <script>
+    
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Contributing · BasicBSpline.jl</title><meta name="title" content="Contributing · BasicBSpline.jl"/><meta property="og:title" content="Contributing · BasicBSpline.jl"/><meta property="twitter:title" content="Contributing · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/contributing/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/contributing/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/contributing/"/><script data-outdated-warner src="../assets/warner.js"></script><link 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So, English corrections would be really helpful. 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diff --git a/previews/PR345/geometricmodeling/index.html b/previews/PR345/geometricmodeling/index.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Geometric modeling · BasicBSpline.jl</title><meta name="title" content="Geometric modeling · BasicBSpline.jl"/><meta property="og:title" content="Geometric modeling · BasicBSpline.jl"/><meta property="twitter:title" content="Geometric modeling · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/geometricmodeling/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/geometricmodeling/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/geometricmodeling/"/><script data-outdated-warner 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manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li class="is-active"><a class="tocitem" href>Geometric modeling</a><ul class="internal"><li><a class="tocitem" href="#Load-packages"><span>Load packages</span></a></li><li><a class="tocitem" href="#Arc"><span>Arc</span></a></li><li><a class="tocitem" href="#Circle"><span>Circle</span></a></li><li><a class="tocitem" href="#Torus"><span>Torus</span></a></li><li><a class="tocitem" href="#Paraboloid"><span>Paraboloid</span></a></li><li><a class="tocitem" href="#Hyperbolic-paraboloid"><span>Hyperbolic paraboloid</span></a></li></ul></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Geometric modeling</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Geometric modeling</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/geometricmodeling.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Geometric-modeling"><a class="docs-heading-anchor" href="#Geometric-modeling">Geometric modeling</a><a id="Geometric-modeling-1"></a><a class="docs-heading-anchor-permalink" href="#Geometric-modeling" title="Permalink"></a></h1><h2 id="Load-packages"><a class="docs-heading-anchor" href="#Load-packages">Load packages</a><a id="Load-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Load-packages" title="Permalink"></a></h2><pre><code class="language-julia hljs">using BasicBSpline
+using StaticArrays
+using Plots
+using LinearAlgebra
+plotly()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Plots.PlotlyBackend()</code></pre><h2 id="Arc"><a class="docs-heading-anchor" href="#Arc">Arc</a><a id="Arc-1"></a><a class="docs-heading-anchor-permalink" href="#Arc" title="Permalink"></a></h2><pre><code class="language-julia hljs">p = 2
+k = KnotVector([0,0,0,1,1,1])
+P = BSplineSpace{p}(k)
+t = 1  # angle in radians
+a = [SVector(1,0), SVector(1,tan(t/2)), SVector(cos(t),sin(t))]
+w = [1,cos(t/2),1]
+M = RationalBSplineManifold(a,w,P)
+plot(M, xlims=(0,1.1), ylims=(0,1.1), aspectratio=1)</code></pre><object type="text/html" data="../geometricmodeling-arc.html" style="width:100%;height:420px;"></object><h2 id="Circle"><a class="docs-heading-anchor" href="#Circle">Circle</a><a id="Circle-1"></a><a class="docs-heading-anchor-permalink" href="#Circle" title="Permalink"></a></h2><pre><code class="language-julia hljs">p = 2
+k = KnotVector([0,0,0,1,1,2,2,3,3,4,4,4])
+P = BSplineSpace{p}(k)
+a = [
+    SVector( 1, 0),
+    SVector( 1, 1),
+    SVector( 0, 1),
+    SVector(-1, 1),
+    SVector(-1, 0),
+    SVector(-1,-1),
+    SVector( 0,-1),
+    SVector( 1,-1),
+    SVector( 1, 0)
+]
+w = [1,1/√2,1,1/√2,1,1/√2,1,1/√2,1]
+M = RationalBSplineManifold(a,w,P)
+plot(M, xlims=(-1.2,1.2), ylims=(-1.2,1.2), aspectratio=1)</code></pre><object type="text/html" data="../geometricmodeling-circle.html" style="width:100%;height:420px;"></object><h2 id="Torus"><a class="docs-heading-anchor" href="#Torus">Torus</a><a id="Torus-1"></a><a class="docs-heading-anchor-permalink" href="#Torus" title="Permalink"></a></h2><pre><code class="language-julia hljs">R = 3
+r = 1
+
+a0 = [
+    SVector( 1, 0, 0),
+    SVector( 1, 1, 0),
+    SVector( 0, 1, 0),
+    SVector(-1, 1, 0),
+    SVector(-1, 0, 0),
+    SVector(-1,-1, 0),
+    SVector( 0,-1, 0),
+    SVector( 1,-1, 0),
+    SVector( 1, 0, 0)
+]
+
+a1 = (R+r)*a0
+a5 = (R-r)*a0
+a2 = [p+r*SVector(0,0,1) for p in a1]
+a3 = [p+r*SVector(0,0,1) for p in R*a0]
+a4 = [p+r*SVector(0,0,1) for p in a5]
+a6 = [p-r*SVector(0,0,1) for p in a5]
+a7 = [p-r*SVector(0,0,1) for p in R*a0]
+a8 = [p-r*SVector(0,0,1) for p in a1]
+a9 = a1
+
+a = hcat(a1,a2,a3,a4,a5,a6,a7,a8,a9)
+M = RationalBSplineManifold(a,w*w&#39;,P,P)
+plot(M)</code></pre><object type="text/html" data="../geometricmodeling-torus.html" style="width:100%;height:420px;"></object><h2 id="Paraboloid"><a class="docs-heading-anchor" href="#Paraboloid">Paraboloid</a><a id="Paraboloid-1"></a><a class="docs-heading-anchor-permalink" href="#Paraboloid" title="Permalink"></a></h2><pre><code class="language-julia hljs">p = 2
+k = KnotVector([-1,-1,-1,1,1,1])
+P = BSplineSpace{p}(k)
+a = [SVector(i,j,2i^2+2j^2-2) for i in -1:1, j in -1:1]
+M = BSplineManifold(a,P,P)
+plot(M)</code></pre><object type="text/html" data="../geometricmodeling-paraboloid.html" style="width:100%;height:420px;"></object><h2 id="Hyperbolic-paraboloid"><a class="docs-heading-anchor" href="#Hyperbolic-paraboloid">Hyperbolic paraboloid</a><a id="Hyperbolic-paraboloid-1"></a><a class="docs-heading-anchor-permalink" href="#Hyperbolic-paraboloid" title="Permalink"></a></h2><pre><code class="language-julia hljs">a = [SVector(i,j,2i^2-2j^2) for i in -1:1, j in -1:1]
+M = BSplineManifold(a,P,P)
+plot(M)</code></pre><object type="text/html" data="../geometricmodeling-hyperbolicparaboloid.html" style="width:100%;height:420px;"></object></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../basicbsplineexporter/">« BasicBSplineExporter.jl</a><a class="docs-footer-nextpage" href="../interpolations/">Interpolations »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR345/helix.html b/previews/PR345/helix.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · BasicBSpline.jl</title><meta name="title" content="Home · BasicBSpline.jl"/><meta property="og:title" content="Home · BasicBSpline.jl"/><meta property="twitter:title" content="Home · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/"/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" 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href>Home</a><ul class="internal"><li><a class="tocitem" href="#Summary"><span>Summary</span></a></li><li><a class="tocitem" href="#Comparison-to-other-Julia-packages-for-B-spline"><span>Comparison to other Julia packages for B-spline</span></a></li><li><a class="tocitem" href="#Installation"><span>Installation</span></a></li><li><a class="tocitem" href="#Example"><span>Example</span></a></li></ul></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="math/">Introduction</a></li><li><a class="tocitem" href="math-knotvector/">Knot vector</a></li><li><a class="tocitem" href="math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="math-derivative/">Derivative</a></li><li><a class="tocitem" href="math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="math-refinement/">Refinement</a></li><li><a class="tocitem" href="math-fitting/">Fitting</a></li><li><a class="tocitem" href="math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="plots/">Plots.jl</a></li><li><a class="tocitem" href="plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="interpolations/">Interpolations</a></li><li><a class="tocitem" href="internal/">Private API</a></li><li><a class="tocitem" href="contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/index.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="BasicBSpline.jl"><a class="docs-heading-anchor" href="#BasicBSpline.jl">BasicBSpline.jl</a><a id="BasicBSpline.jl-1"></a><a class="docs-heading-anchor-permalink" href="#BasicBSpline.jl" title="Permalink"></a></h1><p>Basic (mathematical) operations for B-spline functions and related things with Julia.</p><p><a href="https://hyrodium.github.io/BasicBSpline.jl/stable"><img src="https://img.shields.io/badge/docs-stable-blue.svg" alt="Stable"/></a> <a href="https://hyrodium.github.io/BasicBSpline.jl/dev"><img src="https://img.shields.io/badge/docs-dev-blue.svg" alt="Dev"/></a> <a href="https://github.com/hyrodium/BasicBSpline.jl/actions"><img src="https://github.com/hyrodium/BasicBSpline.jl/workflows/CI/badge.svg" alt="Build Status"/></a> <a href="https://codecov.io/gh/hyrodium/BasicBSpline.jl"><img src="https://codecov.io/gh/hyrodium/BasicBSpline.jl/branch/main/graph/badge.svg" alt="Coverage"/></a> <a href="https://github.com/JuliaTesting/Aqua.jl"><img src="https://raw.githubusercontent.com/JuliaTesting/Aqua.jl/master/badge.svg" alt="Aqua QA"/></a> <a href="https://zenodo.org/badge/latestdoi/258791290"><img src="https://zenodo.org/badge/258791290.svg" alt="DOI"/></a> <a href="https://pkgs.genieframework.com?packages=BasicBSpline"><img src="https://shields.io/endpoint?url=https://pkgs.genieframework.com/api/v1/badge/BasicBSpline" alt="BasicBSpline Downloads"/></a>.</p><p><img src="img/BasicBSplineLogo.png" alt/></p><h2 id="Summary"><a class="docs-heading-anchor" href="#Summary">Summary</a><a id="Summary-1"></a><a class="docs-heading-anchor-permalink" href="#Summary" title="Permalink"></a></h2><p>This package provides basic mathematical operations for <a href="https://en.wikipedia.org/wiki/B-spline">B-spline</a>.</p><ul><li>B-spline basis function</li><li>Some operations for knot vector</li><li>Some operations for B-spline space (piecewise polynomial space)</li><li>B-spline manifold (includes curve, surface and solid)</li><li>Refinement algorithm for B-spline manifold</li><li>Fitting control points for a given function</li></ul><h2 id="Comparison-to-other-Julia-packages-for-B-spline"><a class="docs-heading-anchor" href="#Comparison-to-other-Julia-packages-for-B-spline">Comparison to other Julia packages for B-spline</a><a id="Comparison-to-other-Julia-packages-for-B-spline-1"></a><a class="docs-heading-anchor-permalink" href="#Comparison-to-other-Julia-packages-for-B-spline" title="Permalink"></a></h2><p>There are several Julia packages for B-spline, and this package distinguishes itself with the following key benefits:</p><ul><li>Supports all degrees of polynomials.</li><li>Includes a refinement algorithm for B-spline manifolds.</li><li>Offers a fitting algorithm using least squares. (<a href="https://github.com/hyrodium/BasicBSplineFitting.jl">BasicBSplineFitting.jl</a>)</li><li>Delivers high-speed performance.</li><li>Is mathematically oriented.</li></ul><p>If you have any thoughts, please comment in:</p><ul><li><a href="https://github.com/hyrodium/BasicBSpline.jl/issues/161">Issue#161</a></li><li><a href="https://discourse.julialang.org/t//96323">Discourse post about BasicBSpline.jl</a>.</li></ul><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install this package</p><pre><code class="language-julia hljs">]add BasicBSpline</code></pre><p>To export graphics, use <a href="https://github.com/hyrodium/BasicBSplineExporter.jl">BasicBSplineExporter.jl</a>.</p><pre><code class="language-julia hljs">]add https://github.com/hyrodium/BasicBSplineExporter.jl</code></pre><h2 id="Example"><a class="docs-heading-anchor" href="#Example">Example</a><a id="Example-1"></a><a class="docs-heading-anchor-permalink" href="#Example" title="Permalink"></a></h2><h3 id="B-spline-function"><a class="docs-heading-anchor" href="#B-spline-function">B-spline function</a><a id="B-spline-function-1"></a><a class="docs-heading-anchor-permalink" href="#B-spline-function" title="Permalink"></a></h3><pre><code class="language-julia hljs">using BasicBSpline
+using Plots
+
+k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+P0 = BSplineSpace{0}(k) # 0th degree piecewise polynomial space
+P1 = BSplineSpace{1}(k) # 1st degree piecewise polynomial space
+P2 = BSplineSpace{2}(k) # 2nd degree piecewise polynomial space
+P3 = BSplineSpace{3}(k) # 3rd degree piecewise polynomial space
+plot(
+    plot([t-&gt;bsplinebasis(P0,i,t) for i in 1:dim(P0)], 0, 10, ylims=(0,1), legend=false),
+    plot([t-&gt;bsplinebasis(P1,i,t) for i in 1:dim(P1)], 0, 10, ylims=(0,1), legend=false),
+    plot([t-&gt;bsplinebasis(P2,i,t) for i in 1:dim(P2)], 0, 10, ylims=(0,1), legend=false),
+    plot([t-&gt;bsplinebasis(P3,i,t) for i in 1:dim(P3)], 0, 10, ylims=(0,1), legend=false),
+    layout=(2,2),
+)</code></pre><p><img src="img/cover.png" alt/></p><p>Try <a href="https://www.desmos.com/calculator/ql6jqgdabs">an interactive graph with Desmos graphing calculator</a>!</p><h3 id="B-spline-manifold"><a class="docs-heading-anchor" href="#B-spline-manifold">B-spline manifold</a><a id="B-spline-manifold-1"></a><a class="docs-heading-anchor-permalink" href="#B-spline-manifold" title="Permalink"></a></h3><pre><code class="language-julia hljs">using BasicBSpline
+using BasicBSplineExporter
+using StaticArrays
+
+p = 2 # degree of polynomial
+k1 = KnotVector(1:8)     # knot vector
+k2 = KnotVector(rand(7))+(p+1)*KnotVector([1])
+P1 = BSplineSpace{p}(k1) # B-spline space
+P2 = BSplineSpace{p}(k2)
+n1 = dim(P1) # dimension of B-spline space
+n2 = dim(P2)
+a = [SVector(2i-6.5+rand(),1.5j-6.5+rand()) for i in 1:dim(P1), j in 1:dim(P2)] # random generated control points
+M = BSplineManifold(a,(P1,P2)) # Define B-spline manifold
+save_png(&quot;2dim.png&quot;, M) # save image</code></pre><p><img src="img/2dim.png" alt/></p><h3 id="Refinement"><a class="docs-heading-anchor" href="#Refinement">Refinement</a><a id="Refinement-1"></a><a class="docs-heading-anchor-permalink" href="#Refinement" title="Permalink"></a></h3><h4 id="h-refinement"><a class="docs-heading-anchor" href="#h-refinement">h-refinement</a><a id="h-refinement-1"></a><a class="docs-heading-anchor-permalink" href="#h-refinement" title="Permalink"></a></h4><pre><code class="language-julia hljs">k₊=(KnotVector([3.3,4.2]),KnotVector([0.3,0.5])) # additional knot vectors
+M_h = refinement(M, k₊) # refinement of B-spline manifold
+save_png(&quot;2dim_h-refinement.png&quot;, M_h) # save image</code></pre><p><img src="img/2dim_h-refinement.png" alt/></p><p>Note that this shape and the last shape are equivalent.</p><h4 id="p-refinement"><a class="docs-heading-anchor" href="#p-refinement">p-refinement</a><a id="p-refinement-1"></a><a class="docs-heading-anchor-permalink" href="#p-refinement" title="Permalink"></a></h4><pre><code class="language-julia hljs">p₊=(Val(1),Val(2)) # additional degrees
+M_p = refinement(M, p₊) # refinement of B-spline manifold
+save_png(&quot;2dim_p-refinement.png&quot;, M_p) # save image</code></pre><p><img src="img/2dim_p-refinement.png" alt/></p><p>Note that this shape and the last shape are equivalent.</p><h3 id="Fitting-B-spline-manifold"><a class="docs-heading-anchor" href="#Fitting-B-spline-manifold">Fitting B-spline manifold</a><a id="Fitting-B-spline-manifold-1"></a><a class="docs-heading-anchor-permalink" href="#Fitting-B-spline-manifold" title="Permalink"></a></h3><p><a href="https://www.desmos.com/calculator/2hm3b1fbdf">Try on Desmos graphing calculator!</a></p><pre><code class="language-julia hljs">using BasicBSplineFitting
+
+p1 = 2
+p2 = 2
+k1 = KnotVector(-10:10)+p1*KnotVector([-10,10])
+k2 = KnotVector(-10:10)+p2*KnotVector([-10,10])
+P1 = BSplineSpace{p1}(k1)
+P2 = BSplineSpace{p2}(k2)
+
+f(u1, u2) = SVector(2u1 + sin(u1) + cos(u2) + u2 / 2, 3u2 + sin(u2) + sin(u1) / 2 + u1^2 / 6) / 5
+
+a = fittingcontrolpoints(f, (P1, P2))
+M = BSplineManifold(a, (P1, P2))
+save_png(&quot;fitting.png&quot;, M, unitlength=50, xlims=(-10,10), ylims=(-10,10))</code></pre><p><img src="img/fitting_desmos.png" alt/> <img src="img/fitting.png" alt/></p><p>If the knot vector span is too coarse, the approximation will be coarse.</p><pre><code class="language-julia hljs">p1 = 2
+p2 = 2
+k1 = KnotVector(-10:5:10)+p1*KnotVector([-10,10])
+k2 = KnotVector(-10:5:10)+p2*KnotVector([-10,10])
+P1 = BSplineSpace{p1}(k1)
+P2 = BSplineSpace{p2}(k2)
+
+f(u1, u2) = SVector(2u1 + sin(u1) + cos(u2) + u2 / 2, 3u2 + sin(u2) + sin(u1) / 2 + u1^2 / 6) / 5
+
+a = fittingcontrolpoints(f, (P1, P2))
+M = BSplineManifold(a, (P1, P2))
+save_png(&quot;fitting_coarse.png&quot;, M, unitlength=50, xlims=(-10,10), ylims=(-10,10))</code></pre><p><img src="img/fitting_coarse.png" alt/></p><h3 id="Draw-smooth-vector-graphics"><a class="docs-heading-anchor" href="#Draw-smooth-vector-graphics">Draw smooth vector graphics</a><a id="Draw-smooth-vector-graphics-1"></a><a class="docs-heading-anchor-permalink" href="#Draw-smooth-vector-graphics" title="Permalink"></a></h3><pre><code class="language-julia hljs">p = 3
+k = KnotVector(range(-2π,2π,length=8))+p*KnotVector(-2π,2π)
+P = BSplineSpace{p}(k)
+
+f(u) = SVector(u,sin(u))
+
+a = fittingcontrolpoints(f, P)
+M = BSplineManifold(a, P)
+save_svg(&quot;sine-curve.svg&quot;, M, unitlength=50, xlims=(-2,2), ylims=(-8,8))
+save_svg(&quot;sine-curve_no-points.svg&quot;, M, unitlength=50, xlims=(-2,2), ylims=(-8,8), points=false)</code></pre><p><img src="img/sine-curve.svg" alt/> <img src="img/sine-curve_no-points.svg" alt/></p><p>This is useful when you edit graphs (or curves) with your favorite vector graphics editor.</p><p><img src="img/inkscape.png" alt/></p><p>See <a href="https://forem.julialang.org/hyrodium/plotting-smooth-graphs-with-julia-6mj">Plotting smooth graphs with Julia</a> for more tutorials.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="math/">Introduction »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Private API · BasicBSpline.jl</title><meta name="title" content="Private API · BasicBSpline.jl"/><meta property="og:title" content="Private API · BasicBSpline.jl"/><meta property="twitter:title" content="Private API · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/internal/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/internal/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/internal/"/><script data-outdated-warner src="../assets/warner.js"></script><link 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src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/><link href="../assets/custom.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="BasicBSpline.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="BasicBSpline.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">BasicBSpline.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li><a class="tocitem" href="../math-knotvector/">Knot vector</a></li><li><a class="tocitem" href="../math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="../math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li><a class="tocitem" href="../math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li><a class="tocitem" href="../math-fitting/">Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li class="is-active"><a class="tocitem" href>Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Private API</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Private API</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/internal.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Private-API"><a class="docs-heading-anchor" href="#Private-API">Private API</a><a id="Private-API-1"></a><a class="docs-heading-anchor-permalink" href="#Private-API" title="Permalink"></a></h1><p>Note that the following methods are considered private methods, and changes in their behavior are not considered breaking changes.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.r_nomial" href="#BasicBSpline.r_nomial"><code>BasicBSpline.r_nomial</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Calculate <span>$r$</span>-nomial coefficient</p><pre><code class="nohighlight hljs">r_nomial(n, k, r)</code></pre><p class="math-container">\[(1+x+\cdots+x^r)^n = \sum_{k} a_{n,k,r} x^k\]</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_util.jl#LL9-L17">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline._vec" href="#BasicBSpline._vec"><code>BasicBSpline._vec</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Convert <code>AbstractKnotVector</code> to <code>AbstractVector</code></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL400-L402">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline._lower_R" href="#BasicBSpline._lower_R"><code>BasicBSpline._lower_R</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Internal methods for obtaining a B-spline space with one degree lower.</p><p class="math-container">\[\begin{aligned}
+\mathcal{P}[p,k] &amp;\mapsto \mathcal{P}[p-1,k] \\
+D^r\mathcal{P}[p,k] &amp;\mapsto D^{r-1}\mathcal{P}[p-1,k]
+\end{aligned}\]</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL377-L385">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline._changebasis_sim" href="#BasicBSpline._changebasis_sim"><code>BasicBSpline._changebasis_sim</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Return a coefficient matrix <span>$A$</span> which satisfy</p><p class="math-container">\[B_{(i,p_1,k_1)} = \sum_{j}A_{i,j}B_{(j,p_2,k_2)}\]</p><p>Assumption:</p><ul><li><span>$P_1 ≃ P_2$</span></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_ChangeBasis.jl#LL266-L274">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSplineFitting.innerproduct_R" href="#BasicBSplineFitting.innerproduct_R"><code>BasicBSplineFitting.innerproduct_R</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Calculate a matrix</p><p class="math-container">\[A_{ij}=\int_{\mathbb{R}} B_{(i,p,k)}(t) B_{(j,p,k)}(t) dt\]</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSplineFitting.jl/blob/v0.1.0/src/_Fitting.jl#L35-L40">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSplineFitting.innerproduct_I" href="#BasicBSplineFitting.innerproduct_I"><code>BasicBSplineFitting.innerproduct_I</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Calculate a matrix</p><p class="math-container">\[A_{ij}=\int_{I} B_{(i,p,k)}(t) B_{(j,p,k)}(t) dt\]</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSplineFitting.jl/blob/v0.1.0/src/_Fitting.jl#L3-L8">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../interpolations/">« Interpolations</a><a class="docs-footer-nextpage" href="../contributing/">Contributing »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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diff --git a/previews/PR345/interpolations/index.html b/previews/PR345/interpolations/index.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Interpolations · BasicBSpline.jl</title><meta name="title" content="Interpolations · BasicBSpline.jl"/><meta property="og:title" content="Interpolations · BasicBSpline.jl"/><meta property="twitter:title" content="Interpolations · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/interpolations/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/interpolations/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/interpolations/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/><link href="../assets/custom.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="BasicBSpline.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="BasicBSpline.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">BasicBSpline.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li><a class="tocitem" href="../math-knotvector/">Knot vector</a></li><li><a class="tocitem" href="../math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="../math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li><a class="tocitem" href="../math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li><a class="tocitem" href="../math-fitting/">Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li class="is-active"><a class="tocitem" href>Interpolations</a><ul class="internal"><li><a class="tocitem" href="#Interpolation-with-cubic-B-spline"><span>Interpolation with cubic B-spline</span></a></li><li><a class="tocitem" href="#Interpolation-with-linear-B-spline"><span>Interpolation with linear B-spline</span></a></li><li><a class="tocitem" href="#Interpolation-with-periodic-B-spline"><span>Interpolation with periodic B-spline</span></a></li></ul></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Interpolations</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Interpolations</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/interpolations.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Interpolations"><a class="docs-heading-anchor" href="#Interpolations">Interpolations</a><a id="Interpolations-1"></a><a class="docs-heading-anchor-permalink" href="#Interpolations" title="Permalink"></a></h1><p>Currently, BasicBSpline.jl doesn&#39;t have APIs for interpolations, but it is not hard to implement some basic interpolation algorithms with this package.</p><h2 id="Interpolation-with-cubic-B-spline"><a class="docs-heading-anchor" href="#Interpolation-with-cubic-B-spline">Interpolation with cubic B-spline</a><a id="Interpolation-with-cubic-B-spline-1"></a><a class="docs-heading-anchor-permalink" href="#Interpolation-with-cubic-B-spline" title="Permalink"></a></h2><pre><code class="language-julia hljs">function interpolate(xs::AbstractVector, fs::AbstractVector{T}) where T
+    # Cubic open B-spline space
+    p = 3
+    k = KnotVector(xs) + KnotVector([xs[1],xs[end]]) * p
+    P = BSplineSpace{p}(k)
+
+    # dimensions
+    m = length(xs)
+    n = dim(P)
+
+    # The interpolant function has a f&#39;&#39;=0 property at bounds.
+    ddP = BSplineDerivativeSpace{2}(P)
+    dda = [bsplinebasis(ddP,j,xs[1]) for j in 1:n]
+    ddb = [bsplinebasis(ddP,j,xs[m]) for j in 1:n]
+
+    # Compute the interpolant function (1-dim B-spline manifold)
+    M = [bsplinebasis(P,j,xs[i]) for i in 1:m, j in 1:n]
+    M = vcat(dda&#39;, M, ddb&#39;)
+    y = vcat(zero(T), fs, zero(T))
+    return BSplineManifold(M\y, P)
+end
+
+# Example inputs
+xs = [1, 2, 3, 4, 6, 7]
+fs = [1.3, 1.5, 2, 2.1, 1.9, 1.3]
+f = interpolate(xs,fs)
+
+# Plot
+scatter(xs, fs)
+plot!(t-&gt;f(t))</code></pre><object type="text/html" data="../interpolation_cubic.html" style="width:100%;height:420px;"></object><h2 id="Interpolation-with-linear-B-spline"><a class="docs-heading-anchor" href="#Interpolation-with-linear-B-spline">Interpolation with linear B-spline</a><a id="Interpolation-with-linear-B-spline-1"></a><a class="docs-heading-anchor-permalink" href="#Interpolation-with-linear-B-spline" title="Permalink"></a></h2><pre><code class="language-julia hljs">function interpolate_linear(xs::AbstractVector, fs::AbstractVector{T}) where T
+    # Linear open B-spline space
+    p = 1
+    k = KnotVector(xs) + KnotVector([xs[1],xs[end]])
+    P = BSplineSpace{p}(k)
+
+    # dimensions
+    m = length(xs)
+    n = dim(P)
+
+    # Compute the interpolant function (1-dim B-spline manifold)
+    return BSplineManifold(fs, P)
+end
+
+# Example inputs
+xs = [1, 2, 3, 4, 6, 7]
+fs = [1.3, 1.5, 2, 2.1, 1.9, 1.3]
+
+f = interpolate_linear(xs,fs)
+
+# Plot
+scatter(xs, fs)
+plot!(t-&gt;f(t))</code></pre><object type="text/html" data="../interpolation_linear.html" style="width:100%;height:420px;"></object><h2 id="Interpolation-with-periodic-B-spline"><a class="docs-heading-anchor" href="#Interpolation-with-periodic-B-spline">Interpolation with periodic B-spline</a><a id="Interpolation-with-periodic-B-spline-1"></a><a class="docs-heading-anchor-permalink" href="#Interpolation-with-periodic-B-spline" title="Permalink"></a></h2><pre><code class="language-julia hljs">function interpolate_periodic(xs::AbstractVector, fs::AbstractVector, ::Val{p}) where p
+    # Closed B-spline space, any polynomial degrees can be accepted
+    n = length(xs) - 1
+    period = xs[end]-xs[begin]
+    k = KnotVector(vcat(
+        xs[end-p:end-1] .- period,
+        xs,
+        xs[begin+1:begin+p] .+ period
+    ))
+    P = BSplineSpace{p}(k)
+    A = [bsplinebasis(P,j,xs[i]) for i in 1:n, j in 1:n]
+    for i in 1:p-1, j in 1:i
+        A[n+i-p+1,j] += bsplinebasis(P,j+n,xs[i+n-p+1])
+    end
+    b = A \ fs[begin:end-1]
+    # Compute the interpolant function (1-dim B-spline manifold)
+    return BSplineManifold(vcat(b,b[1:p]), P)
+end
+
+# Example inputs
+xs = [1, 2, 3, 4, 6, 7]
+fs = [1.3, 1.5, 2, 2.1, 1.9, 1.3] # fs[1] == fs[end]
+
+f = interpolate_periodic(xs,fs,Val(2))
+
+# Plot
+scatter(xs, fs)
+plot!(t-&gt;f(mod(t-1,6)+1),1,14)
+plot!(t-&gt;f(t))</code></pre><object type="text/html" data="../interpolation_periodic.html" style="width:100%;height:420px;"></object><p>Note that the periodic interpolation supports any degree of polynomial.</p><pre><code class="language-julia hljs">xs = 2π*rand(10)
+sort!(push!(xs, 0, 2π))
+fs = sin.(xs)
+f1 = interpolate_periodic(xs,fs,Val(1))
+f2 = interpolate_periodic(xs,fs,Val(2))
+f3 = interpolate_periodic(xs,fs,Val(3))
+f4 = interpolate_periodic(xs,fs,Val(4))
+f5 = interpolate_periodic(xs,fs,Val(5))
+scatter(xs, fs, label=&quot;sampling points&quot;)
+plot!(sin, label=&quot;sine curve&quot;, color=:black)
+plot!(t-&gt;f1(t), label=&quot;polynomial degree 1&quot;)
+plot!(t-&gt;f2(t), label=&quot;polynomial degree 2&quot;)
+plot!(t-&gt;f3(t), label=&quot;polynomial degree 3&quot;)
+plot!(t-&gt;f4(t), label=&quot;polynomial degree 4&quot;)
+plot!(t-&gt;f5(t), label=&quot;polynomial degree 5&quot;)</code></pre><object type="text/html" data="../interpolation_periodic_sin.html" style="width:100%;height:420px;"></object></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../geometricmodeling/">« Geometric modeling</a><a class="docs-footer-nextpage" href="../internal/">Private API »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR345/math-bsplinebasis/index.html b/previews/PR345/math-bsplinebasis/index.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>B-spline basis function · BasicBSpline.jl</title><meta name="title" content="B-spline basis function · BasicBSpline.jl"/><meta property="og:title" content="B-spline basis function · BasicBSpline.jl"/><meta property="twitter:title" content="B-spline basis function · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-bsplinebasis/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-bsplinebasis/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/math-bsplinebasis/"/><script data-outdated-warner 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function</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/math-bsplinebasis.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="B-spline-basis-function"><a class="docs-heading-anchor" href="#B-spline-basis-function">B-spline basis function</a><a id="B-spline-basis-function-1"></a><a class="docs-heading-anchor-permalink" href="#B-spline-basis-function" title="Permalink"></a></h1><h2 id="Basic-properties-of-B-spline-basis-function"><a class="docs-heading-anchor" href="#Basic-properties-of-B-spline-basis-function">Basic properties of B-spline basis function</a><a id="Basic-properties-of-B-spline-basis-function-1"></a><a class="docs-heading-anchor-permalink" href="#Basic-properties-of-B-spline-basis-function" title="Permalink"></a></h2><div class="admonition is-success"><header class="admonition-header">Def.  B-spline space</header><div class="admonition-body"><p>B-spline basis function is defined by <a href="https://en.wikipedia.org/wiki/De_Boor%27s_algorithm">Cox–de Boor recursion formula</a>.</p><p class="math-container">\[\begin{aligned}
+{B}_{(i,p,k)}(t)
+&amp;=
+\frac{t-k_{i}}{k_{i+p}-k_{i}}{B}_{(i,p-1,k)}(t)
++\frac{k_{i+p+1}-t}{k_{i+p+1}-k_{i+1}}{B}_{(i+1,p-1,k)}(t) \\
+{B}_{(i,0,k)}(t)
+&amp;=
+\begin{cases}
+    &amp;1\quad (k_{i}\le t&lt; k_{i+1})\\
+    &amp;0\quad (\text{otherwise})
+\end{cases}
+\end{aligned}\]</p><p>If the denominator is zero, then the term is assumed to be zero.</p></div></div><p>The next figure shows the plot of B-spline basis functions. You can manipulate these plots on <a href="https://www.desmos.com/calculator/ql6jqgdabs">desmos graphing calculator</a>!</p><p><img src="../img/bsplinebasis.png" alt/></p><div class="admonition is-info"><header class="admonition-header">Thm.  Basis of B-spline space</header><div class="admonition-body"><p>The set of functions <span>$\{B_{(i,p,k)}\}_i$</span> is a basis of B-spline space <span>$\mathcal{P}[p,k]$</span>.</p></div></div><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; p = 2</code><code class="nohighlight hljs ansi" style="display:block;">2</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])</code><code class="nohighlight hljs ansi" style="display:block;">KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; P = BSplineSpace{p}(k)</code><code class="nohighlight hljs ansi" style="display:block;">BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; plot([t-&gt;bsplinebasis₊₀(P,i,t) for i in 1:dim(P)], 0, 10, ylims=(0,1), label=false)</code><code class="nohighlight hljs ansi" style="display:block;">Plot{Plots.PlotlyBackend() n=5}</code><code class="nohighlight hljs ansi" style="display:block;"></code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><object type="text/html" data="../bsplinebasisplot.html" style="width:100%;height:420px;"></object><p>You can choose the first terms in different ways.</p><p class="math-container">\[\begin{aligned}
+{B}_{(i,0,k)}(t)
+&amp;=
+\begin{cases}
+    &amp;1\quad (k_{i} &lt; t \le k_{i+1}) \\
+    &amp;0\quad (\text{otherwise})
+\end{cases}
+\end{aligned}\]</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; p = 2</code><code class="nohighlight hljs ansi" style="display:block;">2</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])</code><code class="nohighlight hljs ansi" style="display:block;">KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; P = BSplineSpace{p}(k)</code><code class="nohighlight hljs ansi" style="display:block;">BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; plot([t-&gt;bsplinebasis₋₀(P,i,t) for i in 1:dim(P)], 0, 10, ylims=(0,1), label=false)</code><code class="nohighlight hljs ansi" style="display:block;">Plot{Plots.PlotlyBackend() n=5}</code><code class="nohighlight hljs ansi" style="display:block;"></code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><object type="text/html" data="../bsplinebasisplot2.html" style="width:100%;height:420px;"></object><p>In these cases, each B-spline basis function <span>$B_{(i,2,k)}$</span> is coninuous, so <code>bsplinebasis₊₀</code> and <code>bsplinebasis₋₀</code> are equal.</p><h2 id="Support-of-B-spline-basis-function"><a class="docs-heading-anchor" href="#Support-of-B-spline-basis-function">Support of B-spline basis function</a><a id="Support-of-B-spline-basis-function-1"></a><a class="docs-heading-anchor-permalink" href="#Support-of-B-spline-basis-function" title="Permalink"></a></h2><div class="admonition is-info"><header class="admonition-header">Thm.  Support of B-spline basis function</header><div class="admonition-body"><p>If a B-spline space<span>$\mathcal{P}[p,k]$</span> is non-degenerate, the support of its basis function is calculated as follows:</p><p class="math-container">\[\operatorname{supp}(B_{(i,p,k)})=[k_{i},k_{i+p+1}]\]</p></div></div><p>[TODO: fig]</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.bsplinesupport" href="#BasicBSpline.bsplinesupport"><code>BasicBSpline.bsplinesupport</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Return the support of <span>$i$</span>-th B-spline basis function.</p><p class="math-container">\[\operatorname{supp}(B_{(i,p,k)})=[k_{i},k_{i+p+1}]\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+
+julia&gt; P = BSplineSpace{2}(k)
+BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))
+
+julia&gt; bsplinesupport(P,1)
+0.0 .. 5.5
+
+julia&gt; bsplinesupport(P,2)
+1.5 .. 8.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL345-L365">source</a></section></article><h2 id="Partition-of-unity"><a class="docs-heading-anchor" href="#Partition-of-unity">Partition of unity</a><a id="Partition-of-unity-1"></a><a class="docs-heading-anchor-permalink" href="#Partition-of-unity" title="Permalink"></a></h2><div class="admonition is-info"><header class="admonition-header">Thm.  Partition of unity</header><div class="admonition-body"><p>Let <span>$B_{(i,p,k)}$</span> be a B-spline basis function, then the following equation is satisfied.</p><p class="math-container">\[\begin{aligned}
+\sum_{i}B_{(i,p,k)}(t) &amp;= 1 &amp; (k_{p+1} \le t &lt; k_{l-p}) \\
+0 \le B_{(i,p,k)}(t) &amp;\le 1
+\end{aligned}\]</p></div></div><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; p = 2</code><code class="nohighlight hljs ansi" style="display:block;">2</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])</code><code class="nohighlight hljs ansi" style="display:block;">KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; P = BSplineSpace{p}(k)</code><code class="nohighlight hljs ansi" style="display:block;">BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; plot(t-&gt;sum(bsplinebasis₊₀(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)</code><code class="nohighlight hljs ansi" style="display:block;">Plot{Plots.PlotlyBackend() n=1}</code><code class="nohighlight hljs ansi" style="display:block;"></code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><object type="text/html" data="../sumofbsplineplot.html" style="width:100%;height:420px;"></object><p>To satisfy the partition of unity on whole interval <span>$[1,8]$</span>, sometimes more knots will be inserted to the endpoints of the interval.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; p = 2</code><code class="nohighlight hljs ansi" style="display:block;">2</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]) + p * KnotVector([0,10])</code><code class="nohighlight hljs ansi" style="display:block;">KnotVector([0.0, 0.0, 0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0, 10.0, 10.0])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; P = BSplineSpace{p}(k)</code><code class="nohighlight hljs ansi" style="display:block;">BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 0.0, 0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0, 10.0, 10.0]))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; plot(t-&gt;sum(bsplinebasis₊₀(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)</code><code class="nohighlight hljs ansi" style="display:block;">Plot{Plots.PlotlyBackend() n=1}</code><code class="nohighlight hljs ansi" style="display:block;"></code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><object type="text/html" data="../sumofbsplineplot2.html" style="width:100%;height:420px;"></object><p>But, the sum <span>$\sum_{i} B_{(i,p,k)}(t)$</span> is not equal to <span>$1$</span> at <span>$t=8$</span>. Therefore, to satisfy partition of unity on closed interval <span>$[k_{p+1}, k_{l-p}]$</span>, the definition of first terms of B-spline basis functions are sometimes replaced:</p><p class="math-container">\[\begin{aligned}
+{B}_{(i,0,k)}(t)
+&amp;=
+\begin{cases}
+    &amp;1\quad (k_{i} \le t&lt;k_{i+1})\\
+    &amp;1\quad (k_{i} &lt; t = k_{i+1}=k_{l})\\
+    &amp;0\quad (\text{otherwise})
+\end{cases}
+\end{aligned}\]</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; p = 2</code><code class="nohighlight hljs ansi" style="display:block;">2</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]) + p * KnotVector([0,10])</code><code class="nohighlight hljs ansi" style="display:block;">KnotVector([0.0, 0.0, 0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0, 10.0, 10.0])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; P = BSplineSpace{p}(k)</code><code class="nohighlight hljs ansi" style="display:block;">BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 0.0, 0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0, 10.0, 10.0]))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; plot(t-&gt;sum(bsplinebasis(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)</code><code class="nohighlight hljs ansi" style="display:block;">Plot{Plots.PlotlyBackend() n=1}</code><code class="nohighlight hljs ansi" style="display:block;"></code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><object type="text/html" data="../sumofbsplineplot3.html" style="width:100%;height:420px;"></object><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.bsplinebasis₊₀" href="#BasicBSpline.bsplinebasis₊₀"><code>BasicBSpline.bsplinebasis₊₀</code></a> — <span class="docstring-category">Function</span></header><section><div><p><span>$i$</span>-th B-spline basis function. Right-sided limit version.</p><p class="math-container">\[\begin{aligned}
+{B}_{(i,p,k)}(t)
+&amp;=
+\frac{t-k_{i}}{k_{i+p}-k_{i}}{B}_{(i,p-1,k)}(t)
++\frac{k_{i+p+1}-t}{k_{i+p+1}-k_{i+1}}{B}_{(i+1,p-1,k)}(t) \\
+{B}_{(i,0,k)}(t)
+&amp;=
+\begin{cases}
+    &amp;1\quad (k_{i}\le t&lt; k_{i+1})\\
+    &amp;0\quad (\text{otherwise})
+\end{cases}
+\end{aligned}\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; P = BSplineSpace{0}(KnotVector(1:6))
+BSplineSpace{0, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))
+
+julia&gt; bsplinebasis₊₀.(P,1:5,(1:6)&#39;)
+5×6 Matrix{Float64}:
+ 1.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  1.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  1.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  1.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  1.0  0.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineBasis.jl#LL5-L36">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.bsplinebasis₋₀" href="#BasicBSpline.bsplinebasis₋₀"><code>BasicBSpline.bsplinebasis₋₀</code></a> — <span class="docstring-category">Function</span></header><section><div><p><span>$i$</span>-th B-spline basis function. Left-sided limit version.</p><p class="math-container">\[\begin{aligned}
+{B}_{(i,p,k)}(t)
+&amp;=
+\frac{t-k_{i}}{k_{i+p}-k_{i}}{B}_{(i,p-1,k)}(t)
++\frac{k_{i+p+1}-t}{k_{i+p+1}-k_{i+1}}{B}_{(i+1,p-1,k)}(t) \\
+{B}_{(i,0,k)}(t)
+&amp;=
+\begin{cases}
+    &amp;1\quad (k_{i}&lt; t\le k_{i+1})\\
+    &amp;0\quad (\text{otherwise})
+\end{cases}
+\end{aligned}\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; P = BSplineSpace{0}(KnotVector(1:6))
+BSplineSpace{0, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))
+
+julia&gt; bsplinebasis₋₀.(P,1:5,(1:6)&#39;)
+5×6 Matrix{Float64}:
+ 0.0  1.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  1.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  1.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  1.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  1.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineBasis.jl#LL63-L94">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.bsplinebasis" href="#BasicBSpline.bsplinebasis"><code>BasicBSpline.bsplinebasis</code></a> — <span class="docstring-category">Function</span></header><section><div><p><span>$i$</span>-th B-spline basis function. Modified version.</p><p class="math-container">\[\begin{aligned}
+{B}_{(i,p,k)}(t)
+&amp;=
+\frac{t-k_{i}}{k_{i+p}-k_{i}}{B}_{(i,p-1,k)}(t)
++\frac{k_{i+p+1}-t}{k_{i+p+1}-k_{i+1}}{B}_{(i+1,p-1,k)}(t) \\
+{B}_{(i,0,k)}(t)
+&amp;=
+\begin{cases}
+    &amp;1\quad (k_{i} \le t &lt; k_{i+1}) \\
+    &amp;1\quad (k_{i} &lt; t = k_{i+1}=k_{l}) \\
+    &amp;0\quad (\text{otherwise})
+\end{cases}
+\end{aligned}\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; P = BSplineSpace{0}(KnotVector(1:6))
+BSplineSpace{0, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))
+
+julia&gt; bsplinebasis.(P,1:5,(1:6)&#39;)
+5×6 Matrix{Float64}:
+ 1.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  1.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  1.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  1.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  1.0  1.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineBasis.jl#LL121-L153">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.bsplinebasis₋₀I" href="#BasicBSpline.bsplinebasis₋₀I"><code>BasicBSpline.bsplinebasis₋₀I</code></a> — <span class="docstring-category">Function</span></header><section><div><p><span>$i$</span>-th B-spline basis function. Modified version (2).</p><p class="math-container">\[\begin{aligned}
+{B}_{(i,p,k)}(t)
+&amp;=
+\frac{t-k_{i}}{k_{i+p}-k_{i}}{B}_{(i,p-1,k)}(t)
++\frac{k_{i+p+1}-t}{k_{i+p+1}-k_{i+1}}{B}_{(i+1,p-1,k)}(t) \\
+{B}_{(i,0,k)}(t)
+&amp;=
+\begin{cases}
+    &amp;1\quad (k_{1+p} \le k_{i} &lt; t \le k_{i+1}) \\
+    &amp;1\quad (t = k_{1+p} = k_{i} &lt; k_{i+1}) \\
+    &amp;1\quad (k_{i} \le t &lt; k_{i+1} \le k_{1+p}) \\
+    &amp;0\quad (\text{otherwise})
+\end{cases}
+\end{aligned}\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; P = BSplineSpace{0}(KnotVector(1:6))
+BSplineSpace{0, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))
+
+julia&gt; BasicBSpline.bsplinebasis₋₀I.(P,1:5,(1:6)&#39;)
+5×6 Matrix{Float64}:
+ 1.0  1.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  1.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  1.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  1.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  1.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineBasis.jl#LL180-L213">source</a></section></article><h2 id="B-spline-basis-functions-at-specific-point"><a class="docs-heading-anchor" href="#B-spline-basis-functions-at-specific-point">B-spline basis functions at specific point</a><a id="B-spline-basis-functions-at-specific-point-1"></a><a class="docs-heading-anchor-permalink" href="#B-spline-basis-functions-at-specific-point" title="Permalink"></a></h2><p>Sometimes, you may need the non-zero values of B-spline basis functions at specific point. The <code>bsplinebasisall</code> function is much more efficient than evaluating B-spline functions one by one with <code>bsplinebasis</code> function.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using BenchmarkTools, BasicBSpline</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; P = BSplineSpace{2}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))</code><code class="nohighlight hljs ansi" style="display:block;">BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; t = 6.3</code><code class="nohighlight hljs ansi" style="display:block;">6.3</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; (bsplinebasis(P, 2, t), bsplinebasis(P, 3, t), bsplinebasis(P, 4, t))</code><code class="nohighlight hljs ansi" style="display:block;">(0.2101818181818182, 0.7166753246753247, 0.0731428571428571)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; bsplinebasisall(P, 2, t)</code><code class="nohighlight hljs ansi" style="display:block;">3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
+ 0.21018181818181822
+ 0.7166753246753247
+ 0.0731428571428571</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @benchmark (bsplinebasis($P, 2, $t), bsplinebasis($P, 3, $t), bsplinebasis($P, 4, $t))</code><code class="nohighlight hljs ansi" style="display:block;">BenchmarkTools.Trial: 10000 samples with 985 evaluations.
+ Range <span class="sgr90">(</span><span class="sgr36"><span class="sgr1">min</span></span> … <span class="sgr35">max</span><span class="sgr90">):  </span><span class="sgr36"><span class="sgr1">50.457 ns</span></span> … <span class="sgr35"> 1.362 μs</span>  <span class="sgr90">┊</span> GC <span class="sgr90">(</span>min … max<span class="sgr90">): </span>0.00% … 0.00%
+ Time  <span class="sgr90">(</span><span class="sgr34"><span class="sgr1">median</span></span><span class="sgr90">):     </span><span class="sgr34"><span class="sgr1">57.970 ns              </span></span><span class="sgr90">┊</span> GC <span class="sgr90">(</span>median<span class="sgr90">):    </span>0.00%
+ Time  <span class="sgr90">(</span><span class="sgr32"><span class="sgr1">mean</span></span> ± <span class="sgr32">σ</span><span class="sgr90">):   </span><span class="sgr32"><span class="sgr1">58.494 ns</span></span> ± <span class="sgr32">21.444 ns</span>  <span class="sgr90">┊</span> GC <span class="sgr90">(</span>mean ± σ<span class="sgr90">):  </span>0.00% ± 0.00%
+
+  ▄  ▇ ▇▁ <span class="sgr34">█</span><span class="sgr32"> </span>       ▂▂▂▁                                       ▂
+  █▁▃█████<span class="sgr34">█</span><span class="sgr32">▇</span>▆▇▇▆▇▇█████▆▇▆▄▄▆▃▄▆▆▆▆▅▁▄▁▃▄▁▁▃▁▃▁▄▁▃▁▃▄▅▃▃▄▄▄▅▃ █
+  50.5 ns<span class="sgr90">      Histogram: <span class="sgr1">log(</span>frequency<span class="sgr1">)</span> by time</span>       106 ns <span class="sgr1">&lt;</span>
+
+ Memory estimate<span class="sgr90">: </span><span class="sgr33">0 bytes</span>, allocs estimate<span class="sgr90">: </span><span class="sgr33">0</span>.</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @benchmark bsplinebasisall($P, 2, $t)</code><code class="nohighlight hljs ansi" style="display:block;">BenchmarkTools.Trial: 10000 samples with 995 evaluations.
+ Range <span class="sgr90">(</span><span class="sgr36"><span class="sgr1">min</span></span> … <span class="sgr35">max</span><span class="sgr90">):  </span><span class="sgr36"><span class="sgr1">29.046 ns</span></span> … <span class="sgr35"> 2.217 μs</span>  <span class="sgr90">┊</span> GC <span class="sgr90">(</span>min … max<span class="sgr90">): </span>0.00% … 0.00%
+ Time  <span class="sgr90">(</span><span class="sgr34"><span class="sgr1">median</span></span><span class="sgr90">):     </span><span class="sgr34"><span class="sgr1">31.357 ns              </span></span><span class="sgr90">┊</span> GC <span class="sgr90">(</span>median<span class="sgr90">):    </span>0.00%
+ Time  <span class="sgr90">(</span><span class="sgr32"><span class="sgr1">mean</span></span> ± <span class="sgr32">σ</span><span class="sgr90">):   </span><span class="sgr32"><span class="sgr1">33.291 ns</span></span> ± <span class="sgr32">39.700 ns</span>  <span class="sgr90">┊</span> GC <span class="sgr90">(</span>mean ± σ<span class="sgr90">):  </span>0.00% ± 0.00%
+
+  ▆▆ <span class="sgr34">█</span>  <span class="sgr32"> </span>         ▁▂▁                                         ▁
+  ██▁<span class="sgr34">█</span>▄▅<span class="sgr32">▇</span>▆▄▁▄▃▃▅▃▅███▇▅▆▅▆▇▇▄▄▅▁▁▃▅▆▅▄▅▄▄▅▅▄▄▄▄▃▄▄▁▃▁▅▃▃▄▄▅▃▅ █
+  29 ns<span class="sgr90">        Histogram: <span class="sgr1">log(</span>frequency<span class="sgr1">)</span> by time</span>      72.2 ns <span class="sgr1">&lt;</span>
+
+ Memory estimate<span class="sgr90">: </span><span class="sgr33">0 bytes</span>, allocs estimate<span class="sgr90">: </span><span class="sgr33">0</span>.</code></pre><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.intervalindex" href="#BasicBSpline.intervalindex"><code>BasicBSpline.intervalindex</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Return an index of a interval in the domain of B-spline space</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);
+
+
+julia&gt; P = BSplineSpace{2}(k);
+
+
+julia&gt; domain(P)
+2.5 .. 9.0
+
+julia&gt; intervalindex(P,2.6)
+1
+
+julia&gt; intervalindex(P,5.6)
+2
+
+julia&gt; intervalindex(P,8.5)
+3
+
+julia&gt; intervalindex(P,9.5)
+3</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL396-L422">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.bsplinebasisall" href="#BasicBSpline.bsplinebasisall"><code>BasicBSpline.bsplinebasisall</code></a> — <span class="docstring-category">Function</span></header><section><div><p>B-spline basis functions at point <code>t</code> on <code>i</code>-th interval.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+
+julia&gt; p = 2
+2
+
+julia&gt; P = BSplineSpace{p}(k)
+BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))
+
+julia&gt; t = 5.7
+5.7
+
+julia&gt; i = intervalindex(P,t)
+2
+
+julia&gt; bsplinebasisall(P,i,t)
+3-element SVector{3, Float64} with indices SOneTo(3):
+ 0.3847272727272727
+ 0.6107012987012989
+ 0.00457142857142858
+
+julia&gt; bsplinebasis.(P,i:i+p,t)
+3-element Vector{Float64}:
+ 0.38472727272727264
+ 0.6107012987012989
+ 0.00457142857142858</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineBasis.jl#LL263-L295">source</a></section></article><p>The next figures illustlates the relation between <code>domain(P)</code>, <code>intervalindex(P,t)</code> and <code>bsplinebasisall(P,i,t)</code>.</p><pre><code class="language-julia hljs">k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+
+for p in 1:3
+    P = BSplineSpace{p}(k)
+    plot(P, legend=:topleft, label=&quot;B-spline basis (p=1)&quot;)
+    plot!(t-&gt;intervalindex(P,t),0,10, label=&quot;Interval index&quot;)
+    plot!(t-&gt;sum(bsplinebasis(P,i,t) for i in 1:dim(P)),0,10, label=&quot;Sum of B-spline basis&quot;)
+    scatter!(k.vector,zero(k.vector), label=&quot;knot vector&quot;)
+    plot!([t-&gt;bsplinebasisall(P,1,t)[i] for i in 1:p+1],0,10, color=:black, label=&quot;bsplinebasisall (i=1)&quot;, ylim=(-1,8-2p))
+end</code></pre><object type="text/html" data="../bsplinebasisall-1.html" style="width:100%;height:420px;"></object><object type="text/html" data="../bsplinebasisall-2.html" style="width:100%;height:420px;"></object><object type="text/html" data="../bsplinebasisall-3.html" style="width:100%;height:420px;"></object><h2 id="Uniform-B-spline-basis-and-uniform-distribution"><a class="docs-heading-anchor" href="#Uniform-B-spline-basis-and-uniform-distribution">Uniform B-spline basis and uniform distribution</a><a id="Uniform-B-spline-basis-and-uniform-distribution-1"></a><a class="docs-heading-anchor-permalink" href="#Uniform-B-spline-basis-and-uniform-distribution" title="Permalink"></a></h2><p>Let <span>$X_1, \dots, X_n$</span> be i.i.d. random variables with <span>$X_i \sim U(0,1)$</span>, then the probability density function of <span>$X_1+\cdots+X_n$</span> can be obtained via <code>BasicBSpline.uniform_bsplinebasis_kernel(Val(n-1),t)</code>.</p><pre><code class="language-julia hljs">N = 100000
+# polynomial degree 0
+plot1 = histogram([rand() for _ in 1:N], normalize=true, label=false)
+plot!(t-&gt;BasicBSpline.uniform_bsplinebasis_kernel(Val(0),t), label=false)
+# polynomial degree 1
+plot2 = histogram([rand()+rand() for _ in 1:N], normalize=true, label=false)
+plot!(t-&gt;BasicBSpline.uniform_bsplinebasis_kernel(Val(1),t), label=false)
+# polynomial degree 2
+plot3 = histogram([rand()+rand()+rand() for _ in 1:N], normalize=true, label=false)
+plot!(t-&gt;BasicBSpline.uniform_bsplinebasis_kernel(Val(2),t), label=false)
+# polynomial degree 3
+plot4 = histogram([rand()+rand()+rand()+rand() for _ in 1:N], normalize=true, label=false)
+plot!(t-&gt;BasicBSpline.uniform_bsplinebasis_kernel(Val(3),t), label=false)
+# plot all
+plot(plot1,plot2,plot3,plot4)</code></pre><object type="text/html" data="../histogram-uniform.html" style="width:100%;height:420px;"></object></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../math-bsplinespace/">« B-spline space</a><a class="docs-footer-nextpage" href="../math-bsplinemanifold/">B-spline manifold »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>B-spline manifold · BasicBSpline.jl</title><meta name="title" content="B-spline manifold · BasicBSpline.jl"/><meta property="og:title" content="B-spline manifold · BasicBSpline.jl"/><meta property="twitter:title" content="B-spline manifold · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-bsplinemanifold/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-bsplinemanifold/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/math-bsplinemanifold/"/><script data-outdated-warner 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src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/><link href="../assets/custom.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="BasicBSpline.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="BasicBSpline.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">BasicBSpline.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li><a class="tocitem" href="../math-knotvector/">Knot vector</a></li><li><a class="tocitem" href="../math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li class="is-active"><a class="tocitem" href>B-spline manifold</a><ul class="internal"><li><a class="tocitem" href="#Multi-dimensional-B-spline"><span>Multi-dimensional B-spline</span></a></li><li><a class="tocitem" href="#B-spline-manifold-2"><span>B-spline manifold</span></a></li><li><a class="tocitem" href="#Affine-commutativity"><span>Affine commutativity</span></a></li><li><a class="tocitem" href="#Fixing-arguments-(currying)"><span>Fixing arguments (currying)</span></a></li></ul></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li><a class="tocitem" href="../math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li><a class="tocitem" href="../math-fitting/">Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Mathematical properties of B-spline</a></li><li class="is-active"><a href>B-spline manifold</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>B-spline manifold</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/math-bsplinemanifold.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="B-spline-manifold"><a class="docs-heading-anchor" href="#B-spline-manifold">B-spline manifold</a><a id="B-spline-manifold-1"></a><a class="docs-heading-anchor-permalink" href="#B-spline-manifold" title="Permalink"></a></h1><h2 id="Multi-dimensional-B-spline"><a class="docs-heading-anchor" href="#Multi-dimensional-B-spline">Multi-dimensional B-spline</a><a id="Multi-dimensional-B-spline-1"></a><a class="docs-heading-anchor-permalink" href="#Multi-dimensional-B-spline" title="Permalink"></a></h2><div class="admonition is-info"><header class="admonition-header">Thm.  Basis of tensor product of B-spline spaces</header><div class="admonition-body"><p>The tensor product of B-spline spaces <span>$\mathcal{P}[p^1,k^1]\otimes\mathcal{P}[p^2,k^2]$</span> is a linear space with the following basis.</p><p class="math-container">\[\mathcal{P}[p^1,k^1]\otimes\mathcal{P}[p^2,k^2]
+= \operatorname*{span}_{i,j} (B_{(i,p^1,k^1)} \otimes B_{(j,p^2,k^2)})\]</p><p>where the basis are defined as</p><p class="math-container">\[(B_{(i,p^1,k^1)} \otimes B_{(j,p^2,k^2)})(t^1, t^2)
+= B_{(i,p^1,k^1)}(t^1) \cdot B_{(j,p^2,k^2)}(t^2)\]</p></div></div><p>Higher dimensional tensor products <span>$\mathcal{P}[p^1,k^1]\otimes\cdots\otimes\mathcal{P}[p^d,k^d]$</span> are defined similarly.</p><h2 id="B-spline-manifold-2"><a class="docs-heading-anchor" href="#B-spline-manifold-2">B-spline manifold</a><a class="docs-heading-anchor-permalink" href="#B-spline-manifold-2" title="Permalink"></a></h2><p>B-spline manifold is a parametric representation of a shape.</p><div class="admonition is-success"><header class="admonition-header">Def.  B-spline manifold</header><div class="admonition-body"><p>For given <span>$d$</span>-dimensional B-spline basis functions <span>$B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)}$</span> and given points <span>$\bm{a}_{i^1 \dots i^d} \in V$</span>, B-spline manifold is defined by the following equality:</p><p class="math-container">\[\bm{p}(t^1,\dots,t^d;\bm{a}_{i^1 \dots i^d})
+=\sum_{i^1,\dots,i^d}(B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)})(t^1,\dots,t^d) \bm{a}_{i^1 \dots i^d}\]</p><p>Where <span>$\bm{a}_{i^1 \dots i^d}$</span> are called <strong>control points</strong>.</p></div></div><p>We will also write <span>$\bm{p}(t^1,\dots,t^d; \bm{a})$</span>, <span>$\bm{p}(t^1,\dots,t^d)$</span>, <span>$\bm{p}(t; \bm{a})$</span> or <span>$\bm{p}(t)$</span> for simplicity.</p><p>Note that the <code>BSplineManifold</code> objects are callable, and the arguments will be checked if it fits in the domain of <code>BSplineSpace</code>.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.BSplineManifold" href="#BasicBSpline.BSplineManifold"><code>BasicBSpline.BSplineManifold</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Construct B-spline manifold from given control points and B-spline spaces.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; using StaticArrays
+
+julia&gt; P = BSplineSpace{2}(KnotVector([0,0,0,1,1,1]))
+BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 0, 0, 1, 1, 1]))
+
+julia&gt; a = [SVector(1,0), SVector(1,1), SVector(0,1)]
+3-element Vector{SVector{2, Int64}}:
+ [1, 0]
+ [1, 1]
+ [0, 1]
+
+julia&gt; M = BSplineManifold(a, P);
+
+
+julia&gt; M(0.4)
+2-element SVector{2, Float64} with indices SOneTo(2):
+ 0.84
+ 0.64
+
+julia&gt; M(1.2)
+ERROR: DomainError with 1.2:
+The input 1.2 is out of range.
+[...]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineManifold.jl#LL10-L39">source</a></section></article><p>If you need extension of <code>BSplineManifold</code> or don&#39;t need the arguments check, you can call <code>unbounded_mapping</code>.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.unbounded_mapping" href="#BasicBSpline.unbounded_mapping"><code>BasicBSpline.unbounded_mapping</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unbounded_mapping(M::BSplineManifold{Dim}, t::Vararg{Real,Dim})</code></pre><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; P = BSplineSpace{1}(KnotVector([0,0,1,1]))
+BSplineSpace{1, Int64, KnotVector{Int64}}(KnotVector([0, 0, 1, 1]))
+
+julia&gt; domain(P)
+0 .. 1
+
+julia&gt; M = BSplineManifold([0,1], P);
+
+
+julia&gt; unbounded_mapping(M, 0.1)
+0.1
+
+julia&gt; M(0.1)
+0.1
+
+julia&gt; unbounded_mapping(M, 1.2)
+1.2
+
+julia&gt; M(1.2)
+ERROR: DomainError with 1.2:
+The input 1.2 is out of range.
+[...]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineManifold.jl#LL68-L97">source</a></section></article><p><code>unbounded_mapping(M,t...)</code> is a little bit faster than <code>M(t...)</code> because it does not check the domain.</p><h3 id="B-spline-curve"><a class="docs-heading-anchor" href="#B-spline-curve">B-spline curve</a><a id="B-spline-curve-1"></a><a class="docs-heading-anchor-permalink" href="#B-spline-curve" title="Permalink"></a></h3><pre><code class="language-julia hljs">## 1-dim B-spline manifold
+p = 2 # degree of polynomial
+k = KnotVector(1:12) # knot vector
+P = BSplineSpace{p}(k) # B-spline space
+a = [SVector(i-5, 3*sin(i^2)) for i in 1:dim(P)] # control points
+M = BSplineManifold(a, P) # Define B-spline manifold
+plot(M)</code></pre><object type="text/html" data="../1dim-manifold.html" style="width:100%;height:420px;"></object><h3 id="B-spline-surface"><a class="docs-heading-anchor" href="#B-spline-surface">B-spline surface</a><a id="B-spline-surface-1"></a><a class="docs-heading-anchor-permalink" href="#B-spline-surface" title="Permalink"></a></h3><pre><code class="language-julia hljs">## 2-dim B-spline manifold
+p = 2 # degree of polynomial
+k = KnotVector(1:8) # knot vector
+P = BSplineSpace{p}(k) # B-spline space
+rand_a = [SVector(rand(), rand(), rand()) for i in 1:dim(P), j in 1:dim(P)]
+a = [SVector(2*i-6.5, 2*j-6.5, 0) for i in 1:dim(P), j in 1:dim(P)] + rand_a # random generated control points
+M = BSplineManifold(a,(P,P)) # Define B-spline manifold
+plot(M)</code></pre><object type="text/html" data="../2dim-manifold.html" style="width:100%;height:420px;"></object><h2 id="Affine-commutativity"><a class="docs-heading-anchor" href="#Affine-commutativity">Affine commutativity</a><a id="Affine-commutativity-1"></a><a class="docs-heading-anchor-permalink" href="#Affine-commutativity" title="Permalink"></a></h2><div class="admonition is-info"><header class="admonition-header">Thm.  Affine commutativity</header><div class="admonition-body"><p>Let <span>$T$</span> be a affine transform <span>$V \to W$</span>, then the following equality holds.</p><p class="math-container">\[T(\bm{p}(t; \bm{a}))
+=\bm{p}(t; T(\bm{a}))\]</p></div></div><h2 id="Fixing-arguments-(currying)"><a class="docs-heading-anchor" href="#Fixing-arguments-(currying)">Fixing arguments (currying)</a><a id="Fixing-arguments-(currying)-1"></a><a class="docs-heading-anchor-permalink" href="#Fixing-arguments-(currying)" title="Permalink"></a></h2><p>Just like fixing first index such as <code>A[3,:]::Array{1}</code> for matrix <code>A::Array{2}</code>, fixing first argument <code>M(4.3,:)</code> will create <code>BSplineManifold{1}</code> for B-spline surface <code>M::BSplineManifold{2}</code>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; p = 2;</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; k = KnotVector(1:8);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; P = BSplineSpace{p}(k);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; a = [SVector(5i+5j+rand(), 5i-5j+rand(), rand()) for i in 1:dim(P), j in 1:dim(P)];</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; M = BSplineManifold(a,(P,P));</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; M isa BSplineManifold{2}</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; M(:,:) isa BSplineManifold{2}</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; M(4.3,:) isa BSplineManifold{1}  # Fix first argument</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><pre><code class="language-julia hljs">plot(M)
+plot!(M(4.3,:), linewidth = 5, color=:cyan)
+plot!(M(4.4,:), linewidth = 5, color=:red)
+plot!(M(:,5.2), linewidth = 5, color=:green)</code></pre><object type="text/html" data="../2dim-manifold-currying.html" style="width:100%;height:420px;"></object></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../math-bsplinebasis/">« B-spline basis function</a><a class="docs-footer-nextpage" href="../math-derivative/">Derivative »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li><a class="tocitem" href="../math-knotvector/">Knot vector</a></li><li class="is-active"><a class="tocitem" href>B-spline space</a><ul class="internal"><li><a class="tocitem" href="#Defnition"><span>Defnition</span></a></li><li><a class="tocitem" href="#Degeneration"><span>Degeneration</span></a></li><li><a class="tocitem" href="#Dimensions"><span>Dimensions</span></a></li></ul></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="../math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li><a class="tocitem" href="../math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li><a class="tocitem" href="../math-fitting/">Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Mathematical properties of B-spline</a></li><li class="is-active"><a href>B-spline space</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>B-spline space</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/math-bsplinespace.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="B-spline-space"><a class="docs-heading-anchor" href="#B-spline-space">B-spline space</a><a id="B-spline-space-1"></a><a class="docs-heading-anchor-permalink" href="#B-spline-space" title="Permalink"></a></h1><h2 id="Defnition"><a class="docs-heading-anchor" href="#Defnition">Defnition</a><a id="Defnition-1"></a><a class="docs-heading-anchor-permalink" href="#Defnition" title="Permalink"></a></h2><p>Before defining B-spline space, we&#39;ll define polynomial space with degree <span>$p$</span>.</p><div class="admonition is-success"><header class="admonition-header">Def.  Polynomial space</header><div class="admonition-body"><p>Polynomial space with degree <span>$p$</span>.</p><p class="math-container">\[\mathcal{P}[p]
+=\left\{f:\mathbb{R}\to\mathbb{R}\ ;\ t\mapsto a_0+a_1t^1+\cdots+a_pt^p \  \left| \ %
+    a_i\in \mathbb{R}
+    \right.
+\right\}\]</p><p>This space <span>$\mathcal{P}[p]$</span> is a <span>$(p+1)$</span>-dimensional linear space.</p></div></div><p>Note that <span>$\{t\mapsto t^i\}_{0 \le i \le p}$</span> is a basis of <span>$\mathcal{P}[p]$</span>, and also the set of <a href="https://en.wikipedia.org/wiki/Bernstein_polynomial">Bernstein polynomial</a> <span>$\{B_{(i,p)}\}_i$</span> is a basis of <span>$\mathcal{P}[p]$</span>.</p><p class="math-container">\[\begin{aligned}
+B_{(i,p)}(t)
+&amp;=\binom{p}{i-1}t^{i-1}(1-t)^{p-i+1}
+&amp;(i=1, \dots, p+1)
+\end{aligned}\]</p><p>Where <span>$\binom{p}{i-1}$</span> is a <a href="https://en.wikipedia.org/wiki/Binomial_coefficient">binomial coefficient</a>. You can try Bernstein polynomial on <a href="https://www.desmos.com/calculator/yc2qe6j6re">desmos graphing calculator</a>!</p><div class="admonition is-success"><header class="admonition-header">Def.  B-spline space</header><div class="admonition-body"><p>For given polynomial degree <span>$p\ge 0$</span> and knot vector <span>$k=(k_1,\dots,k_l)$</span>, B-spline space <span>$\mathcal{P}[p,k]$</span> is defined as follows:</p><p class="math-container">\[\mathcal{P}[p,k]
+=\left\{f:\mathbb{R}\to\mathbb{R} \  \left| \ %
+    \begin{gathered}
+        \operatorname{supp}(f)\subseteq [k_1, k_l] \\
+        \exists \tilde{f}\in\mathcal{P}[p], f|_{[k_{i}, k_{i+1})} = \tilde{f}|_{[k_{i}, k_{i+1})}  \\
+        \forall t \in \mathbb{R}, \exists \delta &gt; 0, f|_{(t-\delta,t+\delta)}\in C^{p-\mathfrak{n}_k(t)}
+    \end{gathered} \right.
+\right\}\]</p></div></div><p>Note that each element of the space <span>$\mathcal{P}[p,k]$</span> is a piecewise polynomial.</p><p>[TODO: fig]</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.BSplineSpace" href="#BasicBSpline.BSplineSpace"><code>BasicBSpline.BSplineSpace</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Construct B-spline space from given polynominal degree and knot vector.</p><p class="math-container">\[\mathcal{P}[p,k]\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; p = 2
+2
+
+julia&gt; k = KnotVector([1,3,5,6,8,9])
+KnotVector([1, 3, 5, 6, 8, 9])
+
+julia&gt; BSplineSpace{p}(k)
+BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 3, 5, 6, 8, 9]))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL5-L22">source</a></section></article><h2 id="Degeneration"><a class="docs-heading-anchor" href="#Degeneration">Degeneration</a><a id="Degeneration-1"></a><a class="docs-heading-anchor-permalink" href="#Degeneration" title="Permalink"></a></h2><div class="admonition is-success"><header class="admonition-header">Def.  Degeneration</header><div class="admonition-body"><p>A B-spline space is said to be <strong>non-degenerate</strong> if its degree and knot vector satisfies following property:</p><p class="math-container">\[\begin{aligned}
+k_{i}&amp;&lt;k_{i+p+1} &amp; (1 \le i \le l-p-1)
+\end{aligned}\]</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.isnondegenerate" href="#BasicBSpline.isnondegenerate"><code>BasicBSpline.isnondegenerate</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Check if given B-spline space is non-degenerate.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; isnondegenerate(BSplineSpace{2}(KnotVector([1,3,5,6,8,9])))
+true
+
+julia&gt; isnondegenerate(BSplineSpace{1}(KnotVector([1,3,3,3,8,9])))
+false</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL226-L237">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.isdegenerate-Tuple{BSplineSpace}" href="#BasicBSpline.isdegenerate-Tuple{BSplineSpace}"><code>BasicBSpline.isdegenerate</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Check if given B-spline space is degenerate.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; isdegenerate(BSplineSpace{2}(KnotVector([1,3,5,6,8,9])))
+false
+
+julia&gt; isdegenerate(BSplineSpace{1}(KnotVector([1,3,3,3,8,9])))
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL240-L251">source</a></section></article><h2 id="Dimensions"><a class="docs-heading-anchor" href="#Dimensions">Dimensions</a><a id="Dimensions-1"></a><a class="docs-heading-anchor-permalink" href="#Dimensions" title="Permalink"></a></h2><div class="admonition is-info"><header class="admonition-header">Thm.  Dimension of B-spline space</header><div class="admonition-body"><p>The B-spline space is a linear space, and if a B-spline space is non-degenerate, its dimension is calculated by:</p><p class="math-container">\[\dim(\mathcal{P}[p,k])=\# k - p -1\]</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.dim" href="#BasicBSpline.dim"><code>BasicBSpline.dim</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Return dimention of a B-spline space.</p><p class="math-container">\[\dim(\mathcal{P}[p,k])
+=\# k - p -1\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; dim(BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7])))
+5
+
+julia&gt; dim(BSplineSpace{1}(KnotVector([1,2,4,4,4,6,7])))
+5
+
+julia&gt; dim(BSplineSpace{1}(KnotVector([1,2,3,5,5,5,7])))
+5</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL84-L102">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.exactdim_R-Tuple{BSplineSpace}" href="#BasicBSpline.exactdim_R-Tuple{BSplineSpace}"><code>BasicBSpline.exactdim_R</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Exact dimension of a B-spline space.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; exactdim_R(BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7])))
+5
+
+julia&gt; exactdim_R(BSplineSpace{1}(KnotVector([1,2,4,4,4,6,7])))
+4
+
+julia&gt; exactdim_R(BSplineSpace{1}(KnotVector([1,2,3,5,5,5,7])))
+4</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL295-L309">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.exactdim_I-Tuple{BSplineSpace}" href="#BasicBSpline.exactdim_I-Tuple{BSplineSpace}"><code>BasicBSpline.exactdim_I</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Exact dimension of a B-spline space.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; exactdim_I(BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7])))
+5
+
+julia&gt; exactdim_I(BSplineSpace{1}(KnotVector([1,2,4,4,4,6,7])))
+4
+
+julia&gt; exactdim_I(BSplineSpace{1}(KnotVector([1,2,3,5,5,5,7])))
+3</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL319-L333">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../math-knotvector/">« Knot vector</a><a class="docs-footer-nextpage" href="../math-bsplinebasis/">B-spline basis function »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Mathematical properties of B-spline</a></li><li class="is-active"><a href>Derivative</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Derivative</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/math-derivative.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Derivative-of-B-spline"><a class="docs-heading-anchor" href="#Derivative-of-B-spline">Derivative of B-spline</a><a id="Derivative-of-B-spline-1"></a><a class="docs-heading-anchor-permalink" href="#Derivative-of-B-spline" title="Permalink"></a></h1><div class="admonition is-info"><header class="admonition-header">Thm.  Derivative of B-spline basis function</header><div class="admonition-body"><p>The derivative of B-spline basis function can be expressed as follows:</p><p class="math-container">\[\begin{aligned}
+\dot{B}_{(i,p,k)}(t)
+&amp;=\frac{d}{dt}B_{(i,p,k)}(t) \\
+&amp;=p\left(\frac{1}{k_{i+p}-k_{i}}B_{(i,p-1,k)}(t)-\frac{1}{k_{i+p+1}-k_{i+1}}B_{(i+1,p-1,k)}(t)\right)
+\end{aligned}\]</p><p>Note that <span>$\dot{B}_{(i,p,k)}\in\mathcal{P}[p-1,k]$</span>.</p></div></div><pre><code class="language-julia hljs">k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+P = BSplineSpace{3}(k)
+plot(
+    plot(BSplineDerivativeSpace{0}(P), label=&quot;0th derivative&quot;, color=:black),
+    plot(BSplineDerivativeSpace{1}(P), label=&quot;1st derivative&quot;, color=:red),
+    plot(BSplineDerivativeSpace{2}(P), label=&quot;2nd derivative&quot;, color=:green),
+    plot(BSplineDerivativeSpace{3}(P), label=&quot;3rd derivative&quot;, color=:blue),
+)</code></pre><object type="text/html" data="../bsplinebasisderivativeplot.html" style="width:100%;height:420px;"></object><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.BSplineDerivativeSpace" href="#BasicBSpline.BSplineDerivativeSpace"><code>BasicBSpline.BSplineDerivativeSpace</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">BSplineDerivativeSpace{r}(P::BSplineSpace)</code></pre><p>Construct derivative of B-spline space from given differential order and B-spline space.</p><p class="math-container">\[D^{r}(\mathcal{P}[p,k])
+=\left\{t \mapsto \left. \frac{d^r f}{dt^r}(t) \  \right| \ f \in \mathcal{P}[p,k] \right\}\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; P = BSplineSpace{2}(KnotVector([1,2,3,4,5,6]))
+BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))
+
+julia&gt; dP = BSplineDerivativeSpace{1}(P)
+BSplineDerivativeSpace{1, BSplineSpace{2, Int64, KnotVector{Int64}}, Int64}(BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6])))
+
+julia&gt; degree(P), degree(dP)
+(2, 1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_DerivativeSpace.jl#LL2-L22">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.derivative" href="#BasicBSpline.derivative"><code>BasicBSpline.derivative</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">derivative(::BSplineDerivativeSpace{r}) -&gt; BSplineDerivativeSpace{r+1}
+derivative(::BSplineSpace) -&gt; BSplineDerivativeSpace{1}</code></pre><p>Derivative of B-spline related space.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; BSplineSpace{2}(KnotVector(0:5))
+BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 1, 2, 3, 4, 5]))
+
+julia&gt; BasicBSpline.derivative(ans)
+BSplineDerivativeSpace{1, BSplineSpace{2, Int64, KnotVector{Int64}}, Int64}(BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 1, 2, 3, 4, 5])))
+
+julia&gt; BasicBSpline.derivative(ans)
+BSplineDerivativeSpace{2, BSplineSpace{2, Int64, KnotVector{Int64}}, Int64}(BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 1, 2, 3, 4, 5])))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_DerivativeSpace.jl#LL66-L83">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.bsplinebasis′₊₀" href="#BasicBSpline.bsplinebasis′₊₀"><code>BasicBSpline.bsplinebasis′₊₀</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bsplinebasis′₊₀(::AbstractFunctionSpace, ::Integer, ::Real) -&gt; Real</code></pre><p>1st derivative of B-spline basis function. Right-sided limit version.</p><p class="math-container">\[\dot{B}_{(i,p,k)}(t)
+=p\left(\frac{1}{k_{i+p}-k_{i}}B_{(i,p-1,k)}(t)-\frac{1}{k_{i+p+1}-k_{i+1}}B_{(i+1,p-1,k)}(t)\right)\]</p><p><code>bsplinebasis′₊₀(P, i, t)</code> is equivalent to <code>bsplinebasis₊₀(derivative(P), i, t)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_DerivativeBasis.jl#LL111-L121">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.bsplinebasis′₋₀" href="#BasicBSpline.bsplinebasis′₋₀"><code>BasicBSpline.bsplinebasis′₋₀</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bsplinebasis′₋₀(::AbstractFunctionSpace, ::Integer, ::Real) -&gt; Real</code></pre><p>1st derivative of B-spline basis function. Left-sided limit version.</p><p class="math-container">\[\dot{B}_{(i,p,k)}(t)
+=p\left(\frac{1}{k_{i+p}-k_{i}}B_{(i,p-1,k)}(t)-\frac{1}{k_{i+p+1}-k_{i+1}}B_{(i+1,p-1,k)}(t)\right)\]</p><p><code>bsplinebasis′₋₀(P, i, t)</code> is equivalent to <code>bsplinebasis₋₀(derivative(P), i, t)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_DerivativeBasis.jl#LL125-L135">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.bsplinebasis′" href="#BasicBSpline.bsplinebasis′"><code>BasicBSpline.bsplinebasis′</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bsplinebasis′(::AbstractFunctionSpace, ::Integer, ::Real) -&gt; Real</code></pre><p>1st derivative of B-spline basis function. Modified version.</p><p class="math-container">\[\dot{B}_{(i,p,k)}(t)
+=p\left(\frac{1}{k_{i+p}-k_{i}}B_{(i,p-1,k)}(t)-\frac{1}{k_{i+p+1}-k_{i+1}}B_{(i+1,p-1,k)}(t)\right)\]</p><p><code>bsplinebasis′(P, i, t)</code> is equivalent to <code>bsplinebasis(derivative(P), i, t)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_DerivativeBasis.jl#LL138-L148">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../math-bsplinemanifold/">« B-spline manifold</a><a class="docs-footer-nextpage" href="../math-inclusive/">Inclusive relationship »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR345/math-fitting/index.html b/previews/PR345/math-fitting/index.html
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--- /dev/null
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Fitting · BasicBSpline.jl</title><meta name="title" content="Fitting · BasicBSpline.jl"/><meta property="og:title" content="Fitting · BasicBSpline.jl"/><meta property="twitter:title" content="Fitting · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-fitting/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-fitting/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/math-fitting/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/><link href="../assets/custom.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="BasicBSpline.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="BasicBSpline.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">BasicBSpline.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li><a class="tocitem" href="../math-knotvector/">Knot vector</a></li><li><a class="tocitem" href="../math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="../math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li><a class="tocitem" href="../math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li class="is-active"><a class="tocitem" href>Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Mathematical properties of B-spline</a></li><li class="is-active"><a href>Fitting</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Fitting</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/math-fitting.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Fitting-with-B-spline-manifold"><a class="docs-heading-anchor" href="#Fitting-with-B-spline-manifold">Fitting with B-spline manifold</a><a id="Fitting-with-B-spline-manifold-1"></a><a class="docs-heading-anchor-permalink" href="#Fitting-with-B-spline-manifold" title="Permalink"></a></h1><p>The following functions such as <code>fittingcontolpoints</code> were provided from BasicBSpline.jl before v0.9.0. From BasicBSpline v0.9.0, these functions are moved to <a href="https://github.com/hyrodium/BasicBSplineFitting.jl">BasicBSplineFitting</a>.</p><p>Fitting with least squares method.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSplineFitting.fittingcontrolpoints" href="#BasicBSplineFitting.fittingcontrolpoints"><code>BasicBSplineFitting.fittingcontrolpoints</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Fitting controlpoints with least squares method.</p><pre><code class="nohighlight hljs">fittingcontrolpoints(func, Ps::Tuple)</code></pre><p>This function will calculate <span>$\bm{a}_i$</span> to minimize the following integral.</p><p class="math-container">\[\int_I \left\|f(t)-\sum_i B_{(i,p,k)}(t) \bm{a}_i\right\|^2 dt\]</p><p>Similarly, for the two-dimensional case, minimize the following integral.</p><p class="math-container">\[\int_{I^1 \times I^2} \left\|f(t^1, t^2)-\sum_{i,j} B_{(i,p^1,k^1)}(t^1)B_{(j,p^2,k^2)}(t^2) \bm{a}_{ij}\right\|^2 dt^1dt^2\]</p><p>Currently, this function supports up to three dimensions.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; f(t) = SVector(cos(t),sin(t),t);
+
+julia&gt; P = BSplineSpace{3}(KnotVector(range(0,2π,30)) + 3*KnotVector([0,2π]));
+
+julia&gt; a = fittingcontrolpoints(f, P);
+
+julia&gt; M = BSplineManifold(a, P);
+
+julia&gt; norm(M(1) - f(1)) &lt; 1e-5
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSplineFitting.jl/blob/v0.1.0/src/_Fitting.jl#L358-L386">source</a></section></article><pre><code class="language-julia hljs">p1 = 2
+p2 = 2
+k1 = KnotVector(-10:10)+p1*KnotVector([-10,10])
+k2 = KnotVector(-10:10)+p2*KnotVector([-10,10])
+P1 = BSplineSpace{p1}(k1)
+P2 = BSplineSpace{p2}(k2)
+
+f(u1, u2) = SVector(2u1 + sin(u1) + cos(u2) + u2 / 2, 3u2 + sin(u2) + sin(u1) / 2 + u1^2 / 6) / 5
+
+a = fittingcontrolpoints(f, (P1, P2))
+M = BSplineManifold(a, (P1, P2))
+save_png(&quot;fitting.png&quot;, M, unitlength=50, xlims=(-10,10), ylims=(-10,10))</code></pre><p><img src="../fitting.png" alt/></p><p><a href="https://www.desmos.com/calculator/2hm3b1fbdf">Try on Desmos graphing graphing calculator!</a> <img src="../img/fitting_desmos.png" alt/></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../math-refinement/">« Refinement</a><a class="docs-footer-nextpage" href="../math-rationalbsplinemanifold/">Rational B-spline manifold »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR345/math-inclusive/index.html b/previews/PR345/math-inclusive/index.html
new file mode 100644
index 000000000..376120e80
--- /dev/null
+++ b/previews/PR345/math-inclusive/index.html
@@ -0,0 +1,87 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Inclusive relationship · BasicBSpline.jl</title><meta name="title" content="Inclusive relationship · BasicBSpline.jl"/><meta property="og:title" content="Inclusive relationship · BasicBSpline.jl"/><meta property="twitter:title" content="Inclusive relationship · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-inclusive/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-inclusive/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/math-inclusive/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/><link href="../assets/custom.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="BasicBSpline.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="BasicBSpline.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">BasicBSpline.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li><a class="tocitem" href="../math-knotvector/">Knot vector</a></li><li><a class="tocitem" href="../math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="../math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li class="is-active"><a class="tocitem" href>Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li><a class="tocitem" href="../math-fitting/">Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Mathematical properties of B-spline</a></li><li class="is-active"><a href>Inclusive relationship</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Inclusive relationship</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/math-inclusive.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Inclusive-relation-between-B-spline-spaces"><a class="docs-heading-anchor" href="#Inclusive-relation-between-B-spline-spaces">Inclusive relation between B-spline spaces</a><a id="Inclusive-relation-between-B-spline-spaces-1"></a><a class="docs-heading-anchor-permalink" href="#Inclusive-relation-between-B-spline-spaces" title="Permalink"></a></h1><div class="admonition is-info"><header class="admonition-header">Thm.  Inclusive relation between B-spline spaces</header><div class="admonition-body"><p>For non-degenerate B-spline spaces, the following relationship holds.</p><p class="math-container">\[\mathcal{P}[p,k]
+\subseteq \mathcal{P}[p&#39;,k&#39;]
+\Leftrightarrow (m=p&#39;-p \ge 0 \ \text{and} \ k+m\widehat{k}\subseteq k&#39;)\]</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.issubset-Union{Tuple{p′}, Tuple{p}, Tuple{BSplineSpace{p, T} where T&lt;:Real, BSplineSpace{p′, T} where T&lt;:Real}} where {p, p′}" href="#Base.issubset-Union{Tuple{p′}, Tuple{p}, Tuple{BSplineSpace{p, T} where T&lt;:Real, BSplineSpace{p′, T} where T&lt;:Real}} where {p, p′}"><code>Base.issubset</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Check inclusive relationship between B-spline spaces.</p><p class="math-container">\[\mathcal{P}[p,k]
+\subseteq\mathcal{P}[p&#39;,k&#39;]\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; P1 = BSplineSpace{1}(KnotVector([1,3,5,8]));
+
+
+julia&gt; P2 = BSplineSpace{1}(KnotVector([1,3,5,6,8,9]));
+
+
+julia&gt; P3 = BSplineSpace{2}(KnotVector([1,1,3,3,5,5,8,8]));
+
+
+julia&gt; P1 ⊆ P2
+true
+
+julia&gt; P1 ⊆ P3
+true
+
+julia&gt; P2 ⊆ P3
+false
+
+julia&gt; P2 ⊈ P3
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL108-L138">source</a></section></article><p>Here are plots of the B-spline basis functions of the spaces <code>P1</code>, <code>P2</code>, <code>P3</code>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; P1 = BSplineSpace{1}(KnotVector([1,3,5,8]))</code><code class="nohighlight hljs ansi" style="display:block;">BSplineSpace{1, Int64, KnotVector{Int64}}(KnotVector([1, 3, 5, 8]))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; P2 = BSplineSpace{1}(KnotVector([1,3,5,6,8,9]))</code><code class="nohighlight hljs ansi" style="display:block;">BSplineSpace{1, Int64, KnotVector{Int64}}(KnotVector([1, 3, 5, 6, 8, 9]))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; P3 = BSplineSpace{2}(KnotVector([1,1,3,3,5,5,8,8]))</code><code class="nohighlight hljs ansi" style="display:block;">BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 1, 3, 3, 5, 5, 8, 8]))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; plot(
+           plot([t-&gt;bsplinebasis₊₀(P1,i,t) for i in 1:dim(P1)], 1, 9, ylims=(0,1), legend=false),
+           plot([t-&gt;bsplinebasis₊₀(P2,i,t) for i in 1:dim(P2)], 1, 9, ylims=(0,1), legend=false),
+           plot([t-&gt;bsplinebasis₊₀(P3,i,t) for i in 1:dim(P3)], 1, 9, ylims=(0,1), legend=false),
+           layout=(3,1),
+           link=:x
+       )</code><code class="nohighlight hljs ansi" style="display:block;">Plot{Plots.PlotlyBackend() n=11}</code><code class="nohighlight hljs ansi" style="display:block;"></code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><object type="text/html" data="../subbsplineplot.html" style="width:100%;height:420px;"></object><p>This means, there exists a <span>$n \times n&#39;$</span> matrix <span>$A$</span> which holds:</p><p class="math-container">\[\begin{aligned}
+B_{(i,p,k)}
+&amp;=\sum_{j}A_{ij} B_{(j,p&#39;,k&#39;)} \\
+n&amp;=\dim(\mathcal{P}[p,k]) \\
+n&#39;&amp;=\dim(\mathcal{P}[p&#39;,k&#39;])
+\end{aligned}\]</p><p>You can calculate the change of basis matrix <span>$A$</span> with <code>changebasis</code>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; A12 = changebasis(P1,P2)</code><code class="nohighlight hljs ansi" style="display:block;">2×4 SparseArrays.SparseMatrixCSC{Float64, Int32} with 3 stored entries:
+ 1.0   ⋅    ⋅         ⋅
+  ⋅   1.0  0.666667   ⋅</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; A13 = changebasis(P1,P3)</code><code class="nohighlight hljs ansi" style="display:block;">2×5 SparseArrays.SparseMatrixCSC{Float64, Int32} with 6 stored entries:
+ 0.5  1.0  0.5   ⋅    ⋅
+  ⋅    ⋅   0.5  1.0  0.5</code></pre><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; plot(
+           plot([t-&gt;bsplinebasis₊₀(P1,i,t) for i in 1:dim(P1)], 1, 9, ylims=(0,1), legend=false),
+           plot([t-&gt;sum(A12[i,j]*bsplinebasis₊₀(P2,j,t) for j in 1:dim(P2)) for i in 1:dim(P1)], 1, 9, ylims=(0,1), legend=false),
+           plot([t-&gt;sum(A13[i,j]*bsplinebasis₊₀(P3,j,t) for j in 1:dim(P3)) for i in 1:dim(P1)], 1, 9, ylims=(0,1), legend=false),
+           layout=(3,1),
+           link=:x
+       )</code><code class="nohighlight hljs ansi" style="display:block;">Plot{Plots.PlotlyBackend() n=6}</code><code class="nohighlight hljs ansi" style="display:block;"></code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><object type="text/html" data="../subbsplineplot2.html" style="width:100%;height:420px;"></object><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.changebasis_R" href="#BasicBSpline.changebasis_R"><code>BasicBSpline.changebasis_R</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Return a coefficient matrix <span>$A$</span> which satisfy</p><p class="math-container">\[B_{(i,p,k)} = \sum_{j}A_{i,j}B_{(j,p&#39;,k&#39;)}\]</p><p>Assumption:</p><ul><li><span>$P ⊆ P^{\prime}$</span></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_ChangeBasis.jl#LL4-L12">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.changebasis_I" href="#BasicBSpline.changebasis_I"><code>BasicBSpline.changebasis_I</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Return a coefficient matrix <span>$A$</span> which satisfy</p><p class="math-container">\[B_{(i,p,k)} = \sum_{j}A_{i,j}B_{(j,p&#39;,k&#39;)}\]</p><p>Assumption:</p><ul><li><span>$P ⊑ P^{\prime}$</span></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_ChangeBasis.jl#LL339-L347">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.issqsubset" href="#BasicBSpline.issqsubset"><code>BasicBSpline.issqsubset</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Check inclusive relationship between B-spline spaces.</p><p class="math-container">\[\mathcal{P}[p,k]
+\sqsubseteq\mathcal{P}[p&#39;,k&#39;]
+\Leftrightarrow
+\mathcal{P}[p,k]|_{[k_{p+1},k_{l-p}]}
+\subseteq\mathcal{P}[p&#39;,k&#39;]|_{[k&#39;_{p&#39;+1},k&#39;_{l&#39;-p&#39;}]}\]</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL169-L178">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.expandspace" href="#BasicBSpline.expandspace"><code>BasicBSpline.expandspace</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Expand B-spline space with given additional degree and knotvector. The behavior of <code>expandspace</code> is same as <code>expandspace_I</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL522-L525">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.expandspace_R" href="#BasicBSpline.expandspace_R"><code>BasicBSpline.expandspace_R</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Expand B-spline space with given additional degree and knotvector. This function is compatible with <code>issubset</code> (<code>⊆</code>).</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);
+
+
+julia&gt; P = BSplineSpace{2}(k);
+
+
+julia&gt; P′ = expandspace_R(P, Val(1), KnotVector([6.0]))
+BSplineSpace{3, Float64, KnotVector{Float64}}(KnotVector([0.0, 0.0, 1.5, 1.5, 2.5, 2.5, 5.5, 5.5, 6.0, 8.0, 8.0, 9.0, 9.0, 9.5, 9.5, 10.0, 10.0]))
+
+julia&gt; P ⊆ P′
+true
+
+julia&gt; P ⊑ P′
+false
+
+julia&gt; domain(P)
+2.5 .. 9.0
+
+julia&gt; domain(P′)
+1.5 .. 9.5</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL477-L504">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.expandspace_I" href="#BasicBSpline.expandspace_I"><code>BasicBSpline.expandspace_I</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Expand B-spline space with given additional degree and knotvector. This function is compatible with <code>issqsubset</code> (<code>⊑</code>)</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);
+
+
+julia&gt; P = BSplineSpace{2}(k);
+
+
+julia&gt; P′ = expandspace_I(P, Val(1), KnotVector([6.0]))
+BSplineSpace{3, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 2.5, 5.5, 5.5, 6.0, 8.0, 8.0, 9.0, 9.0, 9.5, 10.0]))
+
+julia&gt; P ⊆ P′
+false
+
+julia&gt; P ⊑ P′
+true
+
+julia&gt; domain(P)
+2.5 .. 9.0
+
+julia&gt; domain(P′)
+2.5 .. 9.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_BSplineSpace.jl#LL430-L457">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../math-derivative/">« Derivative</a><a class="docs-footer-nextpage" href="../math-refinement/">Refinement »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR345/math-knotvector/index.html b/previews/PR345/math-knotvector/index.html
new file mode 100644
index 000000000..3f64aba77
--- /dev/null
+++ b/previews/PR345/math-knotvector/index.html
@@ -0,0 +1,80 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Knot vector · BasicBSpline.jl</title><meta name="title" content="Knot vector · BasicBSpline.jl"/><meta property="og:title" content="Knot vector · BasicBSpline.jl"/><meta property="twitter:title" content="Knot vector · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-knotvector/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-knotvector/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/math-knotvector/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/><link href="../assets/custom.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="BasicBSpline.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="BasicBSpline.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">BasicBSpline.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li class="is-active"><a class="tocitem" href>Knot vector</a><ul class="internal"><li><a class="tocitem" href="#Definition"><span>Definition</span></a></li><li><a class="tocitem" href="#Operations-for-knot-vectors"><span>Operations for knot vectors</span></a></li><li><a class="tocitem" href="#KnotVector-with-string-macro"><span><code>KnotVector</code> with string macro</span></a></li></ul></li><li><a class="tocitem" href="../math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="../math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li><a class="tocitem" href="../math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li><a class="tocitem" href="../math-fitting/">Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Mathematical properties of B-spline</a></li><li class="is-active"><a href>Knot vector</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Knot vector</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/math-knotvector.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Knot-vector"><a class="docs-heading-anchor" href="#Knot-vector">Knot vector</a><a id="Knot-vector-1"></a><a class="docs-heading-anchor-permalink" href="#Knot-vector" title="Permalink"></a></h1><h2 id="Definition"><a class="docs-heading-anchor" href="#Definition">Definition</a><a id="Definition-1"></a><a class="docs-heading-anchor-permalink" href="#Definition" title="Permalink"></a></h2><div class="admonition is-success"><header class="admonition-header">Def.  Knot vector</header><div class="admonition-body"><p>A finite sequence</p><p class="math-container">\[k = (k_1, \dots, k_l)\]</p><p>is called <strong>knot vector</strong> if the sequence is broad monotonic increase, i.e. <span>$k_{i} \le k_{i+1}$</span>.</p></div></div><p>[TODO: fig]</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.KnotVector" href="#BasicBSpline.KnotVector"><code>BasicBSpline.KnotVector</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Construct knot vector from given array.</p><p class="math-container">\[k=(k_1,\dots,k_l)\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = KnotVector([1,2,3])
+KnotVector([1, 2, 3])
+
+julia&gt; k = KnotVector(1:3)
+KnotVector([1, 2, 3])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL25-L39">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.UniformKnotVector" href="#BasicBSpline.UniformKnotVector"><code>BasicBSpline.UniformKnotVector</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Construct uniform knot vector from given range.</p><p class="math-container">\[k=(k_1,\dots,k_l)\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = UniformKnotVector(1:8)
+UniformKnotVector(1:8)
+
+julia&gt; UniformKnotVector(8:-1:3)
+UniformKnotVector(3:1:8)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL45-L59">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.SubKnotVector" href="#BasicBSpline.SubKnotVector"><code>BasicBSpline.SubKnotVector</code></a> — <span class="docstring-category">Type</span></header><section><div><p>A type to represetnt sub knot vector.</p><p class="math-container">\[k=(k_1,\dots,k_l)\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = knotvector&quot;1 11 211&quot;
+KnotVector([1, 3, 4, 6, 6, 7, 8])
+
+julia&gt; view(k, 2:5)
+SubKnotVector([3, 4, 6, 6])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL65-L79">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.EmptyKnotVector" href="#BasicBSpline.EmptyKnotVector"><code>BasicBSpline.EmptyKnotVector</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Knot vector with zero-element.</p><p class="math-container">\[k=()\]</p><p>This struct is intended for internal use.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; EmptyKnotVector()
+EmptyKnotVector{Bool}()
+
+julia&gt; EmptyKnotVector{Float64}()
+EmptyKnotVector{Float64}()</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL7-L22">source</a></section></article><h2 id="Operations-for-knot-vectors"><a class="docs-heading-anchor" href="#Operations-for-knot-vectors">Operations for knot vectors</a><a id="Operations-for-knot-vectors-1"></a><a class="docs-heading-anchor-permalink" href="#Operations-for-knot-vectors" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.length-Tuple{AbstractKnotVector}" href="#Base.length-Tuple{AbstractKnotVector}"><code>Base.length</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Length of knot vector</p><p class="math-container">\[\begin{aligned}
+\#{k}
+&amp;=(\text{number of knot elements of} \  k) \\
+\end{aligned}\]</p><p>For example, <span>$\#{(1,2,2,3)}=4$</span>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = KnotVector([1,2,2,3]);
+
+
+julia&gt; length(k)
+4</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL288-L308">source</a></section></article><p>Although a knot vector is <strong>not</strong> a vector in linear algebra, but we introduce <strong>additional operator</strong> <span>$+$</span>.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.:+-Union{Tuple{T}, Tuple{KnotVector{T}, KnotVector{T}}} where T" href="#Base.:+-Union{Tuple{T}, Tuple{KnotVector{T}, KnotVector{T}}} where T"><code>Base.:+</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Sum of knot vectors</p><p class="math-container">\[\begin{aligned}
+k^{(1)}+k^{(2)}
+&amp;=(k^{(1)}_1, \dots, k^{(1)}_{l^{(1)}}) + (k^{(2)}_1, \dots, k^{(2)}_{l^{(2)}}) \\
+&amp;=(\text{sort of union of} \  k^{(1)} \ \text{and} \  k^{(2)} \text{)}
+\end{aligned}\]</p><p>For example, <span>$(1,2,3,5)+(4,5,8)=(1,2,3,4,5,5,8)$</span>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k1 = KnotVector([1,2,3,5]);
+
+
+julia&gt; k2 = KnotVector([4,5,8]);
+
+
+julia&gt; k1 + k2
+KnotVector([1, 2, 3, 4, 5, 5, 8])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL182-L206">source</a></section></article><p>Note that the operator <code>+(::KnotVector, ::KnotVector)</code> is commutative. This is why we choose the <span>$+$</span> sign. We also introduce <strong>product operator</strong> <span>$\cdot$</span> for knot vector.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.:*-Tuple{Integer, AbstractKnotVector}" href="#Base.:*-Tuple{Integer, AbstractKnotVector}"><code>Base.:*</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Product of integer and knot vector</p><p class="math-container">\[\begin{aligned}
+m\cdot k&amp;=\underbrace{k+\cdots+k}_{m}
+\end{aligned}\]</p><p>For example, <span>$2\cdot (1,2,2,5)=(1,1,2,2,2,2,5,5)$</span>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = KnotVector([1,2,2,5]);
+
+
+julia&gt; 2 * k
+KnotVector([1, 1, 2, 2, 2, 2, 5, 5])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL237-L256">source</a></section></article><p>Inclusive relationship between knot vectors.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.issubset-Tuple{KnotVector, KnotVector}" href="#Base.issubset-Tuple{KnotVector, KnotVector}"><code>Base.issubset</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Check a inclusive relationship <span>$k\subseteq k&#39;$</span>, for example:</p><p class="math-container">\[\begin{aligned}
+(1,2) &amp;\subseteq (1,2,3) \\
+(1,2,2) &amp;\not\subseteq (1,2,3) \\
+(1,2,3) &amp;\subseteq (1,2,3) \\
+\end{aligned}\]</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; KnotVector([1,2]) ⊆ KnotVector([1,2,3])
+true
+
+julia&gt; KnotVector([1,2,2]) ⊆ KnotVector([1,2,3])
+false
+
+julia&gt; KnotVector([1,2,3]) ⊆ KnotVector([1,2,3])
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL349-L370">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.unique-Tuple{AbstractKnotVector}" href="#Base.unique-Tuple{AbstractKnotVector}"><code>Base.unique</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Unique elements of knot vector.</p><p class="math-container">\[\begin{aligned}
+\widehat{k}
+&amp;=(\text{unique knot elements of} \  k) \\
+\end{aligned}\]</p><p>For example, <span>$\widehat{(1,2,2,3)}=(1,2,3)$</span>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; k = KnotVector([1,2,2,3]);
+
+
+julia&gt; unique(k)
+KnotVector([1, 2, 3])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL315-L335">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.countknots-Tuple{AbstractKnotVector, Real}" href="#BasicBSpline.countknots-Tuple{AbstractKnotVector, Real}"><code>BasicBSpline.countknots</code></a> — <span class="docstring-category">Method</span></header><section><div><p>For given knot vector <span>$k$</span>, the following function <span>$\mathfrak{n}_k:\mathbb{R}\to\mathbb{Z}$</span> represents the number of knots that duplicate the knot vector <span>$k$</span>.</p><p class="math-container">\[\mathfrak{n}_k(t) = \#\{i \mid k_i=t \}\]</p><p>For example, if <span>$k=(1,2,2,3)$</span>, then <span>$\mathfrak{n}_k(0.3)=0$</span>, <span>$\mathfrak{n}_k(1)=1$</span>, <span>$\mathfrak{n}_k(2)=2$</span>.</p><pre><code class="language-julia-repl hljs">julia&gt; k = KnotVector([1,2,2,3]);
+
+
+julia&gt; countknots(k,0.3)
+0
+
+julia&gt; countknots(k,1.0)
+1
+
+julia&gt; countknots(k,2.0)
+2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL410-L431">source</a></section></article><h2 id="KnotVector-with-string-macro"><a class="docs-heading-anchor" href="#KnotVector-with-string-macro"><code>KnotVector</code> with string macro</a><a id="KnotVector-with-string-macro-1"></a><a class="docs-heading-anchor-permalink" href="#KnotVector-with-string-macro" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.@knotvector_str" href="#BasicBSpline.@knotvector_str"><code>BasicBSpline.@knotvector_str</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@knotvector_str -&gt; KnotVector</code></pre><p>Construct a knotvector by specifying the numbers of duplicates of knots.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; knotvector&quot;11111&quot;
+KnotVector([1, 2, 3, 4, 5])
+
+julia&gt; knotvector&quot;123&quot;
+KnotVector([1, 2, 2, 3, 3, 3])
+
+julia&gt; knotvector&quot; 2 2 2&quot;
+KnotVector([2, 2, 4, 4, 6, 6])
+
+julia&gt; knotvector&quot;     1&quot;
+KnotVector([6])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_KnotVector.jl#LL440-L459">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../math/">« Introduction</a><a class="docs-footer-nextpage" href="../math-bsplinespace/">B-spline space »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR345/math-rationalbsplinemanifold/index.html b/previews/PR345/math-rationalbsplinemanifold/index.html
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+++ b/previews/PR345/math-rationalbsplinemanifold/index.html
@@ -0,0 +1,32 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Rational B-spline manifold · BasicBSpline.jl</title><meta name="title" content="Rational B-spline manifold · BasicBSpline.jl"/><meta property="og:title" content="Rational B-spline manifold · BasicBSpline.jl"/><meta property="twitter:title" content="Rational B-spline manifold · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-rationalbsplinemanifold/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/math-rationalbsplinemanifold/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/math-rationalbsplinemanifold/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script 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alt="BasicBSpline.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">BasicBSpline.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li><a class="tocitem" href="../math-knotvector/">Knot vector</a></li><li><a class="tocitem" href="../math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="../math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li><a class="tocitem" href="../math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li><a class="tocitem" href="../math-fitting/">Fitting</a></li><li class="is-active"><a class="tocitem" href>Rational B-spline manifold</a><ul class="internal"><li><a class="tocitem" href="#Rational-B-spline-manifold-2"><span>Rational B-spline manifold</span></a></li><li><a class="tocitem" href="#Properties"><span>Properties</span></a></li></ul></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Mathematical properties of B-spline</a></li><li class="is-active"><a href>Rational B-spline manifold</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Rational B-spline manifold</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/math-rationalbsplinemanifold.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Rational-B-spline-manifold"><a class="docs-heading-anchor" href="#Rational-B-spline-manifold">Rational B-spline manifold</a><a id="Rational-B-spline-manifold-1"></a><a class="docs-heading-anchor-permalink" href="#Rational-B-spline-manifold" title="Permalink"></a></h1><p><a href="https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline">Non-uniform rational basis spline (NURBS)</a> is also supported in BasicBSpline.jl package.</p><h2 id="Rational-B-spline-manifold-2"><a class="docs-heading-anchor" href="#Rational-B-spline-manifold-2">Rational B-spline manifold</a><a class="docs-heading-anchor-permalink" href="#Rational-B-spline-manifold-2" title="Permalink"></a></h2><p>Rational B-spline manifold is a parametric representation of a shape.</p><div class="admonition is-success"><header class="admonition-header">Def.  Rational B-spline manifold</header><div class="admonition-body"><p>For given <span>$d$</span>-dimensional B-spline basis functions <span>$B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)}$</span>, given points <span>$\bm{a}_{i^1 \dots i^d} \in V$</span> and real numbers <span>$w_{i^1 \dots i^d} &gt; 0$</span>, rational B-spline manifold is defined by the following equality:</p><p class="math-container">\[\bm{p}(t^1,\dots,t^d; \bm{a}_{i^1 \dots i^d}, w_{i^1 \dots i^d})
+=\sum_{i^1,\dots,i^d}
+\frac{(B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)})(t^1,\dots,t^d) w_{i^1 \dots i^d}}
+{\sum\limits_{j^1,\dots,j^d}(B_{(j^1,p^1,k^1)} \otimes \cdots \otimes B_{(j^d,p^d,k^d)})(t^1,\dots,t^d) w_{j^1 \dots j^d}}
+\bm{a}_{i^1 \dots i^d}\]</p><p>Where <span>$\bm{a}_{i^1,\dots,i^d}$</span> are called <strong>control points</strong>, and <span>$w_{i^1 \dots i^d}$</span> are called <strong>weights</strong>.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.RationalBSplineManifold" href="#BasicBSpline.RationalBSplineManifold"><code>BasicBSpline.RationalBSplineManifold</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Construct Rational B-spline manifold from given control points, weights and B-spline spaces.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; using StaticArrays, LinearAlgebra
+
+julia&gt; P = BSplineSpace{2}(KnotVector([0,0,0,1,1,1]))
+BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 0, 0, 1, 1, 1]))
+
+julia&gt; w = [1, 1/√2, 1]
+3-element Vector{Float64}:
+ 1.0
+ 0.7071067811865475
+ 1.0
+
+julia&gt; a = [SVector(1,0), SVector(1,1), SVector(0,1)]
+3-element Vector{SVector{2, Int64}}:
+ [1, 0]
+ [1, 1]
+ [0, 1]
+
+julia&gt; M = RationalBSplineManifold(a,w,P);  # 1/4 arc
+
+
+julia&gt; M(0.3)
+2-element SVector{2, Float64} with indices SOneTo(2):
+ 0.8973756499953727
+ 0.4412674277525845
+
+julia&gt; norm(M(0.3))
+1.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_RationalBSplineManifold.jl#LL2-L35">source</a></section></article><h2 id="Properties"><a class="docs-heading-anchor" href="#Properties">Properties</a><a id="Properties-1"></a><a class="docs-heading-anchor-permalink" href="#Properties" title="Permalink"></a></h2><p>Similar to <code>BSplineManifold</code>, <code>RationalBSplineManifold</code> supports the following methods and properties.</p><ul><li>currying</li><li><code>refinement</code></li><li>Affine commutativity</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../math-fitting/">« Fitting</a><a class="docs-footer-nextpage" href="../plots/">Plots.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR345/math-refinement/index.html b/previews/PR345/math-refinement/index.html
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href="../math-fitting/">Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Mathematical properties of B-spline</a></li><li class="is-active"><a href>Refinement</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Refinement</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/math-refinement.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Refinement"><a class="docs-heading-anchor" href="#Refinement">Refinement</a><a id="Refinement-1"></a><a class="docs-heading-anchor-permalink" href="#Refinement" title="Permalink"></a></h1><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="BasicBSpline.refinement" href="#BasicBSpline.refinement"><code>BasicBSpline.refinement</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Refinement of B-spline manifold with given B-spline spaces.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/hyrodium/BasicBSpline.jl/blob/39af259d668d477132126eda66a0c16195d217cb/src/_Refinement.jl#LL6-L8">source</a></section></article><h2 id="Example"><a class="docs-heading-anchor" href="#Example">Example</a><a id="Example-1"></a><a class="docs-heading-anchor-permalink" href="#Example" title="Permalink"></a></h2><h3 id="Define-original-manifold"><a class="docs-heading-anchor" href="#Define-original-manifold">Define original manifold</a><a id="Define-original-manifold-1"></a><a class="docs-heading-anchor-permalink" href="#Define-original-manifold" title="Permalink"></a></h3><pre><code class="language-julia hljs">p = 2 # degree of polynomial
+k = KnotVector(1:8) # knot vector
+P = BSplineSpace{p}(k) # B-spline space
+rand_a = [SVector(rand(), rand()) for i in 1:dim(P), j in 1:dim(P)]
+a = [SVector(2*i-6.5, 2*j-6.5) for i in 1:dim(P), j in 1:dim(P)] + rand_a # random
+M = BSplineManifold(a,(P,P)) # Define B-spline manifold</code></pre><h3 id="h-refinement"><a class="docs-heading-anchor" href="#h-refinement">h-refinement</a><a id="h-refinement-1"></a><a class="docs-heading-anchor-permalink" href="#h-refinement" title="Permalink"></a></h3><p>Insert additional knots to knot vector.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; k₊ = (KnotVector([3.3,4.2]),KnotVector([3.8,3.2,5.3])) # additional knot vectors</code><code class="nohighlight hljs ansi" style="display:block;">(KnotVector([3.3, 4.2]), KnotVector([3.2, 3.8, 5.3]))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; M_h = refinement(M, k₊) # refinement of B-spline manifold</code><code class="nohighlight hljs ansi" style="display:block;">BSplineManifold{2, (2, 2), StaticArraysCore.SVector{2, Float64}, Tuple{BSplineSpace{2, Float64, KnotVector{Float64}}, BSplineSpace{2, Float64, KnotVector{Float64}}}}((BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([1.0, 2.0, 3.0, 3.3, 4.0, 4.2, 5.0, 6.0, 7.0, 8.0])), BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([1.0, 2.0, 3.0, 3.2, 3.8, 4.0, 5.0, 5.3, 6.0, 7.0, 8.0]))), StaticArraysCore.SVector{2, Float64}[[-3.7392559679567654, -3.783078079871714] [-3.597411705002145, -2.6270907448400385] … [-4.27252342702906, 2.44794251443626] [-4.010429310201709, 3.869439330221196]; [-2.328731904641716, -4.1480930630807595] [-2.275419510381314, -2.8615581264721692] … [-2.736955136391798, 2.5020454657912223] [-2.786990222003846, 3.8096177268489324]; … ; [2.359972932445208, -3.6129585850258517] [2.373167856283101, -2.9446206124413257] … [2.210716406024899, 2.602631448306355] [2.3997592455372003, 3.5324898804805147]; [4.093841853317975, -3.9061261483801712] [3.8504632555175444, -2.9860983311743827] … [4.331025079311032, 2.333423772424664] [4.480663191353303, 4.436571553462537]])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; save_png(&quot;2dim_h-refinement.png&quot;, M_h) # save image</code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><p><img src="../2dim_h-refinement.png" alt/></p><p>Note that this shape and the last shape are equivalent.</p><h3 id="p-refinement"><a class="docs-heading-anchor" href="#p-refinement">p-refinement</a><a id="p-refinement-1"></a><a class="docs-heading-anchor-permalink" href="#p-refinement" title="Permalink"></a></h3><p>Increase the polynomial degree of B-spline manifold.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; p₊ = (Val(1), Val(2)) # additional degrees</code><code class="nohighlight hljs ansi" style="display:block;">(Val{1}(), Val{2}())</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; M_p = refinement(M, p₊) # refinement of B-spline manifold</code><code class="nohighlight hljs ansi" style="display:block;">BSplineManifold{2, (3, 4), StaticArraysCore.SVector{2, Float64}, Tuple{BSplineSpace{3, Int64, KnotVector{Int64}}, BSplineSpace{4, Int64, KnotVector{Int64}}}}((BSplineSpace{3, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8])), BSplineSpace{4, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 8]))), StaticArraysCore.SVector{2, Float64}[[-3.327939887958376, -3.381062882382532] [-3.2286534014564956, -2.3898453165277354] … [-3.857369028126979, 2.625613347418197] [-3.750274329305369, 3.444604754180493]; [-1.9190826728891883, -3.6996476508119316] [-1.8954644502285198, -2.596850938310199] … [-2.320089840437407, 2.6673570013295036] [-2.4013083859231266, 3.417615645058782]; … ; [2.63663136518713, -3.36586722890932] [2.6119919237133846, -2.773961995558151] … [2.5855688772510934, 2.690164285500753] [2.692907338976954, 3.352168201374152]; [3.7212735564249955, -3.491398425417405] [3.554093269596626, -2.759665500609979] … [3.9960174259147876, 2.6027151956081522] [4.087903053338675, 3.7248325816241836]])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; save_png(&quot;2dim_p-refinement.png&quot;, M_p) # save image</code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><p><img src="../2dim_p-refinement.png" alt/></p><p>Note that this shape and the last shape are equivalent.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../math-inclusive/">« Inclusive relationship</a><a class="docs-footer-nextpage" href="../math-fitting/">Fitting »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. 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is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Mathematical properties of B-spline</a></li><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/math.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Mathematical-properties-of-B-spline"><a class="docs-heading-anchor" href="#Mathematical-properties-of-B-spline">Mathematical properties of B-spline</a><a id="Mathematical-properties-of-B-spline-1"></a><a class="docs-heading-anchor-permalink" href="#Mathematical-properties-of-B-spline" title="Permalink"></a></h1><h2 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h2><p><a href="https://en.wikipedia.org/wiki/B-spline">B-spline</a> is a mathematical object, and it has a lot of application. (e.g. Geometric representation: <a href="https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline">NURBS</a>, Interpolation, Numerical analysis: <a href="https://en.wikipedia.org/wiki/Isogeometric_analysis">IGA</a>)</p><p>In this page, we&#39;ll explain the mathematical definitions and properties of B-spline with Julia code. Before running the code in the following section, you need to import packages:</p><pre><code class="language-julia hljs">using BasicBSpline
+using Plots; plotly()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Plots.PlotlyBackend()</code></pre><h2 id="Notice"><a class="docs-heading-anchor" href="#Notice">Notice</a><a id="Notice-1"></a><a class="docs-heading-anchor-permalink" href="#Notice" title="Permalink"></a></h2><p>Some of notations in this page are our original, but these are well-considered results.</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><p>Most of this documentation around B-spline is self-contained. If you want to learn more, the following resources are recommended.</p><ul><li><a href="https://www.routledge.com/p/book/9780367447243">&quot;Geometric Modeling with Splines&quot; by Elaine Cohen, Richard F. Riesenfeld, Gershon Elber</a></li><li><a href="https://www.cambridge.org/core/books/spline-functions-basic-theory/843475201223F90091FFBDDCBF210BFB">&quot;Spline Functions: Basic Theory&quot; by Larry Schumaker</a></li></ul><p>日本語の文献では以下がおすすめです。</p><ul><li><a href="https://www.kyoiku-shuppan.co.jp/book/book/cate5/cate524/sho-463.html">スプライン関数とその応用 by 市田浩三, 吉本富士市</a></li><li><a href="https://hyrodium.github.io/ja/pdf/#NURBS%E5%A4%9A%E6%A7%98%E4%BD%93%E3%81%AB%E3%82%88%E3%82%8B%E5%BD%A2%E7%8A%B6%E8%A1%A8%E7%8F%BE">NURBS多様体による形状表現</a></li><li><a href="https://zenn.dev/hyrodium/articles/5fb08f98d4a918">BasicBSpline.jlを作ったので宣伝です！</a></li><li><a href="https://www.youtube.com/watch?v=GOdY02PA_WI">B-spline入門（線形代数がすこし分かる人向け）</a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../math-knotvector/">Knot vector »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR345/plotlyjs/index.html b/previews/PR345/plotlyjs/index.html
new file mode 100644
index 000000000..9d0537a50
--- /dev/null
+++ b/previews/PR345/plotlyjs/index.html
@@ -0,0 +1,40 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>PlotlyJS.jl · BasicBSpline.jl</title><meta name="title" content="PlotlyJS.jl · BasicBSpline.jl"/><meta property="og:title" content="PlotlyJS.jl · BasicBSpline.jl"/><meta property="twitter:title" content="PlotlyJS.jl · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/plotlyjs/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/plotlyjs/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/plotlyjs/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/><link href="../assets/custom.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="BasicBSpline.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="BasicBSpline.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">BasicBSpline.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li><a class="tocitem" href="../math-knotvector/">Knot vector</a></li><li><a class="tocitem" href="../math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="../math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li><a class="tocitem" href="../math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li><a class="tocitem" href="../math-fitting/">Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li><a class="tocitem" href="../plots/">Plots.jl</a></li><li class="is-active"><a class="tocitem" href>PlotlyJS.jl</a><ul class="internal"><li><a class="tocitem" href="#Cardioid"><span>Cardioid</span></a></li><li><a class="tocitem" href="#Helix"><span>Helix</span></a></li></ul></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Visualization</a></li><li class="is-active"><a href>PlotlyJS.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>PlotlyJS.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/plotlyjs.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="PlotlyJS.jl"><a class="docs-heading-anchor" href="#PlotlyJS.jl">PlotlyJS.jl</a><a id="PlotlyJS.jl-1"></a><a class="docs-heading-anchor-permalink" href="#PlotlyJS.jl" title="Permalink"></a></h1><h2 id="Cardioid"><a class="docs-heading-anchor" href="#Cardioid">Cardioid</a><a id="Cardioid-1"></a><a class="docs-heading-anchor-permalink" href="#Cardioid" title="Permalink"></a></h2><pre><code class="language-julia hljs">using BasicBSpline
+using BasicBSplineFitting
+using StaticArrays
+using PlotlyJS
+f(t) = SVector((1+cos(t))*cos(t),(1+cos(t))*sin(t))
+p = 3
+k = KnotVector(range(0,2π,15)) + p * KnotVector([0,2π]) + 2 * KnotVector([π])
+P = BSplineSpace{p}(k)
+a = fittingcontrolpoints(f, P)
+M = BSplineManifold(a,  P)
+
+ts = range(0,2π,250)
+xs_a = getindex.(a,1)
+ys_a = getindex.(a,2)
+xs_f = getindex.(M.(ts),1)
+ys_f = getindex.(M.(ts),2)
+fig = Plot(scatter(x=xs_a, y=ys_a, name=&quot;control points&quot;, line_color=&quot;blue&quot;, marker_size=8))
+addtraces!(fig, scatter(x=xs_f, y=ys_f, name=&quot;B-spline curve&quot;, mode=&quot;lines&quot;, line_color=&quot;red&quot;))
+relayout!(fig, width=500, height=500)</code></pre><object type="text/html" data="../cardioid.html" style="width:100%;height:550px;"></object><h2 id="Helix"><a class="docs-heading-anchor" href="#Helix">Helix</a><a id="Helix-1"></a><a class="docs-heading-anchor-permalink" href="#Helix" title="Permalink"></a></h2><pre><code class="language-julia hljs">using BasicBSpline
+using BasicBSplineFitting
+using StaticArrays
+using PlotlyJS
+f(t) = SVector(cos(t),sin(t),t)
+p = 3
+k = KnotVector(range(0,6π,15)) + p * KnotVector([0,6π])
+P = BSplineSpace{p}(k)
+a = fittingcontrolpoints(f, P)
+M = BSplineManifold(a,  P)
+
+ts = range(0,6π,250)
+xs_a = getindex.(a,1)
+ys_a = getindex.(a,2)
+zs_a = getindex.(a,3)
+xs_f = getindex.(M.(ts),1)
+ys_f = getindex.(M.(ts),2)
+zs_f = getindex.(M.(ts),3)
+fig = Plot(scatter3d(x=xs_a, y=ys_a, z=zs_a, name=&quot;control points&quot;, line_color=&quot;blue&quot;, marker_size=8))
+addtraces!(fig, scatter3d(x=xs_f, y=ys_f, z=zs_f, name=&quot;B-spline curve&quot;, mode=&quot;lines&quot;, line_color=&quot;red&quot;))
+relayout!(fig, width=500, height=500)</code></pre><object type="text/html" data="../helix.html" style="width:100%;height:550px;"></object></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../plots/">« Plots.jl</a><a class="docs-footer-nextpage" href="../basicbsplineexporter/">BasicBSplineExporter.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR345/plots-arc.html b/previews/PR345/plots-arc.html
new file mode 100644
index 000000000..a2af81554
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diff --git a/previews/PR345/plots/index.html b/previews/PR345/plots/index.html
new file mode 100644
index 000000000..21101ce68
--- /dev/null
+++ b/previews/PR345/plots/index.html
@@ -0,0 +1,61 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Plots.jl · BasicBSpline.jl</title><meta name="title" content="Plots.jl · BasicBSpline.jl"/><meta property="og:title" content="Plots.jl · BasicBSpline.jl"/><meta property="twitter:title" content="Plots.jl · BasicBSpline.jl"/><meta name="description" content="Documentation for BasicBSpline.jl."/><meta property="og:description" content="Documentation for BasicBSpline.jl."/><meta property="twitter:description" content="Documentation for BasicBSpline.jl."/><meta property="og:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/plots/"/><meta property="twitter:url" content="https://hyrodium.github.io/BasicBSpline.jl/stable/plots/"/><link rel="canonical" href="https://hyrodium.github.io/BasicBSpline.jl/stable/plots/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/><link href="../assets/custom.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="BasicBSpline.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="BasicBSpline.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">BasicBSpline.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Mathematical properties of B-spline</span><ul><li><a class="tocitem" href="../math/">Introduction</a></li><li><a class="tocitem" href="../math-knotvector/">Knot vector</a></li><li><a class="tocitem" href="../math-bsplinespace/">B-spline space</a></li><li><a class="tocitem" href="../math-bsplinebasis/">B-spline basis function</a></li><li><a class="tocitem" href="../math-bsplinemanifold/">B-spline manifold</a></li><li><a class="tocitem" href="../math-derivative/">Derivative</a></li><li><a class="tocitem" href="../math-inclusive/">Inclusive relationship</a></li><li><a class="tocitem" href="../math-refinement/">Refinement</a></li><li><a class="tocitem" href="../math-fitting/">Fitting</a></li><li><a class="tocitem" href="../math-rationalbsplinemanifold/">Rational B-spline manifold</a></li></ul></li><li><span class="tocitem">Visualization</span><ul><li class="is-active"><a class="tocitem" href>Plots.jl</a><ul class="internal"><li><a class="tocitem" href="#BSplineSpace"><span><code>BSplineSpace</code></span></a></li><li><a class="tocitem" href="#BSplineDerivativeSpace"><span><code>BSplineDerivativeSpace</code></span></a></li><li><a class="tocitem" href="#BSplineManifold"><span><code>BSplineManifold</code></span></a></li><li><a class="tocitem" href="#RationalBSplineManifold"><span><code>RationalBSplineManifold</code></span></a></li></ul></li><li><a class="tocitem" href="../plotlyjs/">PlotlyJS.jl</a></li><li><a class="tocitem" href="../basicbsplineexporter/">BasicBSplineExporter.jl</a></li></ul></li><li><a class="tocitem" href="../geometricmodeling/">Geometric modeling</a></li><li><a class="tocitem" href="../interpolations/">Interpolations</a></li><li><a class="tocitem" href="../internal/">Private API</a></li><li><a class="tocitem" href="../contributing/">Contributing</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Visualization</a></li><li class="is-active"><a href>Plots.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Plots.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/hyrodium/BasicBSpline.jl/blob/main/docs/src/plots.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Plots.jl"><a class="docs-heading-anchor" href="#Plots.jl">Plots.jl</a><a id="Plots.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Plots.jl" title="Permalink"></a></h1><p><code>BasicBSpline.jl</code> has a dependency on <a href="https://github.com/JuliaPlots/RecipesBase.jl"><code>RecipesBase.jl</code></a>. This means, users can easily visalize instances defined in <code>BasicBSpline</code>. In this section, we will provide some plottig examples.</p><h2 id="BSplineSpace"><a class="docs-heading-anchor" href="#BSplineSpace"><code>BSplineSpace</code></a><a id="BSplineSpace-1"></a><a class="docs-heading-anchor-permalink" href="#BSplineSpace" title="Permalink"></a></h2><pre><code class="language-julia hljs">k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+P0 = BSplineSpace{0}(k) # 0th degree piecewise polynomial space
+P1 = BSplineSpace{1}(k) # 1st degree piecewise polynomial space
+P2 = BSplineSpace{2}(k) # 2nd degree piecewise polynomial space
+P3 = BSplineSpace{3}(k) # 3rd degree piecewise polynomial space
+plot(
+    plot([t-&gt;bsplinebasis(P0,i,t) for i in 1:dim(P0)], 0, 10, ylims=(0,1), legend=false, title=&quot;0th polynomial degree&quot;),
+    plot([t-&gt;bsplinebasis(P1,i,t) for i in 1:dim(P1)], 0, 10, ylims=(0,1), legend=false, title=&quot;1st polynomial degree&quot;),
+    plot([t-&gt;bsplinebasis(P2,i,t) for i in 1:dim(P2)], 0, 10, ylims=(0,1), legend=false, title=&quot;2nd polynomial degree&quot;),
+    plot([t-&gt;bsplinebasis(P3,i,t) for i in 1:dim(P3)], 0, 10, ylims=(0,1), legend=false, title=&quot;3rd polynomial degree&quot;),
+)</code></pre><object type="text/html" data="../plots-bsplinebasis-raw.html" style="width:100%;height:420px;"></object><pre><code class="language-julia hljs">k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+P0 = BSplineSpace{0}(k) # 0th degree piecewise polynomial space
+P1 = BSplineSpace{1}(k) # 1st degree piecewise polynomial space
+P2 = BSplineSpace{2}(k) # 2nd degree piecewise polynomial space
+P3 = BSplineSpace{3}(k) # 3rd degree piecewise polynomial space
+plot(
+    plot(P0, ylims=(0,1), legend=false, title=&quot;0th polynomial degree&quot;),
+    plot(P1, ylims=(0,1), legend=false, title=&quot;1st polynomial degree&quot;),
+    plot(P2, ylims=(0,1), legend=false, title=&quot;2nd polynomial degree&quot;),
+    plot(P3, ylims=(0,1), legend=false, title=&quot;3rd polynomial degree&quot;),
+    layout=(2,2),
+)</code></pre><object type="text/html" data="../plots-bsplinebasis.html" style="width:100%;height:420px;"></object><h2 id="BSplineDerivativeSpace"><a class="docs-heading-anchor" href="#BSplineDerivativeSpace"><code>BSplineDerivativeSpace</code></a><a id="BSplineDerivativeSpace-1"></a><a class="docs-heading-anchor-permalink" href="#BSplineDerivativeSpace" title="Permalink"></a></h2><pre><code class="language-julia hljs">k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+P = BSplineSpace{3}(k)
+plot(
+    plot(BSplineDerivativeSpace{0}(P), label=&quot;0th derivative&quot;, color=:black),
+    plot(BSplineDerivativeSpace{1}(P), label=&quot;1st derivative&quot;, color=:red),
+    plot(BSplineDerivativeSpace{2}(P), label=&quot;2nd derivative&quot;, color=:green),
+    plot(BSplineDerivativeSpace{3}(P), label=&quot;3rd derivative&quot;, color=:blue),
+)</code></pre><object type="text/html" data="../plots-bsplinebasisderivative.html" style="width:100%;height:420px;"></object><h2 id="BSplineManifold"><a class="docs-heading-anchor" href="#BSplineManifold"><code>BSplineManifold</code></a><a id="BSplineManifold-1"></a><a class="docs-heading-anchor-permalink" href="#BSplineManifold" title="Permalink"></a></h2><h3 id="Cardioid-(planar-curve)"><a class="docs-heading-anchor" href="#Cardioid-(planar-curve)">Cardioid (planar curve)</a><a id="Cardioid-(planar-curve)-1"></a><a class="docs-heading-anchor-permalink" href="#Cardioid-(planar-curve)" title="Permalink"></a></h3><pre><code class="language-julia hljs">f(t) = SVector((1+cos(t))*cos(t),(1+cos(t))*sin(t))
+p = 3
+k = KnotVector(range(0,2π,15)) + p * KnotVector([0,2π]) + 2 * KnotVector([π])
+P = BSplineSpace{p}(k)
+a = fittingcontrolpoints(f, P)
+M = BSplineManifold(a, P)
+
+plot(M)</code></pre><object type="text/html" data="../plots-cardioid.html" style="width:100%;height:550px;"></object><h3 id="Helix-(spatial-curve)"><a class="docs-heading-anchor" href="#Helix-(spatial-curve)">Helix (spatial curve)</a><a id="Helix-(spatial-curve)-1"></a><a class="docs-heading-anchor-permalink" href="#Helix-(spatial-curve)" title="Permalink"></a></h3><pre><code class="language-julia hljs">f(t) = SVector(cos(t),sin(t),t)
+p = 3
+k = KnotVector(range(0,6π,15)) + p * KnotVector([0,6π])
+P = BSplineSpace{p}(k)
+a = fittingcontrolpoints(f, P)
+M = BSplineManifold(a, P)
+
+plot(M)</code></pre><object type="text/html" data="../plots-helix.html" style="width:100%;height:550px;"></object><h3 id="B-spline-surface"><a class="docs-heading-anchor" href="#B-spline-surface">B-spline surface</a><a id="B-spline-surface-1"></a><a class="docs-heading-anchor-permalink" href="#B-spline-surface" title="Permalink"></a></h3><pre><code class="language-julia hljs">p1 = 2
+p2 = 3
+k1 = KnotVector(1:10)
+k2 = KnotVector(1:20)
+P1 = BSplineSpace{p1}(k1)
+P2 = BSplineSpace{p2}(k2)
+a = [SVector(i-j^2/20, j+i^2/10, sin((i+j)/2)+randn()) for i in 1:dim(P1), j in 1:dim(P2)]
+M = BSplineManifold(a,(P1,P2))
+plot(M)</code></pre><object type="text/html" data="../plots-surface.html" style="width:100%;height:550px;"></object><h2 id="RationalBSplineManifold"><a class="docs-heading-anchor" href="#RationalBSplineManifold"><code>RationalBSplineManifold</code></a><a id="RationalBSplineManifold-1"></a><a class="docs-heading-anchor-permalink" href="#RationalBSplineManifold" title="Permalink"></a></h2><pre><code class="language-julia hljs">k = KnotVector([0,0,0,1,1,1])
+P = BSplineSpace{2}(k)
+a = [SVector(1,0),SVector(1,1),SVector(0,1)]
+w = [1,1/√2,1]
+M = BSplineManifold(a,P)
+R = RationalBSplineManifold(a,w,P)
+ts = 0:0.01:2
+plot(cospi.(ts),sinpi.(ts), label=&quot;circle&quot;)
+plot!(M, label=&quot;B-spline curve&quot;)
+plot!(R, label=&quot;Rational B-spline curve&quot;)</code></pre><object type="text/html" data="../plots-arc.html" style="width:100%;height:550px;"></object></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../math-rationalbsplinemanifold/">« Rational B-spline manifold</a><a class="docs-footer-nextpage" href="../plotlyjs/">PlotlyJS.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 13 November 2023 14:16">Monday 13 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR345/search_index.js b/previews/PR345/search_index.js
new file mode 100644
index 000000000..9f4959dfb
--- /dev/null
+++ b/previews/PR345/search_index.js
@@ -0,0 +1,3 @@
+var documenterSearchIndex = {"docs":
+[{"location":"math-fitting/#Fitting-with-B-spline-manifold","page":"Fitting","title":"Fitting with B-spline manifold","text":"","category":"section"},{"location":"math-fitting/","page":"Fitting","title":"Fitting","text":"The following functions such as fittingcontolpoints were provided from BasicBSpline.jl before v0.9.0. From BasicBSpline v0.9.0, these functions are moved to BasicBSplineFitting.","category":"page"},{"location":"math-fitting/","page":"Fitting","title":"Fitting","text":"using BasicBSpline\nusing BasicBSplineFitting\nusing BasicBSplineExporter\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"math-fitting/","page":"Fitting","title":"Fitting","text":"Fitting with least squares method.","category":"page"},{"location":"math-fitting/","page":"Fitting","title":"Fitting","text":"fittingcontrolpoints","category":"page"},{"location":"math-fitting/#BasicBSplineFitting.fittingcontrolpoints","page":"Fitting","title":"BasicBSplineFitting.fittingcontrolpoints","text":"Fitting controlpoints with least squares method.\n\nfittingcontrolpoints(func, Ps::Tuple)\n\nThis function will calculate bma_i to minimize the following integral.\n\nint_I leftf(t)-sum_i B_(ipk)(t) bma_iright^2 dt\n\nSimilarly, for the two-dimensional case, minimize the following integral.\n\nint_I^1 times I^2 leftf(t^1 t^2)-sum_ij B_(ip^1k^1)(t^1)B_(jp^2k^2)(t^2) bma_ijright^2 dt^1dt^2\n\nCurrently, this function supports up to three dimensions.\n\nExamples\n\njulia> f(t) = SVector(cos(t),sin(t),t);\n\njulia> P = BSplineSpace{3}(KnotVector(range(0,2π,30)) + 3*KnotVector([0,2π]));\n\njulia> a = fittingcontrolpoints(f, P);\n\njulia> M = BSplineManifold(a, P);\n\njulia> norm(M(1) - f(1)) < 1e-5\ntrue\n\n\n\n\n\n","category":"function"},{"location":"math-fitting/","page":"Fitting","title":"Fitting","text":"p1 = 2\np2 = 2\nk1 = KnotVector(-10:10)+p1*KnotVector([-10,10])\nk2 = KnotVector(-10:10)+p2*KnotVector([-10,10])\nP1 = BSplineSpace{p1}(k1)\nP2 = BSplineSpace{p2}(k2)\n\nf(u1, u2) = SVector(2u1 + sin(u1) + cos(u2) + u2 / 2, 3u2 + sin(u2) + sin(u1) / 2 + u1^2 / 6) / 5\n\na = fittingcontrolpoints(f, (P1, P2))\nM = BSplineManifold(a, (P1, P2))\nsave_png(\"fitting.png\", M, unitlength=50, xlims=(-10,10), ylims=(-10,10))","category":"page"},{"location":"math-fitting/","page":"Fitting","title":"Fitting","text":"(Image: )","category":"page"},{"location":"math-fitting/","page":"Fitting","title":"Fitting","text":"Try on Desmos graphing graphing calculator! (Image: )","category":"page"},{"location":"math-refinement/#Refinement","page":"Refinement","title":"Refinement","text":"","category":"section"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"math-refinement/#Documentation","page":"Refinement","title":"Documentation","text":"","category":"section"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"refinement","category":"page"},{"location":"math-refinement/#BasicBSpline.refinement","page":"Refinement","title":"BasicBSpline.refinement","text":"Refinement of B-spline manifold with given B-spline spaces.\n\n\n\n\n\n","category":"function"},{"location":"math-refinement/#Example","page":"Refinement","title":"Example","text":"","category":"section"},{"location":"math-refinement/#Define-original-manifold","page":"Refinement","title":"Define original manifold","text":"","category":"section"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"p = 2 # degree of polynomial\nk = KnotVector(1:8) # knot vector\nP = BSplineSpace{p}(k) # B-spline space\nrand_a = [SVector(rand(), rand()) for i in 1:dim(P), j in 1:dim(P)]\na = [SVector(2*i-6.5, 2*j-6.5) for i in 1:dim(P), j in 1:dim(P)] + rand_a # random \nM = BSplineManifold(a,(P,P)) # Define B-spline manifold\nnothing # hide","category":"page"},{"location":"math-refinement/#h-refinement","page":"Refinement","title":"h-refinement","text":"","category":"section"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"Insert additional knots to knot vector.","category":"page"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"k₊ = (KnotVector([3.3,4.2]),KnotVector([3.8,3.2,5.3])) # additional knot vectors\nM_h = refinement(M, k₊) # refinement of B-spline manifold\nsave_png(\"2dim_h-refinement.png\", M_h) # save image","category":"page"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"(Image: )","category":"page"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"Note that this shape and the last shape are equivalent.","category":"page"},{"location":"math-refinement/#p-refinement","page":"Refinement","title":"p-refinement","text":"","category":"section"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"Increase the polynomial degree of B-spline manifold.","category":"page"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"p₊ = (Val(1), Val(2)) # additional degrees\nM_p = refinement(M, p₊) # refinement of B-spline manifold\nsave_png(\"2dim_p-refinement.png\", M_p) # save image","category":"page"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"(Image: )","category":"page"},{"location":"math-refinement/","page":"Refinement","title":"Refinement","text":"Note that this shape and the last shape are equivalent.","category":"page"},{"location":"interpolations/#Interpolations","page":"Interpolations","title":"Interpolations","text":"","category":"section"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"Currently, BasicBSpline.jl doesn't have APIs for interpolations, but it is not hard to implement some basic interpolation algorithms with this package.","category":"page"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"using BasicBSpline\nusing IntervalSets\nusing Random; Random.seed!(42)\nusing Plots; plotly()","category":"page"},{"location":"interpolations/#Interpolation-with-cubic-B-spline","page":"Interpolations","title":"Interpolation with cubic B-spline","text":"","category":"section"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"function interpolate(xs::AbstractVector, fs::AbstractVector{T}) where T\n    # Cubic open B-spline space\n    p = 3\n    k = KnotVector(xs) + KnotVector([xs[1],xs[end]]) * p\n    P = BSplineSpace{p}(k)\n\n    # dimensions\n    m = length(xs)\n    n = dim(P)\n\n    # The interpolant function has a f''=0 property at bounds.\n    ddP = BSplineDerivativeSpace{2}(P)\n    dda = [bsplinebasis(ddP,j,xs[1]) for j in 1:n]\n    ddb = [bsplinebasis(ddP,j,xs[m]) for j in 1:n]\n\n    # Compute the interpolant function (1-dim B-spline manifold)\n    M = [bsplinebasis(P,j,xs[i]) for i in 1:m, j in 1:n]\n    M = vcat(dda', M, ddb')\n    y = vcat(zero(T), fs, zero(T))\n    return BSplineManifold(M\\y, P)\nend\n\n# Example inputs\nxs = [1, 2, 3, 4, 6, 7]\nfs = [1.3, 1.5, 2, 2.1, 1.9, 1.3]\nf = interpolate(xs,fs)\n\n# Plot\nscatter(xs, fs)\nplot!(t->f(t))\nsavefig(\"interpolation_cubic.html\") # hide\nnothing # hide","category":"page"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"<object type=\"text/html\" data=\"../interpolation_cubic.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"interpolations/#Interpolation-with-linear-B-spline","page":"Interpolations","title":"Interpolation with linear B-spline","text":"","category":"section"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"function interpolate_linear(xs::AbstractVector, fs::AbstractVector{T}) where T\n    # Linear open B-spline space\n    p = 1\n    k = KnotVector(xs) + KnotVector([xs[1],xs[end]])\n    P = BSplineSpace{p}(k)\n\n    # dimensions\n    m = length(xs)\n    n = dim(P)\n\n    # Compute the interpolant function (1-dim B-spline manifold)\n    return BSplineManifold(fs, P)\nend\n\n# Example inputs\nxs = [1, 2, 3, 4, 6, 7]\nfs = [1.3, 1.5, 2, 2.1, 1.9, 1.3]\n\nf = interpolate_linear(xs,fs)\n\n# Plot\nscatter(xs, fs)\nplot!(t->f(t))\nsavefig(\"interpolation_linear.html\") # hide\nnothing # hide","category":"page"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"<object type=\"text/html\" data=\"../interpolation_linear.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"interpolations/#Interpolation-with-periodic-B-spline","page":"Interpolations","title":"Interpolation with periodic B-spline","text":"","category":"section"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"function interpolate_periodic(xs::AbstractVector, fs::AbstractVector, ::Val{p}) where p\n    # Closed B-spline space, any polynomial degrees can be accepted\n    n = length(xs) - 1\n    period = xs[end]-xs[begin]\n    k = KnotVector(vcat(\n        xs[end-p:end-1] .- period,\n        xs,\n        xs[begin+1:begin+p] .+ period\n    ))\n    P = BSplineSpace{p}(k)\n    A = [bsplinebasis(P,j,xs[i]) for i in 1:n, j in 1:n]\n    for i in 1:p-1, j in 1:i\n        A[n+i-p+1,j] += bsplinebasis(P,j+n,xs[i+n-p+1])\n    end\n    b = A \\ fs[begin:end-1]\n    # Compute the interpolant function (1-dim B-spline manifold)\n    return BSplineManifold(vcat(b,b[1:p]), P)\nend\n\n# Example inputs\nxs = [1, 2, 3, 4, 6, 7]\nfs = [1.3, 1.5, 2, 2.1, 1.9, 1.3] # fs[1] == fs[end]\n\nf = interpolate_periodic(xs,fs,Val(2))\n\n# Plot\nscatter(xs, fs)\nplot!(t->f(mod(t-1,6)+1),1,14)\nplot!(t->f(t))\nsavefig(\"interpolation_periodic.html\") # hide\nnothing # hide","category":"page"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"<object type=\"text/html\" data=\"../interpolation_periodic.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"Note that the periodic interpolation supports any degree of polynomial.","category":"page"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"xs = 2π*rand(10)\nsort!(push!(xs, 0, 2π))\nfs = sin.(xs)\nf1 = interpolate_periodic(xs,fs,Val(1))\nf2 = interpolate_periodic(xs,fs,Val(2))\nf3 = interpolate_periodic(xs,fs,Val(3))\nf4 = interpolate_periodic(xs,fs,Val(4))\nf5 = interpolate_periodic(xs,fs,Val(5))\nscatter(xs, fs, label=\"sampling points\")\nplot!(sin, label=\"sine curve\", color=:black)\nplot!(t->f1(t), label=\"polynomial degree 1\")\nplot!(t->f2(t), label=\"polynomial degree 2\")\nplot!(t->f3(t), label=\"polynomial degree 3\")\nplot!(t->f4(t), label=\"polynomial degree 4\")\nplot!(t->f5(t), label=\"polynomial degree 5\")\nsavefig(\"interpolation_periodic_sin.html\") # hide\nnothing # hide","category":"page"},{"location":"interpolations/","page":"Interpolations","title":"Interpolations","text":"<object type=\"text/html\" data=\"../interpolation_periodic_sin.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"contributing/#Contributing","page":"Contributing","title":"Contributing","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"The main contributer Hyrodium is not native English speaker. So, English corrections would be really helpful. Of course, other code improvement are welcomed!","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Feel free to open issues and pull requests!","category":"page"},{"location":"math-bsplinebasis/#B-spline-basis-function","page":"B-spline basis function","title":"B-spline basis function","text":"","category":"section"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"math-bsplinebasis/#Basic-properties-of-B-spline-basis-function","page":"B-spline basis function","title":"Basic properties of B-spline basis function","text":"","category":"section"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"tip: Def.  B-spline space\nB-spline basis function is defined by Cox–de Boor recursion formula.beginaligned\nB_(ipk)(t)\n=\nfract-k_ik_i+p-k_iB_(ip-1k)(t)\n+frack_i+p+1-tk_i+p+1-k_i+1B_(i+1p-1k)(t) \nB_(i0k)(t)\n=\nbegincases\n    1quad (k_ile t k_i+1)\n    0quad (textotherwise)\nendcases\nendalignedIf the denominator is zero, then the term is assumed to be zero.","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"The next figure shows the plot of B-spline basis functions. You can manipulate these plots on desmos graphing calculator!","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"(Image: )","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"info: Thm.  Basis of B-spline space\nThe set of functions B_(ipk)_i is a basis of B-spline space mathcalPpk.","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"p = 2\nk = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\nP = BSplineSpace{p}(k)\nplot([t->bsplinebasis₊₀(P,i,t) for i in 1:dim(P)], 0, 10, ylims=(0,1), label=false)\nsavefig(\"bsplinebasisplot.html\") # hide\nnothing # hide","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"<object type=\"text/html\" data=\"../bsplinebasisplot.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"You can choose the first terms in different ways.","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"beginaligned\nB_(i0k)(t)\n=\nbegincases\n    1quad (k_i  t le k_i+1) \n    0quad (textotherwise)\nendcases\nendaligned","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"p = 2\nk = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\nP = BSplineSpace{p}(k)\nplot([t->bsplinebasis₋₀(P,i,t) for i in 1:dim(P)], 0, 10, ylims=(0,1), label=false)\nsavefig(\"bsplinebasisplot2.html\") # hide\nnothing # hide","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"<object type=\"text/html\" data=\"../bsplinebasisplot2.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"In these cases, each B-spline basis function B_(i2k) is coninuous, so bsplinebasis₊₀ and bsplinebasis₋₀ are equal.","category":"page"},{"location":"math-bsplinebasis/#Support-of-B-spline-basis-function","page":"B-spline basis function","title":"Support of B-spline basis function","text":"","category":"section"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"info: Thm.  Support of B-spline basis function\nIf a B-spline spacemathcalPpk is non-degenerate, the support of its basis function is calculated as follows:operatornamesupp(B_(ipk))=k_ik_i+p+1","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"[TODO: fig]","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"bsplinesupport","category":"page"},{"location":"math-bsplinebasis/#BasicBSpline.bsplinesupport","page":"B-spline basis function","title":"BasicBSpline.bsplinesupport","text":"Return the support of i-th B-spline basis function.\n\noperatornamesupp(B_(ipk))=k_ik_i+p+1\n\nExamples\n\njulia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\nKnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\n\njulia> P = BSplineSpace{2}(k)\nBSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))\n\njulia> bsplinesupport(P,1)\n0.0 .. 5.5\n\njulia> bsplinesupport(P,2)\n1.5 .. 8.0\n\n\n\n\n\n","category":"function"},{"location":"math-bsplinebasis/#Partition-of-unity","page":"B-spline basis function","title":"Partition of unity","text":"","category":"section"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"info: Thm.  Partition of unity\nLet B_(ipk) be a B-spline basis function, then the following equation is satisfied.beginaligned\nsum_iB_(ipk)(t) = 1  (k_p+1 le t  k_l-p) \n0 le B_(ipk)(t) le 1\nendaligned","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"p = 2\nk = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\nP = BSplineSpace{p}(k)\nplot(t->sum(bsplinebasis₊₀(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)\nsavefig(\"sumofbsplineplot.html\") # hide\nnothing # hide","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"<object type=\"text/html\" data=\"../sumofbsplineplot.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"To satisfy the partition of unity on whole interval 18, sometimes more knots will be inserted to the endpoints of the interval.","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"p = 2\nk = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]) + p * KnotVector([0,10])\nP = BSplineSpace{p}(k)\nplot(t->sum(bsplinebasis₊₀(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)\nsavefig(\"sumofbsplineplot2.html\") # hide\nnothing # hide","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"<object type=\"text/html\" data=\"../sumofbsplineplot2.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"But, the sum sum_i B_(ipk)(t) is not equal to 1 at t=8. Therefore, to satisfy partition of unity on closed interval k_p+1 k_l-p, the definition of first terms of B-spline basis functions are sometimes replaced:","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"beginaligned\nB_(i0k)(t)\n=\nbegincases\n    1quad (k_i le tk_i+1)\n    1quad (k_i  t = k_i+1=k_l)\n    0quad (textotherwise)\nendcases\nendaligned","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"p = 2\nk = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]) + p * KnotVector([0,10])\nP = BSplineSpace{p}(k)\nplot(t->sum(bsplinebasis(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)\nsavefig(\"sumofbsplineplot3.html\") # hide\nnothing # hide","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"<object type=\"text/html\" data=\"../sumofbsplineplot3.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"bsplinebasis₊₀","category":"page"},{"location":"math-bsplinebasis/#BasicBSpline.bsplinebasis₊₀","page":"B-spline basis function","title":"BasicBSpline.bsplinebasis₊₀","text":"i-th B-spline basis function. Right-sided limit version.\n\nbeginaligned\nB_(ipk)(t)\n=\nfract-k_ik_i+p-k_iB_(ip-1k)(t)\n+frack_i+p+1-tk_i+p+1-k_i+1B_(i+1p-1k)(t) \nB_(i0k)(t)\n=\nbegincases\n    1quad (k_ile t k_i+1)\n    0quad (textotherwise)\nendcases\nendaligned\n\nExamples\n\njulia> P = BSplineSpace{0}(KnotVector(1:6))\nBSplineSpace{0, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))\n\njulia> bsplinebasis₊₀.(P,1:5,(1:6)')\n5×6 Matrix{Float64}:\n 1.0  0.0  0.0  0.0  0.0  0.0\n 0.0  1.0  0.0  0.0  0.0  0.0\n 0.0  0.0  1.0  0.0  0.0  0.0\n 0.0  0.0  0.0  1.0  0.0  0.0\n 0.0  0.0  0.0  0.0  1.0  0.0\n\n\n\n\n\n","category":"function"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"bsplinebasis₋₀","category":"page"},{"location":"math-bsplinebasis/#BasicBSpline.bsplinebasis₋₀","page":"B-spline basis function","title":"BasicBSpline.bsplinebasis₋₀","text":"i-th B-spline basis function. Left-sided limit version.\n\nbeginaligned\nB_(ipk)(t)\n=\nfract-k_ik_i+p-k_iB_(ip-1k)(t)\n+frack_i+p+1-tk_i+p+1-k_i+1B_(i+1p-1k)(t) \nB_(i0k)(t)\n=\nbegincases\n    1quad (k_i tle k_i+1)\n    0quad (textotherwise)\nendcases\nendaligned\n\nExamples\n\njulia> P = BSplineSpace{0}(KnotVector(1:6))\nBSplineSpace{0, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))\n\njulia> bsplinebasis₋₀.(P,1:5,(1:6)')\n5×6 Matrix{Float64}:\n 0.0  1.0  0.0  0.0  0.0  0.0\n 0.0  0.0  1.0  0.0  0.0  0.0\n 0.0  0.0  0.0  1.0  0.0  0.0\n 0.0  0.0  0.0  0.0  1.0  0.0\n 0.0  0.0  0.0  0.0  0.0  1.0\n\n\n\n\n\n","category":"function"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"bsplinebasis","category":"page"},{"location":"math-bsplinebasis/#BasicBSpline.bsplinebasis","page":"B-spline basis function","title":"BasicBSpline.bsplinebasis","text":"i-th B-spline basis function. Modified version.\n\nbeginaligned\nB_(ipk)(t)\n=\nfract-k_ik_i+p-k_iB_(ip-1k)(t)\n+frack_i+p+1-tk_i+p+1-k_i+1B_(i+1p-1k)(t) \nB_(i0k)(t)\n=\nbegincases\n    1quad (k_i le t  k_i+1) \n    1quad (k_i  t = k_i+1=k_l) \n    0quad (textotherwise)\nendcases\nendaligned\n\nExamples\n\njulia> P = BSplineSpace{0}(KnotVector(1:6))\nBSplineSpace{0, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))\n\njulia> bsplinebasis.(P,1:5,(1:6)')\n5×6 Matrix{Float64}:\n 1.0  0.0  0.0  0.0  0.0  0.0\n 0.0  1.0  0.0  0.0  0.0  0.0\n 0.0  0.0  1.0  0.0  0.0  0.0\n 0.0  0.0  0.0  1.0  0.0  0.0\n 0.0  0.0  0.0  0.0  1.0  1.0\n\n\n\n\n\n","category":"function"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"BasicBSpline.bsplinebasis₋₀I","category":"page"},{"location":"math-bsplinebasis/#BasicBSpline.bsplinebasis₋₀I","page":"B-spline basis function","title":"BasicBSpline.bsplinebasis₋₀I","text":"i-th B-spline basis function. Modified version (2).\n\nbeginaligned\nB_(ipk)(t)\n=\nfract-k_ik_i+p-k_iB_(ip-1k)(t)\n+frack_i+p+1-tk_i+p+1-k_i+1B_(i+1p-1k)(t) \nB_(i0k)(t)\n=\nbegincases\n    1quad (k_1+p le k_i  t le k_i+1) \n    1quad (t = k_1+p = k_i  k_i+1) \n    1quad (k_i le t  k_i+1 le k_1+p) \n    0quad (textotherwise)\nendcases\nendaligned\n\nExamples\n\njulia> P = BSplineSpace{0}(KnotVector(1:6))\nBSplineSpace{0, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))\n\njulia> BasicBSpline.bsplinebasis₋₀I.(P,1:5,(1:6)')\n5×6 Matrix{Float64}:\n 1.0  1.0  0.0  0.0  0.0  0.0\n 0.0  0.0  1.0  0.0  0.0  0.0\n 0.0  0.0  0.0  1.0  0.0  0.0\n 0.0  0.0  0.0  0.0  1.0  0.0\n 0.0  0.0  0.0  0.0  0.0  1.0\n\n\n\n\n\n","category":"function"},{"location":"math-bsplinebasis/#B-spline-basis-functions-at-specific-point","page":"B-spline basis function","title":"B-spline basis functions at specific point","text":"","category":"section"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"Sometimes, you may need the non-zero values of B-spline basis functions at specific point. The bsplinebasisall function is much more efficient than evaluating B-spline functions one by one with bsplinebasis function.","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"using BenchmarkTools, BasicBSpline\nP = BSplineSpace{2}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))\nt = 6.3\n(bsplinebasis(P, 2, t), bsplinebasis(P, 3, t), bsplinebasis(P, 4, t))\nbsplinebasisall(P, 2, t)\n@benchmark (bsplinebasis($P, 2, $t), bsplinebasis($P, 3, $t), bsplinebasis($P, 4, $t))\n@benchmark bsplinebasisall($P, 2, $t)","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"intervalindex","category":"page"},{"location":"math-bsplinebasis/#BasicBSpline.intervalindex","page":"B-spline basis function","title":"BasicBSpline.intervalindex","text":"Return an index of a interval in the domain of B-spline space\n\nExamples\n\njulia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);\n\n\njulia> P = BSplineSpace{2}(k);\n\n\njulia> domain(P)\n2.5 .. 9.0\n\njulia> intervalindex(P,2.6)\n1\n\njulia> intervalindex(P,5.6)\n2\n\njulia> intervalindex(P,8.5)\n3\n\njulia> intervalindex(P,9.5)\n3\n\n\n\n\n\n","category":"function"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"bsplinebasisall","category":"page"},{"location":"math-bsplinebasis/#BasicBSpline.bsplinebasisall","page":"B-spline basis function","title":"BasicBSpline.bsplinebasisall","text":"B-spline basis functions at point t on i-th interval.\n\nExamples\n\njulia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\nKnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\n\njulia> p = 2\n2\n\njulia> P = BSplineSpace{p}(k)\nBSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))\n\njulia> t = 5.7\n5.7\n\njulia> i = intervalindex(P,t)\n2\n\njulia> bsplinebasisall(P,i,t)\n3-element SVector{3, Float64} with indices SOneTo(3):\n 0.3847272727272727\n 0.6107012987012989\n 0.00457142857142858\n\njulia> bsplinebasis.(P,i:i+p,t)\n3-element Vector{Float64}:\n 0.38472727272727264\n 0.6107012987012989\n 0.00457142857142858\n\n\n\n\n\n","category":"function"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"The next figures illustlates the relation between domain(P), intervalindex(P,t) and bsplinebasisall(P,i,t).","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\n\nfor p in 1:3\n    P = BSplineSpace{p}(k)\n    plot(P, legend=:topleft, label=\"B-spline basis (p=1)\")\n    plot!(t->intervalindex(P,t),0,10, label=\"Interval index\")\n    plot!(t->sum(bsplinebasis(P,i,t) for i in 1:dim(P)),0,10, label=\"Sum of B-spline basis\")\n    scatter!(k.vector,zero(k.vector), label=\"knot vector\")\n    plot!([t->bsplinebasisall(P,1,t)[i] for i in 1:p+1],0,10, color=:black, label=\"bsplinebasisall (i=1)\", ylim=(-1,8-2p))\n    savefig(\"bsplinebasisall-$(p).html\") # hide\n    nothing # hide\nend","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"<object type=\"text/html\" data=\"../bsplinebasisall-1.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"<object type=\"text/html\" data=\"../bsplinebasisall-2.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"<object type=\"text/html\" data=\"../bsplinebasisall-3.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinebasis/#Uniform-B-spline-basis-and-uniform-distribution","page":"B-spline basis function","title":"Uniform B-spline basis and uniform distribution","text":"","category":"section"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"Let X_1 dots X_n be i.i.d. random variables with X_i sim U(01), then the probability density function of X_1+cdots+X_n can be obtained via BasicBSpline.uniform_bsplinebasis_kernel(Val(n-1),t).","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"N = 100000\n# polynomial degree 0\nplot1 = histogram([rand() for _ in 1:N], normalize=true, label=false)\nplot!(t->BasicBSpline.uniform_bsplinebasis_kernel(Val(0),t), label=false)\n# polynomial degree 1\nplot2 = histogram([rand()+rand() for _ in 1:N], normalize=true, label=false)\nplot!(t->BasicBSpline.uniform_bsplinebasis_kernel(Val(1),t), label=false)\n# polynomial degree 2\nplot3 = histogram([rand()+rand()+rand() for _ in 1:N], normalize=true, label=false)\nplot!(t->BasicBSpline.uniform_bsplinebasis_kernel(Val(2),t), label=false)\n# polynomial degree 3\nplot4 = histogram([rand()+rand()+rand()+rand() for _ in 1:N], normalize=true, label=false)\nplot!(t->BasicBSpline.uniform_bsplinebasis_kernel(Val(3),t), label=false)\n# plot all\nplot(plot1,plot2,plot3,plot4)\nsavefig(\"histogram-uniform.html\") # hide\nnothing # hide","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"<object type=\"text/html\" data=\"../histogram-uniform.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-knotvector/#Knot-vector","page":"Knot vector","title":"Knot vector","text":"","category":"section"},{"location":"math-knotvector/#Definition","page":"Knot vector","title":"Definition","text":"","category":"section"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"tip: Def.  Knot vector\nA finite sequencek = (k_1 dots k_l)is called knot vector if the sequence is broad monotonic increase, i.e. k_i le k_i+1.","category":"page"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"[TODO: fig]","category":"page"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"KnotVector","category":"page"},{"location":"math-knotvector/#BasicBSpline.KnotVector","page":"Knot vector","title":"BasicBSpline.KnotVector","text":"Construct knot vector from given array.\n\nk=(k_1dotsk_l)\n\nExamples\n\njulia> k = KnotVector([1,2,3])\nKnotVector([1, 2, 3])\n\njulia> k = KnotVector(1:3)\nKnotVector([1, 2, 3])\n\n\n\n\n\n","category":"type"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"UniformKnotVector","category":"page"},{"location":"math-knotvector/#BasicBSpline.UniformKnotVector","page":"Knot vector","title":"BasicBSpline.UniformKnotVector","text":"Construct uniform knot vector from given range.\n\nk=(k_1dotsk_l)\n\nExamples\n\njulia> k = UniformKnotVector(1:8)\nUniformKnotVector(1:8)\n\njulia> UniformKnotVector(8:-1:3)\nUniformKnotVector(3:1:8)\n\n\n\n\n\n","category":"type"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"SubKnotVector","category":"page"},{"location":"math-knotvector/#BasicBSpline.SubKnotVector","page":"Knot vector","title":"BasicBSpline.SubKnotVector","text":"A type to represetnt sub knot vector.\n\nk=(k_1dotsk_l)\n\nExamples\n\njulia> k = knotvector\"1 11 211\"\nKnotVector([1, 3, 4, 6, 6, 7, 8])\n\njulia> view(k, 2:5)\nSubKnotVector([3, 4, 6, 6])\n\n\n\n\n\n","category":"type"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"EmptyKnotVector","category":"page"},{"location":"math-knotvector/#BasicBSpline.EmptyKnotVector","page":"Knot vector","title":"BasicBSpline.EmptyKnotVector","text":"Knot vector with zero-element.\n\nk=()\n\nThis struct is intended for internal use.\n\nExamples\n\njulia> EmptyKnotVector()\nEmptyKnotVector{Bool}()\n\njulia> EmptyKnotVector{Float64}()\nEmptyKnotVector{Float64}()\n\n\n\n\n\n","category":"type"},{"location":"math-knotvector/#Operations-for-knot-vectors","page":"Knot vector","title":"Operations for knot vectors","text":"","category":"section"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"length(k::AbstractKnotVector)","category":"page"},{"location":"math-knotvector/#Base.length-Tuple{AbstractKnotVector}","page":"Knot vector","title":"Base.length","text":"Length of knot vector\n\nbeginaligned\nk\n=(textnumber of knot elements of   k) \nendaligned\n\nFor example, (1223)=4.\n\nExamples\n\njulia> k = KnotVector([1,2,2,3]);\n\n\njulia> length(k)\n4\n\n\n\n\n\n","category":"method"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"Although a knot vector is not a vector in linear algebra, but we introduce additional operator +.","category":"page"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"Base.:+(k1::KnotVector{T}, k2::KnotVector{T}) where T","category":"page"},{"location":"math-knotvector/#Base.:+-Union{Tuple{T}, Tuple{KnotVector{T}, KnotVector{T}}} where T","page":"Knot vector","title":"Base.:+","text":"Sum of knot vectors\n\nbeginaligned\nk^(1)+k^(2)\n=(k^(1)_1 dots k^(1)_l^(1)) + (k^(2)_1 dots k^(2)_l^(2)) \n=(textsort of union of   k^(1)  textand   k^(2) text)\nendaligned\n\nFor example, (1235)+(458)=(1234558).\n\nExamples\n\njulia> k1 = KnotVector([1,2,3,5]);\n\n\njulia> k2 = KnotVector([4,5,8]);\n\n\njulia> k1 + k2\nKnotVector([1, 2, 3, 4, 5, 5, 8])\n\n\n\n\n\n","category":"method"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"Note that the operator +(::KnotVector, ::KnotVector) is commutative. This is why we choose the + sign. We also introduce product operator cdot for knot vector.","category":"page"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"*(m::Integer, k::AbstractKnotVector)","category":"page"},{"location":"math-knotvector/#Base.:*-Tuple{Integer, AbstractKnotVector}","page":"Knot vector","title":"Base.:*","text":"Product of integer and knot vector\n\nbeginaligned\nmcdot k=underbracek+cdots+k_m\nendaligned\n\nFor example, 2cdot (1225)=(11222255).\n\nExamples\n\njulia> k = KnotVector([1,2,2,5]);\n\n\njulia> 2 * k\nKnotVector([1, 1, 2, 2, 2, 2, 5, 5])\n\n\n\n\n\n","category":"method"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"Inclusive relationship between knot vectors.","category":"page"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"Base.issubset(k::KnotVector, k′::KnotVector)","category":"page"},{"location":"math-knotvector/#Base.issubset-Tuple{KnotVector, KnotVector}","page":"Knot vector","title":"Base.issubset","text":"Check a inclusive relationship ksubseteq k, for example:\n\nbeginaligned\n(12) subseteq (123) \n(122) notsubseteq (123) \n(123) subseteq (123) \nendaligned\n\nExamples\n\njulia> KnotVector([1,2]) ⊆ KnotVector([1,2,3])\ntrue\n\njulia> KnotVector([1,2,2]) ⊆ KnotVector([1,2,3])\nfalse\n\njulia> KnotVector([1,2,3]) ⊆ KnotVector([1,2,3])\ntrue\n\n\n\n\n\n","category":"method"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"unique(k::AbstractKnotVector)","category":"page"},{"location":"math-knotvector/#Base.unique-Tuple{AbstractKnotVector}","page":"Knot vector","title":"Base.unique","text":"Unique elements of knot vector.\n\nbeginaligned\nwidehatk\n=(textunique knot elements of   k) \nendaligned\n\nFor example, widehat(1223)=(123).\n\nExamples\n\njulia> k = KnotVector([1,2,2,3]);\n\n\njulia> unique(k)\nKnotVector([1, 2, 3])\n\n\n\n\n\n","category":"method"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"countknots(k::AbstractKnotVector, t::Real)","category":"page"},{"location":"math-knotvector/#BasicBSpline.countknots-Tuple{AbstractKnotVector, Real}","page":"Knot vector","title":"BasicBSpline.countknots","text":"For given knot vector k, the following function mathfrakn_kmathbbRtomathbbZ represents the number of knots that duplicate the knot vector k.\n\nmathfrakn_k(t) = i mid k_i=t \n\nFor example, if k=(1223), then mathfrakn_k(03)=0, mathfrakn_k(1)=1, mathfrakn_k(2)=2.\n\njulia> k = KnotVector([1,2,2,3]);\n\n\njulia> countknots(k,0.3)\n0\n\njulia> countknots(k,1.0)\n1\n\njulia> countknots(k,2.0)\n2\n\n\n\n\n\n","category":"method"},{"location":"math-knotvector/#KnotVector-with-string-macro","page":"Knot vector","title":"KnotVector with string macro","text":"","category":"section"},{"location":"math-knotvector/","page":"Knot vector","title":"Knot vector","text":"@knotvector_str","category":"page"},{"location":"math-knotvector/#BasicBSpline.@knotvector_str","page":"Knot vector","title":"BasicBSpline.@knotvector_str","text":"@knotvector_str -> KnotVector\n\nConstruct a knotvector by specifying the numbers of duplicates of knots.\n\nExamples\n\njulia> knotvector\"11111\"\nKnotVector([1, 2, 3, 4, 5])\n\njulia> knotvector\"123\"\nKnotVector([1, 2, 2, 3, 3, 3])\n\njulia> knotvector\" 2 2 2\"\nKnotVector([2, 2, 4, 4, 6, 6])\n\njulia> knotvector\"     1\"\nKnotVector([6])\n\n\n\n\n\n","category":"macro"},{"location":"plots/#Plots.jl","page":"Plots.jl","title":"Plots.jl","text":"","category":"section"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"BasicBSpline.jl has a dependency on RecipesBase.jl. This means, users can easily visalize instances defined in BasicBSpline. In this section, we will provide some plottig examples.","category":"page"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"using BasicBSpline\nusing BasicBSplineFitting\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"plots/#BSplineSpace","page":"Plots.jl","title":"BSplineSpace","text":"","category":"section"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\nP0 = BSplineSpace{0}(k) # 0th degree piecewise polynomial space\nP1 = BSplineSpace{1}(k) # 1st degree piecewise polynomial space\nP2 = BSplineSpace{2}(k) # 2nd degree piecewise polynomial space\nP3 = BSplineSpace{3}(k) # 3rd degree piecewise polynomial space\nplot(\n    plot([t->bsplinebasis(P0,i,t) for i in 1:dim(P0)], 0, 10, ylims=(0,1), legend=false, title=\"0th polynomial degree\"),\n    plot([t->bsplinebasis(P1,i,t) for i in 1:dim(P1)], 0, 10, ylims=(0,1), legend=false, title=\"1st polynomial degree\"),\n    plot([t->bsplinebasis(P2,i,t) for i in 1:dim(P2)], 0, 10, ylims=(0,1), legend=false, title=\"2nd polynomial degree\"),\n    plot([t->bsplinebasis(P3,i,t) for i in 1:dim(P3)], 0, 10, ylims=(0,1), legend=false, title=\"3rd polynomial degree\"),\n)\nsavefig(\"plots-bsplinebasis-raw.html\") # hide\nnothing # hide","category":"page"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"<object type=\"text/html\" data=\"../plots-bsplinebasis-raw.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\nP0 = BSplineSpace{0}(k) # 0th degree piecewise polynomial space\nP1 = BSplineSpace{1}(k) # 1st degree piecewise polynomial space\nP2 = BSplineSpace{2}(k) # 2nd degree piecewise polynomial space\nP3 = BSplineSpace{3}(k) # 3rd degree piecewise polynomial space\nplot(\n    plot(P0, ylims=(0,1), legend=false, title=\"0th polynomial degree\"),\n    plot(P1, ylims=(0,1), legend=false, title=\"1st polynomial degree\"),\n    plot(P2, ylims=(0,1), legend=false, title=\"2nd polynomial degree\"),\n    plot(P3, ylims=(0,1), legend=false, title=\"3rd polynomial degree\"),\n    layout=(2,2),\n)\nsavefig(\"plots-bsplinebasis.html\") # hide\nnothing # hide","category":"page"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"<object type=\"text/html\" data=\"../plots-bsplinebasis.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"plots/#BSplineDerivativeSpace","page":"Plots.jl","title":"BSplineDerivativeSpace","text":"","category":"section"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\nP = BSplineSpace{3}(k)\nplot(\n    plot(BSplineDerivativeSpace{0}(P), label=\"0th derivative\", color=:black),\n    plot(BSplineDerivativeSpace{1}(P), label=\"1st derivative\", color=:red),\n    plot(BSplineDerivativeSpace{2}(P), label=\"2nd derivative\", color=:green),\n    plot(BSplineDerivativeSpace{3}(P), label=\"3rd derivative\", color=:blue),\n)\nsavefig(\"plots-bsplinebasisderivative.html\") # hide\nnothing # hide","category":"page"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"<object type=\"text/html\" data=\"../plots-bsplinebasisderivative.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"plots/#BSplineManifold","page":"Plots.jl","title":"BSplineManifold","text":"","category":"section"},{"location":"plots/#Cardioid-(planar-curve)","page":"Plots.jl","title":"Cardioid (planar curve)","text":"","category":"section"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"f(t) = SVector((1+cos(t))*cos(t),(1+cos(t))*sin(t))\np = 3\nk = KnotVector(range(0,2π,15)) + p * KnotVector([0,2π]) + 2 * KnotVector([π])\nP = BSplineSpace{p}(k)\na = fittingcontrolpoints(f, P)\nM = BSplineManifold(a, P)\n\nplot(M)\nsavefig(\"plots-cardioid.html\") # hide\nnothing # hide","category":"page"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"<object type=\"text/html\" data=\"../plots-cardioid.html\" style=\"width:100%;height:550px;\"></object>","category":"page"},{"location":"plots/#Helix-(spatial-curve)","page":"Plots.jl","title":"Helix (spatial curve)","text":"","category":"section"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"f(t) = SVector(cos(t),sin(t),t)\np = 3\nk = KnotVector(range(0,6π,15)) + p * KnotVector([0,6π])\nP = BSplineSpace{p}(k)\na = fittingcontrolpoints(f, P)\nM = BSplineManifold(a, P)\n\nplot(M)\nsavefig(\"plots-helix.html\") # hide\nnothing # hide","category":"page"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"<object type=\"text/html\" data=\"../plots-helix.html\" style=\"width:100%;height:550px;\"></object>","category":"page"},{"location":"plots/#B-spline-surface","page":"Plots.jl","title":"B-spline surface","text":"","category":"section"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"p1 = 2\np2 = 3\nk1 = KnotVector(1:10)\nk2 = KnotVector(1:20)\nP1 = BSplineSpace{p1}(k1)\nP2 = BSplineSpace{p2}(k2)\na = [SVector(i-j^2/20, j+i^2/10, sin((i+j)/2)+randn()) for i in 1:dim(P1), j in 1:dim(P2)]\nM = BSplineManifold(a,(P1,P2))\nplot(M)\n\nsavefig(\"plots-surface.html\") # hide\nnothing # hide","category":"page"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"<object type=\"text/html\" data=\"../plots-surface.html\" style=\"width:100%;height:550px;\"></object>","category":"page"},{"location":"plots/#RationalBSplineManifold","page":"Plots.jl","title":"RationalBSplineManifold","text":"","category":"section"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"k = KnotVector([0,0,0,1,1,1])\nP = BSplineSpace{2}(k)\na = [SVector(1,0),SVector(1,1),SVector(0,1)]\nw = [1,1/√2,1]\nM = BSplineManifold(a,P)\nR = RationalBSplineManifold(a,w,P)\nts = 0:0.01:2\nplot(cospi.(ts),sinpi.(ts), label=\"circle\")\nplot!(M, label=\"B-spline curve\")\nplot!(R, label=\"Rational B-spline curve\")\n\nsavefig(\"plots-arc.html\") # hide\nnothing # hide","category":"page"},{"location":"plots/","page":"Plots.jl","title":"Plots.jl","text":"<object type=\"text/html\" data=\"../plots-arc.html\" style=\"width:100%;height:550px;\"></object>","category":"page"},{"location":"basicbsplineexporter/#BasicBSplineExporter.jl","page":"BasicBSplineExporter.jl","title":"BasicBSplineExporter.jl","text":"","category":"section"},{"location":"basicbsplineexporter/","page":"BasicBSplineExporter.jl","title":"BasicBSplineExporter.jl","text":"BasicBSplineExporter.jl supports export BasicBSpline.BSplineManifold{Dim,Deg,<:StaticVector} to:","category":"page"},{"location":"basicbsplineexporter/","page":"BasicBSplineExporter.jl","title":"BasicBSplineExporter.jl","text":"PNG image (.png)\nSVG image (.png)\nPOV-Ray mesh (.inc)","category":"page"},{"location":"basicbsplineexporter/#Installation","page":"BasicBSplineExporter.jl","title":"Installation","text":"","category":"section"},{"location":"basicbsplineexporter/","page":"BasicBSplineExporter.jl","title":"BasicBSplineExporter.jl","text":"] add https://github.com/hyrodium/BasicBSplineExporter.jl","category":"page"},{"location":"basicbsplineexporter/#First-example","page":"BasicBSplineExporter.jl","title":"First example","text":"","category":"section"},{"location":"basicbsplineexporter/","page":"BasicBSplineExporter.jl","title":"BasicBSplineExporter.jl","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\n\np = 2 # degree of polynomial\nk1 = KnotVector(1:8) # knot vector\nk2 = KnotVector(rand(7))+(p+1)*KnotVector([1])\nP1 = BSplineSpace{p}(k1) # B-spline space\nP2 = BSplineSpace{p}(k2)\nn1 = dim(P1) # dimension of B-spline space\nn2 = dim(P2)\na = [SVector(2i-6.5+rand(),1.5j-6.5+rand()) for i in 1:dim(P1), j in 1:dim(P2)] # random generated control points\nM = BSplineManifold(a,(P1,P2)) # Define B-spline manifold\nsave_png(\"BasicBSplineExporter_2dim.png\", M) # save image","category":"page"},{"location":"basicbsplineexporter/","page":"BasicBSplineExporter.jl","title":"BasicBSplineExporter.jl","text":"(Image: )","category":"page"},{"location":"basicbsplineexporter/#Other-examples","page":"BasicBSplineExporter.jl","title":"Other examples","text":"","category":"section"},{"location":"basicbsplineexporter/","page":"BasicBSplineExporter.jl","title":"BasicBSplineExporter.jl","text":"Here are some images rendared with POV-Ray.","category":"page"},{"location":"basicbsplineexporter/","page":"BasicBSplineExporter.jl","title":"BasicBSplineExporter.jl","text":"(Image: ) (Image: ) (Image: )","category":"page"},{"location":"basicbsplineexporter/","page":"BasicBSplineExporter.jl","title":"BasicBSplineExporter.jl","text":"See BasicBSplineExporter.jl/test for more examples.","category":"page"},{"location":"math-derivative/#Derivative-of-B-spline","page":"Derivative","title":"Derivative of B-spline","text":"","category":"section"},{"location":"math-derivative/","page":"Derivative","title":"Derivative","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"math-derivative/","page":"Derivative","title":"Derivative","text":"info: Thm.  Derivative of B-spline basis function\nThe derivative of B-spline basis function can be expressed as follows:beginaligned\ndotB_(ipk)(t)\n=fracddtB_(ipk)(t) \n=pleft(frac1k_i+p-k_iB_(ip-1k)(t)-frac1k_i+p+1-k_i+1B_(i+1p-1k)(t)right)\nendalignedNote that dotB_(ipk)inmathcalPp-1k.","category":"page"},{"location":"math-derivative/","page":"Derivative","title":"Derivative","text":"k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\nP = BSplineSpace{3}(k)\nplot(\n    plot(BSplineDerivativeSpace{0}(P), label=\"0th derivative\", color=:black),\n    plot(BSplineDerivativeSpace{1}(P), label=\"1st derivative\", color=:red),\n    plot(BSplineDerivativeSpace{2}(P), label=\"2nd derivative\", color=:green),\n    plot(BSplineDerivativeSpace{3}(P), label=\"3rd derivative\", color=:blue),\n)\nsavefig(\"bsplinebasisderivativeplot.html\") # hide\nnothing # hide","category":"page"},{"location":"math-derivative/","page":"Derivative","title":"Derivative","text":"<object type=\"text/html\" data=\"../bsplinebasisderivativeplot.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-derivative/","page":"Derivative","title":"Derivative","text":"BSplineDerivativeSpace","category":"page"},{"location":"math-derivative/#BasicBSpline.BSplineDerivativeSpace","page":"Derivative","title":"BasicBSpline.BSplineDerivativeSpace","text":"BSplineDerivativeSpace{r}(P::BSplineSpace)\n\nConstruct derivative of B-spline space from given differential order and B-spline space.\n\nD^r(mathcalPpk)\n=leftt mapsto left fracd^r fdt^r(t)   right  f in mathcalPpk right\n\nExamples\n\njulia> P = BSplineSpace{2}(KnotVector([1,2,3,4,5,6]))\nBSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))\n\njulia> dP = BSplineDerivativeSpace{1}(P)\nBSplineDerivativeSpace{1, BSplineSpace{2, Int64, KnotVector{Int64}}, Int64}(BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6])))\n\njulia> degree(P), degree(dP)\n(2, 1)\n\n\n\n\n\n","category":"type"},{"location":"math-derivative/","page":"Derivative","title":"Derivative","text":"BasicBSpline.derivative","category":"page"},{"location":"math-derivative/#BasicBSpline.derivative","page":"Derivative","title":"BasicBSpline.derivative","text":"derivative(::BSplineDerivativeSpace{r}) -> BSplineDerivativeSpace{r+1}\nderivative(::BSplineSpace) -> BSplineDerivativeSpace{1}\n\nDerivative of B-spline related space.\n\nExamples\n\njulia> BSplineSpace{2}(KnotVector(0:5))\nBSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 1, 2, 3, 4, 5]))\n\njulia> BasicBSpline.derivative(ans)\nBSplineDerivativeSpace{1, BSplineSpace{2, Int64, KnotVector{Int64}}, Int64}(BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 1, 2, 3, 4, 5])))\n\njulia> BasicBSpline.derivative(ans)\nBSplineDerivativeSpace{2, BSplineSpace{2, Int64, KnotVector{Int64}}, Int64}(BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 1, 2, 3, 4, 5])))\n\n\n\n\n\n","category":"function"},{"location":"math-derivative/","page":"Derivative","title":"Derivative","text":"bsplinebasis′₊₀","category":"page"},{"location":"math-derivative/#BasicBSpline.bsplinebasis′₊₀","page":"Derivative","title":"BasicBSpline.bsplinebasis′₊₀","text":"bsplinebasis′₊₀(::AbstractFunctionSpace, ::Integer, ::Real) -> Real\n\n1st derivative of B-spline basis function. Right-sided limit version.\n\ndotB_(ipk)(t)\n=pleft(frac1k_i+p-k_iB_(ip-1k)(t)-frac1k_i+p+1-k_i+1B_(i+1p-1k)(t)right)\n\nbsplinebasis′₊₀(P, i, t) is equivalent to bsplinebasis₊₀(derivative(P), i, t).\n\n\n\n\n\n","category":"function"},{"location":"math-derivative/","page":"Derivative","title":"Derivative","text":"bsplinebasis′₋₀","category":"page"},{"location":"math-derivative/#BasicBSpline.bsplinebasis′₋₀","page":"Derivative","title":"BasicBSpline.bsplinebasis′₋₀","text":"bsplinebasis′₋₀(::AbstractFunctionSpace, ::Integer, ::Real) -> Real\n\n1st derivative of B-spline basis function. Left-sided limit version.\n\ndotB_(ipk)(t)\n=pleft(frac1k_i+p-k_iB_(ip-1k)(t)-frac1k_i+p+1-k_i+1B_(i+1p-1k)(t)right)\n\nbsplinebasis′₋₀(P, i, t) is equivalent to bsplinebasis₋₀(derivative(P), i, t).\n\n\n\n\n\n","category":"function"},{"location":"math-derivative/","page":"Derivative","title":"Derivative","text":"bsplinebasis′","category":"page"},{"location":"math-derivative/#BasicBSpline.bsplinebasis′","page":"Derivative","title":"BasicBSpline.bsplinebasis′","text":"bsplinebasis′(::AbstractFunctionSpace, ::Integer, ::Real) -> Real\n\n1st derivative of B-spline basis function. Modified version.\n\ndotB_(ipk)(t)\n=pleft(frac1k_i+p-k_iB_(ip-1k)(t)-frac1k_i+p+1-k_i+1B_(i+1p-1k)(t)right)\n\nbsplinebasis′(P, i, t) is equivalent to bsplinebasis(derivative(P), i, t).\n\n\n\n\n\n","category":"function"},{"location":"plotlyjs/#PlotlyJS.jl","page":"PlotlyJS.jl","title":"PlotlyJS.jl","text":"","category":"section"},{"location":"plotlyjs/#Cardioid","page":"PlotlyJS.jl","title":"Cardioid","text":"","category":"section"},{"location":"plotlyjs/","page":"PlotlyJS.jl","title":"PlotlyJS.jl","text":"using BasicBSpline\nusing BasicBSplineFitting\nusing StaticArrays\nusing PlotlyJS\nf(t) = SVector((1+cos(t))*cos(t),(1+cos(t))*sin(t))\np = 3\nk = KnotVector(range(0,2π,15)) + p * KnotVector([0,2π]) + 2 * KnotVector([π])\nP = BSplineSpace{p}(k)\na = fittingcontrolpoints(f, P)\nM = BSplineManifold(a,  P)\n\nts = range(0,2π,250)\nxs_a = getindex.(a,1)\nys_a = getindex.(a,2)\nxs_f = getindex.(M.(ts),1)\nys_f = getindex.(M.(ts),2)\nfig = Plot(scatter(x=xs_a, y=ys_a, name=\"control points\", line_color=\"blue\", marker_size=8))\naddtraces!(fig, scatter(x=xs_f, y=ys_f, name=\"B-spline curve\", mode=\"lines\", line_color=\"red\"))\nrelayout!(fig, width=500, height=500)\nsavefig(fig,\"cardioid.html\") # hide\nnothing # hide","category":"page"},{"location":"plotlyjs/","page":"PlotlyJS.jl","title":"PlotlyJS.jl","text":"<object type=\"text/html\" data=\"../cardioid.html\" style=\"width:100%;height:550px;\"></object>","category":"page"},{"location":"plotlyjs/#Helix","page":"PlotlyJS.jl","title":"Helix","text":"","category":"section"},{"location":"plotlyjs/","page":"PlotlyJS.jl","title":"PlotlyJS.jl","text":"using BasicBSpline\nusing BasicBSplineFitting\nusing StaticArrays\nusing PlotlyJS\nf(t) = SVector(cos(t),sin(t),t)\np = 3\nk = KnotVector(range(0,6π,15)) + p * KnotVector([0,6π])\nP = BSplineSpace{p}(k)\na = fittingcontrolpoints(f, P)\nM = BSplineManifold(a,  P)\n\nts = range(0,6π,250)\nxs_a = getindex.(a,1)\nys_a = getindex.(a,2)\nzs_a = getindex.(a,3)\nxs_f = getindex.(M.(ts),1)\nys_f = getindex.(M.(ts),2)\nzs_f = getindex.(M.(ts),3)\nfig = Plot(scatter3d(x=xs_a, y=ys_a, z=zs_a, name=\"control points\", line_color=\"blue\", marker_size=8))\naddtraces!(fig, scatter3d(x=xs_f, y=ys_f, z=zs_f, name=\"B-spline curve\", mode=\"lines\", line_color=\"red\"))\nrelayout!(fig, width=500, height=500)\nsavefig(fig,\"helix.html\") # hide\nnothing # hide","category":"page"},{"location":"plotlyjs/","page":"PlotlyJS.jl","title":"PlotlyJS.jl","text":"<object type=\"text/html\" data=\"../helix.html\" style=\"width:100%;height:550px;\"></object>","category":"page"},{"location":"internal/#Private-API","page":"Private API","title":"Private API","text":"","category":"section"},{"location":"internal/","page":"Private API","title":"Private API","text":"Note that the following methods are considered private methods, and changes in their behavior are not considered breaking changes.","category":"page"},{"location":"internal/","page":"Private API","title":"Private API","text":"BasicBSpline.r_nomial","category":"page"},{"location":"internal/#BasicBSpline.r_nomial","page":"Private API","title":"BasicBSpline.r_nomial","text":"Calculate r-nomial coefficient\n\nr_nomial(n, k, r)\n\n(1+x+cdots+x^r)^n = sum_k a_nkr x^k\n\n\n\n\n\n","category":"function"},{"location":"internal/","page":"Private API","title":"Private API","text":"BasicBSpline._vec","category":"page"},{"location":"internal/#BasicBSpline._vec","page":"Private API","title":"BasicBSpline._vec","text":"Convert AbstractKnotVector to AbstractVector\n\n\n\n\n\n","category":"function"},{"location":"internal/","page":"Private API","title":"Private API","text":"BasicBSpline._lower_R","category":"page"},{"location":"internal/#BasicBSpline._lower_R","page":"Private API","title":"BasicBSpline._lower_R","text":"Internal methods for obtaining a B-spline space with one degree lower.\n\nbeginaligned\nmathcalPpk mapsto mathcalPp-1k \nD^rmathcalPpk mapsto D^r-1mathcalPp-1k\nendaligned\n\n\n\n\n\n","category":"function"},{"location":"internal/","page":"Private API","title":"Private API","text":"BasicBSpline._changebasis_sim","category":"page"},{"location":"internal/#BasicBSpline._changebasis_sim","page":"Private API","title":"BasicBSpline._changebasis_sim","text":"Return a coefficient matrix A which satisfy\n\nB_(ip_1k_1) = sum_jA_ijB_(jp_2k_2)\n\nAssumption:\n\nP_1  P_2\n\n\n\n\n\n","category":"function"},{"location":"internal/","page":"Private API","title":"Private API","text":"BasicBSplineFitting.innerproduct_R","category":"page"},{"location":"internal/#BasicBSplineFitting.innerproduct_R","page":"Private API","title":"BasicBSplineFitting.innerproduct_R","text":"Calculate a matrix\n\nA_ij=int_mathbbR B_(ipk)(t) B_(jpk)(t) dt\n\n\n\n\n\n","category":"function"},{"location":"internal/","page":"Private API","title":"Private API","text":"BasicBSplineFitting.innerproduct_I","category":"page"},{"location":"internal/#BasicBSplineFitting.innerproduct_I","page":"Private API","title":"BasicBSplineFitting.innerproduct_I","text":"Calculate a matrix\n\nA_ij=int_I B_(ipk)(t) B_(jpk)(t) dt\n\n\n\n\n\n","category":"function"},{"location":"math-inclusive/#Inclusive-relation-between-B-spline-spaces","page":"Inclusive relationship","title":"Inclusive relation between B-spline spaces","text":"","category":"section"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"info: Thm.  Inclusive relation between B-spline spaces\nFor non-degenerate B-spline spaces, the following relationship holds.mathcalPpk\nsubseteq mathcalPpk\nLeftrightarrow (m=p-p ge 0  textand  k+mwidehatksubseteq k)","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"Base.issubset(P::BSplineSpace{p}, P′::BSplineSpace{p′}) where {p, p′}","category":"page"},{"location":"math-inclusive/#Base.issubset-Union{Tuple{p′}, Tuple{p}, Tuple{BSplineSpace{p, T} where T<:Real, BSplineSpace{p′, T} where T<:Real}} where {p, p′}","page":"Inclusive relationship","title":"Base.issubset","text":"Check inclusive relationship between B-spline spaces.\n\nmathcalPpk\nsubseteqmathcalPpk\n\nExamples\n\njulia> P1 = BSplineSpace{1}(KnotVector([1,3,5,8]));\n\n\njulia> P2 = BSplineSpace{1}(KnotVector([1,3,5,6,8,9]));\n\n\njulia> P3 = BSplineSpace{2}(KnotVector([1,1,3,3,5,5,8,8]));\n\n\njulia> P1 ⊆ P2\ntrue\n\njulia> P1 ⊆ P3\ntrue\n\njulia> P2 ⊆ P3\nfalse\n\njulia> P2 ⊈ P3\ntrue\n\n\n\n\n\n","category":"method"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"Here are plots of the B-spline basis functions of the spaces P1, P2, P3.","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"P1 = BSplineSpace{1}(KnotVector([1,3,5,8]))\nP2 = BSplineSpace{1}(KnotVector([1,3,5,6,8,9]))\nP3 = BSplineSpace{2}(KnotVector([1,1,3,3,5,5,8,8]))\nplot(\n    plot([t->bsplinebasis₊₀(P1,i,t) for i in 1:dim(P1)], 1, 9, ylims=(0,1), legend=false),\n    plot([t->bsplinebasis₊₀(P2,i,t) for i in 1:dim(P2)], 1, 9, ylims=(0,1), legend=false),\n    plot([t->bsplinebasis₊₀(P3,i,t) for i in 1:dim(P3)], 1, 9, ylims=(0,1), legend=false),\n    layout=(3,1),\n    link=:x\n)\nsavefig(\"subbsplineplot.html\") # hide\nnothing # hide","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"<object type=\"text/html\" data=\"../subbsplineplot.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"This means, there exists a n times n matrix A which holds:","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"beginaligned\nB_(ipk)\n=sum_jA_ij B_(jpk) \nn=dim(mathcalPpk) \nn=dim(mathcalPpk)\nendaligned","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"You can calculate the change of basis matrix A with changebasis.","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"A12 = changebasis(P1,P2)\nA13 = changebasis(P1,P3)","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"plot(\n    plot([t->bsplinebasis₊₀(P1,i,t) for i in 1:dim(P1)], 1, 9, ylims=(0,1), legend=false),\n    plot([t->sum(A12[i,j]*bsplinebasis₊₀(P2,j,t) for j in 1:dim(P2)) for i in 1:dim(P1)], 1, 9, ylims=(0,1), legend=false),\n    plot([t->sum(A13[i,j]*bsplinebasis₊₀(P3,j,t) for j in 1:dim(P3)) for i in 1:dim(P1)], 1, 9, ylims=(0,1), legend=false),\n    layout=(3,1),\n    link=:x\n)\nsavefig(\"subbsplineplot2.html\") # hide\nnothing # hide","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"<object type=\"text/html\" data=\"../subbsplineplot2.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"changebasis_R","category":"page"},{"location":"math-inclusive/#BasicBSpline.changebasis_R","page":"Inclusive relationship","title":"BasicBSpline.changebasis_R","text":"Return a coefficient matrix A which satisfy\n\nB_(ipk) = sum_jA_ijB_(jpk)\n\nAssumption:\n\nP  P^prime\n\n\n\n\n\n","category":"function"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"changebasis_I","category":"page"},{"location":"math-inclusive/#BasicBSpline.changebasis_I","page":"Inclusive relationship","title":"BasicBSpline.changebasis_I","text":"Return a coefficient matrix A which satisfy\n\nB_(ipk) = sum_jA_ijB_(jpk)\n\nAssumption:\n\nP  P^prime\n\n\n\n\n\n","category":"function"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"issqsubset","category":"page"},{"location":"math-inclusive/#BasicBSpline.issqsubset","page":"Inclusive relationship","title":"BasicBSpline.issqsubset","text":"Check inclusive relationship between B-spline spaces.\n\nmathcalPpk\nsqsubseteqmathcalPpk\nLeftrightarrow\nmathcalPpk_k_p+1k_l-p\nsubseteqmathcalPpk_k_p+1k_l-p\n\n\n\n\n\n","category":"function"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"expandspace","category":"page"},{"location":"math-inclusive/#BasicBSpline.expandspace","page":"Inclusive relationship","title":"BasicBSpline.expandspace","text":"Expand B-spline space with given additional degree and knotvector. The behavior of expandspace is same as expandspace_I.\n\n\n\n\n\n","category":"function"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"expandspace_R","category":"page"},{"location":"math-inclusive/#BasicBSpline.expandspace_R","page":"Inclusive relationship","title":"BasicBSpline.expandspace_R","text":"Expand B-spline space with given additional degree and knotvector. This function is compatible with issubset (⊆).\n\nExamples\n\njulia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);\n\n\njulia> P = BSplineSpace{2}(k);\n\n\njulia> P′ = expandspace_R(P, Val(1), KnotVector([6.0]))\nBSplineSpace{3, Float64, KnotVector{Float64}}(KnotVector([0.0, 0.0, 1.5, 1.5, 2.5, 2.5, 5.5, 5.5, 6.0, 8.0, 8.0, 9.0, 9.0, 9.5, 9.5, 10.0, 10.0]))\n\njulia> P ⊆ P′\ntrue\n\njulia> P ⊑ P′\nfalse\n\njulia> domain(P)\n2.5 .. 9.0\n\njulia> domain(P′)\n1.5 .. 9.5\n\n\n\n\n\n","category":"function"},{"location":"math-inclusive/","page":"Inclusive relationship","title":"Inclusive relationship","text":"expandspace_I","category":"page"},{"location":"math-inclusive/#BasicBSpline.expandspace_I","page":"Inclusive relationship","title":"BasicBSpline.expandspace_I","text":"Expand B-spline space with given additional degree and knotvector. This function is compatible with issqsubset (⊑)\n\nExamples\n\njulia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);\n\n\njulia> P = BSplineSpace{2}(k);\n\n\njulia> P′ = expandspace_I(P, Val(1), KnotVector([6.0]))\nBSplineSpace{3, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 2.5, 5.5, 5.5, 6.0, 8.0, 8.0, 9.0, 9.0, 9.5, 10.0]))\n\njulia> P ⊆ P′\nfalse\n\njulia> P ⊑ P′\ntrue\n\njulia> domain(P)\n2.5 .. 9.0\n\njulia> domain(P′)\n2.5 .. 9.0\n\n\n\n\n\n","category":"function"},{"location":"math/#Mathematical-properties-of-B-spline","page":"Introduction","title":"Mathematical properties of B-spline","text":"","category":"section"},{"location":"math/#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"math/","page":"Introduction","title":"Introduction","text":"B-spline is a mathematical object, and it has a lot of application. (e.g. Geometric representation: NURBS, Interpolation, Numerical analysis: IGA)","category":"page"},{"location":"math/","page":"Introduction","title":"Introduction","text":"In this page, we'll explain the mathematical definitions and properties of B-spline with Julia code. Before running the code in the following section, you need to import packages:","category":"page"},{"location":"math/","page":"Introduction","title":"Introduction","text":"using BasicBSpline\nusing Plots; plotly()","category":"page"},{"location":"math/#Notice","page":"Introduction","title":"Notice","text":"","category":"section"},{"location":"math/","page":"Introduction","title":"Introduction","text":"Some of notations in this page are our original, but these are well-considered results.","category":"page"},{"location":"math/#References","page":"Introduction","title":"References","text":"","category":"section"},{"location":"math/","page":"Introduction","title":"Introduction","text":"Most of this documentation around B-spline is self-contained. If you want to learn more, the following resources are recommended.","category":"page"},{"location":"math/","page":"Introduction","title":"Introduction","text":"\"Geometric Modeling with Splines\" by Elaine Cohen, Richard F. Riesenfeld, Gershon Elber\n\"Spline Functions: Basic Theory\" by Larry Schumaker","category":"page"},{"location":"math/","page":"Introduction","title":"Introduction","text":"日本語の文献では以下がおすすめです。","category":"page"},{"location":"math/","page":"Introduction","title":"Introduction","text":"スプライン関数とその応用 by 市田浩三, 吉本富士市\nNURBS多様体による形状表現\nBasicBSpline.jlを作ったので宣伝です！\nB-spline入門（線形代数がすこし分かる人向け）","category":"page"},{"location":"math-bsplinemanifold/#B-spline-manifold","page":"B-spline manifold","title":"B-spline manifold","text":"","category":"section"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"math-bsplinemanifold/#Multi-dimensional-B-spline","page":"B-spline manifold","title":"Multi-dimensional B-spline","text":"","category":"section"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"info: Thm.  Basis of tensor product of B-spline spaces\nThe tensor product of B-spline spaces mathcalPp^1k^1otimesmathcalPp^2k^2 is a linear space with the following basis.mathcalPp^1k^1otimesmathcalPp^2k^2\n= operatorname*span_ij (B_(ip^1k^1) otimes B_(jp^2k^2))where the basis are defined as(B_(ip^1k^1) otimes B_(jp^2k^2))(t^1 t^2)\n= B_(ip^1k^1)(t^1) cdot B_(jp^2k^2)(t^2)","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"Higher dimensional tensor products mathcalPp^1k^1otimescdotsotimesmathcalPp^dk^d are defined similarly.","category":"page"},{"location":"math-bsplinemanifold/#B-spline-manifold-2","page":"B-spline manifold","title":"B-spline manifold","text":"","category":"section"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"B-spline manifold is a parametric representation of a shape.","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"tip: Def.  B-spline manifold\nFor given d-dimensional B-spline basis functions B_(i^1p^1k^1) otimes cdots otimes B_(i^dp^dk^d) and given points bma_i^1 dots i^d in V, B-spline manifold is defined by the following equality:bmp(t^1dotst^dbma_i^1 dots i^d)\n=sum_i^1dotsi^d(B_(i^1p^1k^1) otimes cdots otimes B_(i^dp^dk^d))(t^1dotst^d) bma_i^1 dots i^dWhere bma_i^1 dots i^d are called control points.","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"We will also write bmp(t^1dotst^d bma), bmp(t^1dotst^d), bmp(t bma) or bmp(t) for simplicity.","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"Note that the BSplineManifold objects are callable, and the arguments will be checked if it fits in the domain of BSplineSpace.","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"BSplineManifold","category":"page"},{"location":"math-bsplinemanifold/#BasicBSpline.BSplineManifold","page":"B-spline manifold","title":"BasicBSpline.BSplineManifold","text":"Construct B-spline manifold from given control points and B-spline spaces.\n\nExamples\n\njulia> using StaticArrays\n\njulia> P = BSplineSpace{2}(KnotVector([0,0,0,1,1,1]))\nBSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 0, 0, 1, 1, 1]))\n\njulia> a = [SVector(1,0), SVector(1,1), SVector(0,1)]\n3-element Vector{SVector{2, Int64}}:\n [1, 0]\n [1, 1]\n [0, 1]\n\njulia> M = BSplineManifold(a, P);\n\n\njulia> M(0.4)\n2-element SVector{2, Float64} with indices SOneTo(2):\n 0.84\n 0.64\n\njulia> M(1.2)\nERROR: DomainError with 1.2:\nThe input 1.2 is out of range.\n[...]\n\n\n\n\n\n","category":"type"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"If you need extension of BSplineManifold or don't need the arguments check, you can call unbounded_mapping.","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"unbounded_mapping","category":"page"},{"location":"math-bsplinemanifold/#BasicBSpline.unbounded_mapping","page":"B-spline manifold","title":"BasicBSpline.unbounded_mapping","text":"unbounded_mapping(M::BSplineManifold{Dim}, t::Vararg{Real,Dim})\n\nExamples\n\njulia> P = BSplineSpace{1}(KnotVector([0,0,1,1]))\nBSplineSpace{1, Int64, KnotVector{Int64}}(KnotVector([0, 0, 1, 1]))\n\njulia> domain(P)\n0 .. 1\n\njulia> M = BSplineManifold([0,1], P);\n\n\njulia> unbounded_mapping(M, 0.1)\n0.1\n\njulia> M(0.1)\n0.1\n\njulia> unbounded_mapping(M, 1.2)\n1.2\n\njulia> M(1.2)\nERROR: DomainError with 1.2:\nThe input 1.2 is out of range.\n[...]\n\n\n\n\n\n","category":"function"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"unbounded_mapping(M,t...) is a little bit faster than M(t...) because it does not check the domain.","category":"page"},{"location":"math-bsplinemanifold/#B-spline-curve","page":"B-spline manifold","title":"B-spline curve","text":"","category":"section"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"## 1-dim B-spline manifold\np = 2 # degree of polynomial\nk = KnotVector(1:12) # knot vector\nP = BSplineSpace{p}(k) # B-spline space\na = [SVector(i-5, 3*sin(i^2)) for i in 1:dim(P)] # control points\nM = BSplineManifold(a, P) # Define B-spline manifold\nplot(M)\nsavefig(\"1dim-manifold.html\") # hide\nnothing # hide","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"<object type=\"text/html\" data=\"../1dim-manifold.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinemanifold/#B-spline-surface","page":"B-spline manifold","title":"B-spline surface","text":"","category":"section"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"## 2-dim B-spline manifold\np = 2 # degree of polynomial\nk = KnotVector(1:8) # knot vector\nP = BSplineSpace{p}(k) # B-spline space\nrand_a = [SVector(rand(), rand(), rand()) for i in 1:dim(P), j in 1:dim(P)]\na = [SVector(2*i-6.5, 2*j-6.5, 0) for i in 1:dim(P), j in 1:dim(P)] + rand_a # random generated control points\nM = BSplineManifold(a,(P,P)) # Define B-spline manifold\nplot(M)\nsavefig(\"2dim-manifold.html\") # hide\nnothing # hide","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"<object type=\"text/html\" data=\"../2dim-manifold.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinemanifold/#Affine-commutativity","page":"B-spline manifold","title":"Affine commutativity","text":"","category":"section"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"info: Thm.  Affine commutativity\nLet T be a affine transform V to W, then the following equality holds.T(bmp(t bma))\n=bmp(t T(bma))","category":"page"},{"location":"math-bsplinemanifold/#Fixing-arguments-(currying)","page":"B-spline manifold","title":"Fixing arguments (currying)","text":"","category":"section"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"Just like fixing first index such as A[3,:]::Array{1} for matrix A::Array{2}, fixing first argument M(4.3,:) will create BSplineManifold{1} for B-spline surface M::BSplineManifold{2}.","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"p = 2;\nk = KnotVector(1:8);\nP = BSplineSpace{p}(k);\na = [SVector(5i+5j+rand(), 5i-5j+rand(), rand()) for i in 1:dim(P), j in 1:dim(P)];\nM = BSplineManifold(a,(P,P));\nM isa BSplineManifold{2}\nM(:,:) isa BSplineManifold{2}\nM(4.3,:) isa BSplineManifold{1}  # Fix first argument","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"plot(M)\nplot!(M(4.3,:), linewidth = 5, color=:cyan)\nplot!(M(4.4,:), linewidth = 5, color=:red)\nplot!(M(:,5.2), linewidth = 5, color=:green)\nsavefig(\"2dim-manifold-currying.html\") # hide\nnothing # hide","category":"page"},{"location":"math-bsplinemanifold/","page":"B-spline manifold","title":"B-spline manifold","text":"<object type=\"text/html\" data=\"../2dim-manifold-currying.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"math-bsplinespace/#B-spline-space","page":"B-spline space","title":"B-spline space","text":"","category":"section"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"math-bsplinespace/#Defnition","page":"B-spline space","title":"Defnition","text":"","category":"section"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"Before defining B-spline space, we'll define polynomial space with degree p.","category":"page"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"tip: Def.  Polynomial space\nPolynomial space with degree p.mathcalPp\n=leftfmathbbRtomathbbR  tmapsto a_0+a_1t^1+cdots+a_pt^p   left  \n    a_iin mathbbR\n    right\nrightThis space mathcalPp is a (p+1)-dimensional linear space.","category":"page"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"Note that tmapsto t^i_0 le i le p is a basis of mathcalPp, and also the set of Bernstein polynomial B_(ip)_i is a basis of mathcalPp.","category":"page"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"beginaligned\nB_(ip)(t)\n=binompi-1t^i-1(1-t)^p-i+1\n(i=1 dots p+1)\nendaligned","category":"page"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"Where binompi-1 is a binomial coefficient. You can try Bernstein polynomial on desmos graphing calculator!","category":"page"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"tip: Def.  B-spline space\nFor given polynomial degree pge 0 and knot vector k=(k_1dotsk_l), B-spline space mathcalPpk is defined as follows:mathcalPpk\n=leftfmathbbRtomathbbR   left  \n    begingathered\n        operatornamesupp(f)subseteq k_1 k_l \n        exists tildefinmathcalPp f_k_i k_i+1) = tildef_k_i k_i+1)  \n        forall t in mathbbR exists delta  0 f_(t-deltat+delta)in C^p-mathfrakn_k(t)\n    endgathered right\nright","category":"page"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"Note that each element of the space mathcalPpk is a piecewise polynomial.","category":"page"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"[TODO: fig]","category":"page"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"BSplineSpace","category":"page"},{"location":"math-bsplinespace/#BasicBSpline.BSplineSpace","page":"B-spline space","title":"BasicBSpline.BSplineSpace","text":"Construct B-spline space from given polynominal degree and knot vector.\n\nmathcalPpk\n\nExamples\n\njulia> p = 2\n2\n\njulia> k = KnotVector([1,3,5,6,8,9])\nKnotVector([1, 3, 5, 6, 8, 9])\n\njulia> BSplineSpace{p}(k)\nBSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 3, 5, 6, 8, 9]))\n\n\n\n\n\n","category":"type"},{"location":"math-bsplinespace/#Degeneration","page":"B-spline space","title":"Degeneration","text":"","category":"section"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"tip: Def.  Degeneration\nA B-spline space is said to be non-degenerate if its degree and knot vector satisfies following property:beginaligned\nk_ik_i+p+1  (1 le i le l-p-1)\nendaligned","category":"page"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"isnondegenerate","category":"page"},{"location":"math-bsplinespace/#BasicBSpline.isnondegenerate","page":"B-spline space","title":"BasicBSpline.isnondegenerate","text":"Check if given B-spline space is non-degenerate.\n\nExamples\n\njulia> isnondegenerate(BSplineSpace{2}(KnotVector([1,3,5,6,8,9])))\ntrue\n\njulia> isnondegenerate(BSplineSpace{1}(KnotVector([1,3,3,3,8,9])))\nfalse\n\n\n\n\n\n","category":"function"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"isdegenerate(P::BSplineSpace)","category":"page"},{"location":"math-bsplinespace/#BasicBSpline.isdegenerate-Tuple{BSplineSpace}","page":"B-spline space","title":"BasicBSpline.isdegenerate","text":"Check if given B-spline space is degenerate.\n\nExamples\n\njulia> isdegenerate(BSplineSpace{2}(KnotVector([1,3,5,6,8,9])))\nfalse\n\njulia> isdegenerate(BSplineSpace{1}(KnotVector([1,3,3,3,8,9])))\ntrue\n\n\n\n\n\n","category":"method"},{"location":"math-bsplinespace/#Dimensions","page":"B-spline space","title":"Dimensions","text":"","category":"section"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"info: Thm.  Dimension of B-spline space\nThe B-spline space is a linear space, and if a B-spline space is non-degenerate, its dimension is calculated by:dim(mathcalPpk)= k - p -1","category":"page"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"dim","category":"page"},{"location":"math-bsplinespace/#BasicBSpline.dim","page":"B-spline space","title":"BasicBSpline.dim","text":"Return dimention of a B-spline space.\n\ndim(mathcalPpk)\n= k - p -1\n\nExamples\n\njulia> dim(BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7])))\n5\n\njulia> dim(BSplineSpace{1}(KnotVector([1,2,4,4,4,6,7])))\n5\n\njulia> dim(BSplineSpace{1}(KnotVector([1,2,3,5,5,5,7])))\n5\n\n\n\n\n\n","category":"function"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"exactdim_R(P::BSplineSpace)","category":"page"},{"location":"math-bsplinespace/#BasicBSpline.exactdim_R-Tuple{BSplineSpace}","page":"B-spline space","title":"BasicBSpline.exactdim_R","text":"Exact dimension of a B-spline space.\n\nExamples\n\njulia> exactdim_R(BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7])))\n5\n\njulia> exactdim_R(BSplineSpace{1}(KnotVector([1,2,4,4,4,6,7])))\n4\n\njulia> exactdim_R(BSplineSpace{1}(KnotVector([1,2,3,5,5,5,7])))\n4\n\n\n\n\n\n","category":"method"},{"location":"math-bsplinespace/","page":"B-spline space","title":"B-spline space","text":"exactdim_I(P::BSplineSpace)","category":"page"},{"location":"math-bsplinespace/#BasicBSpline.exactdim_I-Tuple{BSplineSpace}","page":"B-spline space","title":"BasicBSpline.exactdim_I","text":"Exact dimension of a B-spline space.\n\nExamples\n\njulia> exactdim_I(BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7])))\n5\n\njulia> exactdim_I(BSplineSpace{1}(KnotVector([1,2,4,4,4,6,7])))\n4\n\njulia> exactdim_I(BSplineSpace{1}(KnotVector([1,2,3,5,5,5,7])))\n3\n\n\n\n\n\n","category":"method"},{"location":"math-rationalbsplinemanifold/#Rational-B-spline-manifold","page":"Rational B-spline manifold","title":"Rational B-spline manifold","text":"","category":"section"},{"location":"math-rationalbsplinemanifold/","page":"Rational B-spline manifold","title":"Rational B-spline manifold","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\nusing Plots; plotly()","category":"page"},{"location":"math-rationalbsplinemanifold/","page":"Rational B-spline manifold","title":"Rational B-spline manifold","text":"Non-uniform rational basis spline (NURBS) is also supported in BasicBSpline.jl package.","category":"page"},{"location":"math-rationalbsplinemanifold/#Rational-B-spline-manifold-2","page":"Rational B-spline manifold","title":"Rational B-spline manifold","text":"","category":"section"},{"location":"math-rationalbsplinemanifold/","page":"Rational B-spline manifold","title":"Rational B-spline manifold","text":"Rational B-spline manifold is a parametric representation of a shape.","category":"page"},{"location":"math-rationalbsplinemanifold/","page":"Rational B-spline manifold","title":"Rational B-spline manifold","text":"tip: Def.  Rational B-spline manifold\nFor given d-dimensional B-spline basis functions B_(i^1p^1k^1) otimes cdots otimes B_(i^dp^dk^d), given points bma_i^1 dots i^d in V and real numbers w_i^1 dots i^d  0, rational B-spline manifold is defined by the following equality:bmp(t^1dotst^d bma_i^1 dots i^d w_i^1 dots i^d)\n=sum_i^1dotsi^d\nfrac(B_(i^1p^1k^1) otimes cdots otimes B_(i^dp^dk^d))(t^1dotst^d) w_i^1 dots i^d\nsumlimits_j^1dotsj^d(B_(j^1p^1k^1) otimes cdots otimes B_(j^dp^dk^d))(t^1dotst^d) w_j^1 dots j^d\nbma_i^1 dots i^dWhere bma_i^1dotsi^d are called control points, and w_i^1 dots i^d are called weights.","category":"page"},{"location":"math-rationalbsplinemanifold/","page":"Rational B-spline manifold","title":"Rational B-spline manifold","text":"RationalBSplineManifold","category":"page"},{"location":"math-rationalbsplinemanifold/#BasicBSpline.RationalBSplineManifold","page":"Rational B-spline manifold","title":"BasicBSpline.RationalBSplineManifold","text":"Construct Rational B-spline manifold from given control points, weights and B-spline spaces.\n\nExamples\n\njulia> using StaticArrays, LinearAlgebra\n\njulia> P = BSplineSpace{2}(KnotVector([0,0,0,1,1,1]))\nBSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 0, 0, 1, 1, 1]))\n\njulia> w = [1, 1/√2, 1]\n3-element Vector{Float64}:\n 1.0\n 0.7071067811865475\n 1.0\n\njulia> a = [SVector(1,0), SVector(1,1), SVector(0,1)]\n3-element Vector{SVector{2, Int64}}:\n [1, 0]\n [1, 1]\n [0, 1]\n\njulia> M = RationalBSplineManifold(a,w,P);  # 1/4 arc\n\n\njulia> M(0.3)\n2-element SVector{2, Float64} with indices SOneTo(2):\n 0.8973756499953727\n 0.4412674277525845\n\njulia> norm(M(0.3))\n1.0\n\n\n\n\n\n","category":"type"},{"location":"math-rationalbsplinemanifold/#Properties","page":"Rational B-spline manifold","title":"Properties","text":"","category":"section"},{"location":"math-rationalbsplinemanifold/","page":"Rational B-spline manifold","title":"Rational B-spline manifold","text":"Similar to BSplineManifold, RationalBSplineManifold supports the following methods and properties.","category":"page"},{"location":"math-rationalbsplinemanifold/","page":"Rational B-spline manifold","title":"Rational B-spline manifold","text":"currying\nrefinement\nAffine commutativity","category":"page"},{"location":"geometricmodeling/#Geometric-modeling","page":"Geometric modeling","title":"Geometric modeling","text":"","category":"section"},{"location":"geometricmodeling/#Load-packages","page":"Geometric modeling","title":"Load packages","text":"","category":"section"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"using BasicBSpline\nusing StaticArrays\nusing Plots\nusing LinearAlgebra\nplotly()","category":"page"},{"location":"geometricmodeling/#Arc","page":"Geometric modeling","title":"Arc","text":"","category":"section"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"p = 2\nk = KnotVector([0,0,0,1,1,1])\nP = BSplineSpace{p}(k)\nt = 1  # angle in radians\na = [SVector(1,0), SVector(1,tan(t/2)), SVector(cos(t),sin(t))]\nw = [1,cos(t/2),1]\nM = RationalBSplineManifold(a,w,P)\nplot(M, xlims=(0,1.1), ylims=(0,1.1), aspectratio=1)\nsavefig(\"geometricmodeling-arc.html\") # hide\nnothing # hide","category":"page"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"<object type=\"text/html\" data=\"../geometricmodeling-arc.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"geometricmodeling/#Circle","page":"Geometric modeling","title":"Circle","text":"","category":"section"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"p = 2\nk = KnotVector([0,0,0,1,1,2,2,3,3,4,4,4])\nP = BSplineSpace{p}(k)\na = [\n    SVector( 1, 0),\n    SVector( 1, 1),\n    SVector( 0, 1),\n    SVector(-1, 1),\n    SVector(-1, 0),\n    SVector(-1,-1),\n    SVector( 0,-1),\n    SVector( 1,-1),\n    SVector( 1, 0)\n]\nw = [1,1/√2,1,1/√2,1,1/√2,1,1/√2,1]\nM = RationalBSplineManifold(a,w,P)\nplot(M, xlims=(-1.2,1.2), ylims=(-1.2,1.2), aspectratio=1)\nsavefig(\"geometricmodeling-circle.html\") # hide\nnothing # hide","category":"page"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"<object type=\"text/html\" data=\"../geometricmodeling-circle.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"geometricmodeling/#Torus","page":"Geometric modeling","title":"Torus","text":"","category":"section"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"R = 3\nr = 1\n\na0 = [\n    SVector( 1, 0, 0),\n    SVector( 1, 1, 0),\n    SVector( 0, 1, 0),\n    SVector(-1, 1, 0),\n    SVector(-1, 0, 0),\n    SVector(-1,-1, 0),\n    SVector( 0,-1, 0),\n    SVector( 1,-1, 0),\n    SVector( 1, 0, 0)\n]\n\na1 = (R+r)*a0\na5 = (R-r)*a0\na2 = [p+r*SVector(0,0,1) for p in a1]\na3 = [p+r*SVector(0,0,1) for p in R*a0]\na4 = [p+r*SVector(0,0,1) for p in a5]\na6 = [p-r*SVector(0,0,1) for p in a5]\na7 = [p-r*SVector(0,0,1) for p in R*a0]\na8 = [p-r*SVector(0,0,1) for p in a1]\na9 = a1\n\na = hcat(a1,a2,a3,a4,a5,a6,a7,a8,a9)\nM = RationalBSplineManifold(a,w*w',P,P)\nplot(M)\nsavefig(\"geometricmodeling-torus.html\") # hide\nnothing # hide","category":"page"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"<object type=\"text/html\" data=\"../geometricmodeling-torus.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"geometricmodeling/#Paraboloid","page":"Geometric modeling","title":"Paraboloid","text":"","category":"section"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"p = 2\nk = KnotVector([-1,-1,-1,1,1,1])\nP = BSplineSpace{p}(k)\na = [SVector(i,j,2i^2+2j^2-2) for i in -1:1, j in -1:1]\nM = BSplineManifold(a,P,P)\nplot(M)\nsavefig(\"geometricmodeling-paraboloid.html\") # hide\nnothing # hide","category":"page"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"<object type=\"text/html\" data=\"../geometricmodeling-paraboloid.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"geometricmodeling/#Hyperbolic-paraboloid","page":"Geometric modeling","title":"Hyperbolic paraboloid","text":"","category":"section"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"a = [SVector(i,j,2i^2-2j^2) for i in -1:1, j in -1:1]\nM = BSplineManifold(a,P,P)\nplot(M)\nsavefig(\"geometricmodeling-hyperbolicparaboloid.html\") # hide\nnothing # hide","category":"page"},{"location":"geometricmodeling/","page":"Geometric modeling","title":"Geometric modeling","text":"<object type=\"text/html\" data=\"../geometricmodeling-hyperbolicparaboloid.html\" style=\"width:100%;height:420px;\"></object>","category":"page"},{"location":"#BasicBSpline.jl","page":"Home","title":"BasicBSpline.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Basic (mathematical) operations for B-spline functions and related things with Julia.","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: Stable) (Image: Dev) (Image: Build Status) (Image: Coverage) (Image: Aqua QA) (Image: DOI) (Image: BasicBSpline Downloads).","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: )","category":"page"},{"location":"#Summary","page":"Home","title":"Summary","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides basic mathematical operations for B-spline.","category":"page"},{"location":"","page":"Home","title":"Home","text":"B-spline basis function\nSome operations for knot vector\nSome operations for B-spline space (piecewise polynomial space)\nB-spline manifold (includes curve, surface and solid)\nRefinement algorithm for B-spline manifold\nFitting control points for a given function","category":"page"},{"location":"#Comparison-to-other-Julia-packages-for-B-spline","page":"Home","title":"Comparison to other Julia packages for B-spline","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"There are several Julia packages for B-spline, and this package distinguishes itself with the following key benefits:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Supports all degrees of polynomials.\nIncludes a refinement algorithm for B-spline manifolds.\nOffers a fitting algorithm using least squares. (BasicBSplineFitting.jl)\nDelivers high-speed performance.\nIs mathematically oriented.","category":"page"},{"location":"","page":"Home","title":"Home","text":"If you have any thoughts, please comment in:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Issue#161\nDiscourse post about BasicBSpline.jl.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Install this package","category":"page"},{"location":"","page":"Home","title":"Home","text":"]add BasicBSpline","category":"page"},{"location":"","page":"Home","title":"Home","text":"To export graphics, use BasicBSplineExporter.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"]add https://github.com/hyrodium/BasicBSplineExporter.jl","category":"page"},{"location":"#Example","page":"Home","title":"Example","text":"","category":"section"},{"location":"#B-spline-function","page":"Home","title":"B-spline function","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"using BasicBSpline\nusing Plots\n\nk = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])\nP0 = BSplineSpace{0}(k) # 0th degree piecewise polynomial space\nP1 = BSplineSpace{1}(k) # 1st degree piecewise polynomial space\nP2 = BSplineSpace{2}(k) # 2nd degree piecewise polynomial space\nP3 = BSplineSpace{3}(k) # 3rd degree piecewise polynomial space\nplot(\n    plot([t->bsplinebasis(P0,i,t) for i in 1:dim(P0)], 0, 10, ylims=(0,1), legend=false),\n    plot([t->bsplinebasis(P1,i,t) for i in 1:dim(P1)], 0, 10, ylims=(0,1), legend=false),\n    plot([t->bsplinebasis(P2,i,t) for i in 1:dim(P2)], 0, 10, ylims=(0,1), legend=false),\n    plot([t->bsplinebasis(P3,i,t) for i in 1:dim(P3)], 0, 10, ylims=(0,1), legend=false),\n    layout=(2,2),\n)","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: )","category":"page"},{"location":"","page":"Home","title":"Home","text":"Try an interactive graph with Desmos graphing calculator!","category":"page"},{"location":"#B-spline-manifold","page":"Home","title":"B-spline manifold","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"using BasicBSpline\nusing BasicBSplineExporter\nusing StaticArrays\n\np = 2 # degree of polynomial\nk1 = KnotVector(1:8)     # knot vector\nk2 = KnotVector(rand(7))+(p+1)*KnotVector([1])\nP1 = BSplineSpace{p}(k1) # B-spline space\nP2 = BSplineSpace{p}(k2)\nn1 = dim(P1) # dimension of B-spline space\nn2 = dim(P2)\na = [SVector(2i-6.5+rand(),1.5j-6.5+rand()) for i in 1:dim(P1), j in 1:dim(P2)] # random generated control points\nM = BSplineManifold(a,(P1,P2)) # Define B-spline manifold\nsave_png(\"2dim.png\", M) # save image","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: )","category":"page"},{"location":"#Refinement","page":"Home","title":"Refinement","text":"","category":"section"},{"location":"#h-refinement","page":"Home","title":"h-refinement","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"k₊=(KnotVector([3.3,4.2]),KnotVector([0.3,0.5])) # additional knot vectors\nM_h = refinement(M, k₊) # refinement of B-spline manifold\nsave_png(\"2dim_h-refinement.png\", M_h) # save image","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: )","category":"page"},{"location":"","page":"Home","title":"Home","text":"Note that this shape and the last shape are equivalent.","category":"page"},{"location":"#p-refinement","page":"Home","title":"p-refinement","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"p₊=(Val(1),Val(2)) # additional degrees\nM_p = refinement(M, p₊) # refinement of B-spline manifold\nsave_png(\"2dim_p-refinement.png\", M_p) # save image","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: )","category":"page"},{"location":"","page":"Home","title":"Home","text":"Note that this shape and the last shape are equivalent.","category":"page"},{"location":"#Fitting-B-spline-manifold","page":"Home","title":"Fitting B-spline manifold","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Try on Desmos graphing calculator!","category":"page"},{"location":"","page":"Home","title":"Home","text":"using BasicBSplineFitting\n\np1 = 2\np2 = 2\nk1 = KnotVector(-10:10)+p1*KnotVector([-10,10])\nk2 = KnotVector(-10:10)+p2*KnotVector([-10,10])\nP1 = BSplineSpace{p1}(k1)\nP2 = BSplineSpace{p2}(k2)\n\nf(u1, u2) = SVector(2u1 + sin(u1) + cos(u2) + u2 / 2, 3u2 + sin(u2) + sin(u1) / 2 + u1^2 / 6) / 5\n\na = fittingcontrolpoints(f, (P1, P2))\nM = BSplineManifold(a, (P1, P2))\nsave_png(\"fitting.png\", M, unitlength=50, xlims=(-10,10), ylims=(-10,10))","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: ) (Image: )","category":"page"},{"location":"","page":"Home","title":"Home","text":"If the knot vector span is too coarse, the approximation will be coarse.","category":"page"},{"location":"","page":"Home","title":"Home","text":"p1 = 2\np2 = 2\nk1 = KnotVector(-10:5:10)+p1*KnotVector([-10,10])\nk2 = KnotVector(-10:5:10)+p2*KnotVector([-10,10])\nP1 = BSplineSpace{p1}(k1)\nP2 = BSplineSpace{p2}(k2)\n\nf(u1, u2) = SVector(2u1 + sin(u1) + cos(u2) + u2 / 2, 3u2 + sin(u2) + sin(u1) / 2 + u1^2 / 6) / 5\n\na = fittingcontrolpoints(f, (P1, P2))\nM = BSplineManifold(a, (P1, P2))\nsave_png(\"fitting_coarse.png\", M, unitlength=50, xlims=(-10,10), ylims=(-10,10))","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: )","category":"page"},{"location":"#Draw-smooth-vector-graphics","page":"Home","title":"Draw smooth vector graphics","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"p = 3\nk = KnotVector(range(-2π,2π,length=8))+p*KnotVector(-2π,2π)\nP = BSplineSpace{p}(k)\n\nf(u) = SVector(u,sin(u))\n\na = fittingcontrolpoints(f, P)\nM = BSplineManifold(a, P)\nsave_svg(\"sine-curve.svg\", M, unitlength=50, xlims=(-2,2), ylims=(-8,8))\nsave_svg(\"sine-curve_no-points.svg\", M, unitlength=50, xlims=(-2,2), ylims=(-8,8), points=false)","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: ) (Image: )","category":"page"},{"location":"","page":"Home","title":"Home","text":"This is useful when you edit graphs (or curves) with your favorite vector graphics editor.","category":"page"},{"location":"","page":"Home","title":"Home","text":"(Image: )","category":"page"},{"location":"","page":"Home","title":"Home","text":"See Plotting smooth graphs with Julia for more tutorials.","category":"page"}]
+}
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