From cb5da831c8f802476d28fc2850dbcd5979d58005 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Tue, 19 Sep 2023 03:42:13 +0000 Subject: [PATCH] build based on d10dab0 --- dev/.documenter-siteinfo.json | 1 + dev/1dim-manifold.html | 4 +- dev/2dim-manifold-currying.html | 4 +- dev/2dim-manifold.html | 4 +- dev/2dim_h-refinement.png | Bin 108969 -> 108953 bytes dev/2dim_p-refinement.png | Bin 115373 -> 115445 bytes dev/BasicBSplineExporter_2dim.png | Bin 125953 -> 125913 bytes dev/assets/documenter.js | 737 +- dev/assets/search.js | 267 - dev/assets/themes/documenter-dark.css | 7699 +--------------- dev/assets/themes/documenter-light.css | 7733 +---------------- dev/assets/themeswap.js | 82 +- dev/assets/warner.js | 87 +- dev/basicbsplineexporter/index.html | 4 +- dev/bsplinebasisall-1.html | 4 +- dev/bsplinebasisall-2.html | 4 +- dev/bsplinebasisall-3.html | 4 +- dev/bsplinebasisderivativeplot.html | 4 +- dev/bsplinebasisplot.html | 4 +- dev/bsplinebasisplot2.html | 4 +- dev/cardioid.html | 6 +- dev/contributing/index.html | 2 +- dev/fitting.png | Bin 227321 -> 227485 bytes dev/geometricmodeling-arc.html | 4 +- dev/geometricmodeling-circle.html | 4 +- ...eometricmodeling-hyperbolicparaboloid.html | 4 +- dev/geometricmodeling-paraboloid.html | 4 +- dev/geometricmodeling-torus.html | 4 +- dev/geometricmodeling/index.html | 4 +- dev/helix.html | 6 +- dev/histogram-uniform.html | 66 +- dev/index.html | 4 +- dev/internal/index.html | 4 +- dev/interpolation_cubic.html | 4 +- dev/interpolation_linear.html | 4 +- dev/interpolation_periodic.html | 4 +- dev/interpolation_periodic_sin.html | 4 +- dev/interpolations/index.html | 4 +- dev/math-bsplinebasis/index.html | 75 +- dev/math-bsplinemanifold/index.html | 16 +- dev/math-bsplinespace/index.html | 23 +- dev/math-derivative/index.html | 14 +- dev/math-fitting/index.html | 6 +- dev/math-inclusive/index.html | 25 +- dev/math-knotvector/index.html | 40 +- dev/math-rationalbsplinemanifold/index.html | 7 +- dev/math-refinement/index.html | 4 +- dev/math/index.html | 4 +- dev/plotlyjs/index.html | 4 +- dev/plots-arc.html | 4 +- dev/plots-bsplinebasis-raw.html | 4 +- dev/plots-bsplinebasis.html | 4 +- dev/plots-bsplinebasisderivative.html | 4 +- dev/plots-cardioid.html | 4 +- dev/plots-helix.html | 4 +- dev/plots-surface.html | 4 +- dev/plots/index.html | 4 +- dev/search/index.html | 2 - dev/search_index.js | 2 +- dev/subbsplineplot.html | 4 +- dev/subbsplineplot2.html | 4 +- dev/sumofbsplineplot.html | 4 +- dev/sumofbsplineplot2.html | 4 +- dev/sumofbsplineplot3.html | 4 +- 64 files changed, 993 insertions(+), 16059 deletions(-) create mode 100644 dev/.documenter-siteinfo.json delete mode 100644 dev/assets/search.js delete mode 100644 dev/search/index.html diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json new file mode 100644 index 000000000..398e963e8 --- /dev/null +++ b/dev/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-09-19T03:41:16","documenter_version":"1.0.1"}} \ No newline at end of file diff --git a/dev/1dim-manifold.html b/dev/1dim-manifold.html index 7d04ebc3a..83875207e 100644 --- a/dev/1dim-manifold.html +++ b/dev/1dim-manifold.html @@ -1,7 +1,7 @@ -
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Edit on GitHub

BasicBSplineExporter.jl

BasicBSplineExporter.jl supports export BasicBSpline.BSplineManifold{Dim,Deg,<:StaticVector} to:

  • PNG image (.png)
  • SVG image (.png)
  • POV-Ray mesh (.inc)

Installation

] add https://github.com/hyrodium/BasicBSplineExporter.jl

First example

using BasicBSpline
+BasicBSplineExporter.jl · BasicBSpline.jl
GitHub
BasicBSplineExporter.jl
BasicBSplineExporter.jl supports export BasicBSpline.BSplineManifold{Dim,Deg,<:StaticVector} to:
  • PNG image (.png)
  • SVG image (.png)
  • POV-Ray mesh (.inc)
Installation
] add https://github.com/hyrodium/BasicBSplineExporter.jl
First example
using BasicBSpline
 using BasicBSplineExporter
 using StaticArrays
 
@@ -12,4 +12,4 @@
 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("BasicBSplineExporter_2dim.png", M) # save image
Other examples
Here are some images rendared with POV-Ray.
  
See BasicBSplineExporter.jl/test for more examples.

+save_png("BasicBSplineExporter_2dim.png", M) # save image

Other examples

Here are some images rendared with POV-Ray.

See BasicBSplineExporter.jl/test for more examples.

diff --git a/dev/bsplinebasisall-1.html b/dev/bsplinebasisall-1.html index 0274eab64..3fd1bc0d8 100644 --- a/dev/bsplinebasisall-1.html +++ b/dev/bsplinebasisall-1.html @@ -1,7 +1,7 @@ -
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@@ -17,9 +17,9 @@ window.PLOTLYENV = window.PLOTLYENV || {} - if (document.getElementById('d38325cf-284c-4911-81f4-8abedd839f49')) { + if (document.getElementById('75e2ff16-2b87-4bf5-99d6-e05da58db803')) { Plotly.newPlot( - 'd38325cf-284c-4911-81f4-8abedd839f49', + '75e2ff16-2b87-4bf5-99d6-e05da58db803', 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{"editable":false,"responsive":true,"staticPlot":false,"scrollZoom":true}, diff --git a/dev/contributing/index.html b/dev/contributing/index.html index 015e8c648..415e61c21 100644 --- a/dev/contributing/index.html +++ b/dev/contributing/index.html @@ -1,2 +1,2 @@ -Contributing · BasicBSpline.jl
Edit on GitHub

Contributing

The main contributer Hyrodium is not native English speaker. So, English corrections would be really helpful. Of course, other code improvement are welcomed!

Feel free to open issues and pull requests!

+Contributing · BasicBSpline.jl
GitHub

Contributing

The main contributer Hyrodium is not native English speaker. So, English corrections would be really helpful. Of course, other code improvement are welcomed!

Feel free to open issues and pull requests!

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+
Edit on GitHub

BasicBSpline.jl

Basic (mathematical) operations for B-spline functions and related things with Julia.

Stable Dev Build Status Coverage Aqua QA DOI BasicBSpline Downloads.

Summary

This package provides basic mathematical operations for B-spline.

  • B-spline basis function
  • Some operations for knot vector
  • Some operations for B-spline space (piecewise polynomial space)
  • B-spline manifold (includes curve, surface and solid)
  • Refinement algorithm for B-spline manifold
  • Fitting control points for a given function

Comparison to other Julia packages for B-spline

There are several Julia packages for B-spline, and this package distinguishes itself with the following key benefits:

  • Supports all degrees of polynomials.
  • Includes a refinement algorithm for B-spline manifolds.
  • Offers a fitting algorithm using least squares. (BasicBSplineFitting.jl)
  • Delivers high-speed performance.
  • Is mathematically oriented.

If you have any thoughts, please comment in:

Installation

Install this package

]add BasicBSpline

To export graphics, use BasicBSplineExporter.jl.

]add https://github.com/hyrodium/BasicBSplineExporter.jl

Example

B-spline function

using BasicBSpline
+Home · BasicBSpline.jl
GitHub
BasicBSpline.jl
Basic (mathematical) operations for B-spline functions and related things with Julia.
Stable Dev Build Status Coverage Aqua QA DOI BasicBSpline Downloads.
Summary
This package provides basic mathematical operations for B-spline.
  • B-spline basis function
  • Some operations for knot vector
  • Some operations for B-spline space (piecewise polynomial space)
  • B-spline manifold (includes curve, surface and solid)
  • Refinement algorithm for B-spline manifold
  • Fitting control points for a given function
Comparison to other Julia packages for B-spline
There are several Julia packages for B-spline, and this package distinguishes itself with the following key benefits:
  • Supports all degrees of polynomials.
  • Includes a refinement algorithm for B-spline manifolds.
  • Offers a fitting algorithm using least squares. (BasicBSplineFitting.jl)
  • Delivers high-speed performance.
  • Is mathematically oriented.
If you have any thoughts, please comment in:
Installation
Install this package
]add BasicBSpline
To export graphics, use BasicBSplineExporter.jl.
]add https://github.com/hyrodium/BasicBSplineExporter.jl
Example
B-spline function
using BasicBSpline
 using Plots
 
 k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
@@ -63,4 +63,4 @@
 a = fittingcontrolpoints(f, P)
 M = BSplineManifold(a, P)
 save_svg("sine-curve.svg", M, unitlength=50, xlims=(-2,2), ylims=(-8,8))
-save_svg("sine-curve_no-points.svg", M, unitlength=50, xlims=(-2,2), ylims=(-8,8), points=false)
 
This is useful when you edit graphs (or curves) with your favorite vector graphics editor.
See Plotting smooth graphs with Julia for more tutorials.

+save_svg("sine-curve_no-points.svg", M, unitlength=50, xlims=(-2,2), ylims=(-8,8), points=false)

This is useful when you edit graphs (or curves) with your favorite vector graphics editor.

See Plotting smooth graphs with Julia for more tutorials.

diff --git a/dev/internal/index.html b/dev/internal/index.html index 5824b9148..e0d71016a 100644 --- a/dev/internal/index.html +++ b/dev/internal/index.html @@ -1,5 +1,5 @@ -Private API · BasicBSpline.jl
Edit on GitHub

Private API

Note that the following methods are considered private methods, and changes in their behavior are not considered breaking changes.

BasicBSpline.r_nomial — Function

Calculate $r$-nomial coefficient

r_nomial(n, k, r)

\[(1+x+\cdots+x^r)^n = \sum_{k} a_{n,k,r} x^k\]

source
BasicBSpline._lower_R — Function

Internal methods for obtaining a B-spline space with one degree lower.

\[\begin{aligned} +Private API · BasicBSpline.jl

GitHub

Private API

Note that the following methods are considered private methods, and changes in their behavior are not considered breaking changes.

BasicBSpline.r_nomial — Function

Calculate $r$-nomial coefficient

r_nomial(n, k, r)

\[(1+x+\cdots+x^r)^n = \sum_{k} a_{n,k,r} x^k\]

source
BasicBSpline._lower_R — Function

Internal methods for obtaining a B-spline space with one degree lower.

\[\begin{aligned} \mathcal{P}[p,k] &\mapsto \mathcal{P}[p-1,k] \\ D^r\mathcal{P}[p,k] &\mapsto D^{r-1}\mathcal{P}[p-1,k] -\end{aligned}\]

source
Missing docstring.

Missing docstring for BasicBSpline._changebasis_R. Check Documenter's build log for details.

Missing docstring.

Missing docstring for BasicBSpline._changebasis_I. Check Documenter's build log for details.

+\end{aligned}\]

source
diff --git a/dev/interpolation_cubic.html b/dev/interpolation_cubic.html index 9d7888aaa..c6eac1835 100644 --- a/dev/interpolation_cubic.html +++ b/dev/interpolation_cubic.html @@ -1,7 +1,7 @@ -
+
+
+
+
Edit on GitHub

Interpolations

Currently, BasicBSpline.jl doesn't have APIs for interpolations, but it is not hard to implement some basic interpolation algorithms with this package.

Interpolation with cubic B-spline

function interpolate(xs::AbstractVector, fs::AbstractVector{T}) where T
+Interpolations · BasicBSpline.jl
GitHub
Interpolations
Currently, BasicBSpline.jl doesn't have APIs for interpolations, but it is not hard to implement some basic interpolation algorithms with this package.
Interpolation with cubic B-spline
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
@@ -92,4 +92,4 @@
 plot!(t->f2(t), label="polynomial degree 2")
 plot!(t->f3(t), label="polynomial degree 3")
 plot!(t->f4(t), label="polynomial degree 4")
-plot!(t->f5(t), label="polynomial degree 5")

+plot!(t->f5(t), label="polynomial degree 5")
diff --git a/dev/math-bsplinebasis/index.html b/dev/math-bsplinebasis/index.html index 470e7de33..a3f44f293 100644 --- a/dev/math-bsplinebasis/index.html +++ b/dev/math-bsplinebasis/index.html @@ -1,5 +1,5 @@ -B-spline basis function · BasicBSpline.jl
Edit on GitHub

B-spline basis function

Basic properties of B-spline basis function

Def. B-spline space

B-spline basis function is defined by Cox–de Boor recursion formula.

\[\begin{aligned} +B-spline basis function · BasicBSpline.jl

GitHub

B-spline basis function

Basic properties of B-spline basis function

Def. B-spline space

B-spline basis function is defined by Cox–de Boor recursion formula.

\[\begin{aligned} {B}_{(i,p,k)}(t) &= \frac{t-k_{i}}{k_{i+p}-k_{i}}{B}_{(i,p-1,k)}(t) @@ -17,17 +17,17 @@ &1\quad (k_{i} < t \le k_{i+1}) \\ &0\quad (\text{otherwise}) \end{cases} -\end{aligned}\]

julia> p = 22
julia> 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> 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> plot([t->bsplinebasis₋₀(P,i,t) for i in 1:dim(P)], 0, 10, ylims=(0,1), label=false)Plot{Plots.PlotlyBackend() n=5}

In these cases, each B-spline basis function $B_{(i,2,k)}$ is coninuous, so bsplinebasis₊₀ and bsplinebasis₋₀ are equal.

Support of B-spline basis function

Thm. Support of B-spline basis function

If a B-spline space$\mathcal{P}[p,k]$ is non-degenerate, the support of its basis function is calculated as follows:

\[\operatorname{supp}(B_{(i,p,k)})=[k_{i},k_{i+p+1}]\]

[TODO: fig]

BasicBSpline.bsplinesupport — Function

Return the support of $i$-th B-spline basis function.

\[\operatorname{supp}(B_{(i,p,k)})=[k_{i},k_{i+p+1}]\]

Examples

julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+\end{aligned}\]
julia> p = 22
julia> 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> 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> plot([t->bsplinebasis₋₀(P,i,t) for i in 1:dim(P)], 0, 10, ylims=(0,1), label=false)Plot{Plots.PlotlyBackend() n=5}
In these cases, each B-spline basis function $B_{(i,2,k)}$ is coninuous, so bsplinebasis₊₀ and bsplinebasis₋₀ are equal.
Support of B-spline basis function
Thm.  Support of B-spline basis function
If a B-spline space$\mathcal{P}[p,k]$ is non-degenerate, the support of its basis function is calculated as follows:
\[\operatorname{supp}(B_{(i,p,k)})=[k_{i},k_{i+p+1}]\]
[TODO: fig]
BasicBSpline.bsplinesupport — Function
Return the support of $i$-th B-spline basis function.
\[\operatorname{supp}(B_{(i,p,k)})=[k_{i},k_{i+p+1}]\]
Examples
julia> 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> 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> bsplinesupport(P,1)
-0.0..5.5
+0.0 .. 5.5
 
 julia> bsplinesupport(P,2)
-1.5..8.0
source
Partition of unity
Thm.  Partition of unity
Let $B_{(i,p,k)}$ be a B-spline basis function, then the following equation is satisfied.
\[\begin{aligned}
+1.5 .. 8.0
source

Partition of unity

Thm. Partition of unity

Let $B_{(i,p,k)}$ be a B-spline basis function, then the following equation is satisfied.

\[\begin{aligned} \sum_{i}B_{(i,p,k)}(t) &= 1 & (k_{p+1} \le t < k_{l-p}) \\ 0 \le B_{(i,p,k)}(t) &\le 1 \end{aligned}\]

julia> p = 22
julia> 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> 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> plot(t->sum(bsplinebasis₊₀(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)Plot{Plots.PlotlyBackend() n=1}

To satisfy the partition of unity on whole interval $[1,8]$, sometimes more knots will be inserted to the endpoints of the interval.

julia> p = 22
julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]) + p * KnotVector([0,10])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])
julia> P = BSplineSpace{p}(k)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]))
julia> plot(t->sum(bsplinebasis₊₀(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)Plot{Plots.PlotlyBackend() n=1}

But, the sum $\sum_{i} B_{(i,p,k)}(t)$ 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:

\[\begin{aligned} @@ -38,7 +38,7 @@ &1\quad (k_{i} < t = k_{i+1}=k_{l})\\ &0\quad (\text{otherwise}) \end{cases} -\end{aligned}\]

julia> p = 22
julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]) + p * KnotVector([0,10])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])
julia> P = BSplineSpace{p}(k)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]))
julia> plot(t->sum(bsplinebasis(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)Plot{Plots.PlotlyBackend() n=1}
BasicBSpline.bsplinebasis₊₀ — Function

$i$-th B-spline basis function. Right-sided limit version.

\[\begin{aligned} +\end{aligned}\]

julia> p = 22
julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]) + p * KnotVector([0,10])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])
julia> P = BSplineSpace{p}(k)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]))
julia> plot(t->sum(bsplinebasis(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)Plot{Plots.PlotlyBackend() n=1}
BasicBSpline.bsplinebasis₊₀ — Function

$i$-th B-spline basis function. Right-sided limit version.

\[\begin{aligned} {B}_{(i,p,k)}(t) &= \frac{t-k_{i}}{k_{i+p}-k_{i}}{B}_{(i,p-1,k)}(t) @@ -58,7 +58,7 @@ 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

source
BasicBSpline.bsplinebasis₋₀ — Function

$i$-th B-spline basis function. Left-sided limit version.

\[\begin{aligned} {B}_{(i,p,k)}(t) &= \frac{t-k_{i}}{k_{i+p}-k_{i}}{B}_{(i,p-1,k)}(t) @@ -78,7 +78,7 @@ 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

source
BasicBSpline.bsplinebasis — Function

$i$-th B-spline basis function. Modified version.

\[\begin{aligned} {B}_{(i,p,k)}(t) &= \frac{t-k_{i}}{k_{i+p}-k_{i}}{B}_{(i,p-1,k)}(t) @@ -99,33 +99,56 @@ 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

source

B-spline basis functions at specific point

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.

julia> using BenchmarkTools, BasicBSpline
julia> P = BSplineSpace{2}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))
julia> t = 6.36.3
julia> (bsplinebasis(P, 2, t), bsplinebasis(P, 3, t), bsplinebasis(P, 4, t))(0.2101818181818182, 0.7166753246753247, 0.0731428571428571)
julia> bsplinebasisall(P, 2, t)3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
+ 0.0  0.0  0.0  0.0  1.0  1.0
source
BasicBSpline.bsplinebasis₋₀I — Function

$i$-th B-spline basis function. Modified version (2).

\[\begin{aligned} +{B}_{(i,p,k)}(t) +&= +\frac{t-k_{i}}{k_{i+p}-k_{i}}{B}_{(i,p-1,k)}(t) ++\frac{k_{i+p+1}-t}{k_{i+p+1}-k_{i+1}}{B}_{(i+1,p-1,k)}(t) \\ +{B}_{(i,0,k)}(t) +&= +\begin{cases} + &1\quad (k_{i} \le t<k_{i+1})\\ + &1\quad (k_{i} < t = k_{i+1}=k_{l})\\ + &0\quad (\text{otherwise}) +\end{cases} +\end{aligned}\]

Examples

julia> P = BSplineSpace{0}(KnotVector(1:6))
+BSplineSpace{0, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))
+
+julia> BasicBSpline.bsplinebasis₋₀I.(P,1:5,(1:6)')
+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
source

B-spline basis functions at specific point

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.

julia> using BenchmarkTools, BasicBSpline
julia> P = BSplineSpace{2}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))BSplineSpace{2, Float64, KnotVector{Float64}}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))
julia> t = 6.36.3
julia> (bsplinebasis(P, 2, t), bsplinebasis(P, 3, t), bsplinebasis(P, 4, t))(0.2101818181818182, 0.7166753246753247, 0.0731428571428571)
julia> bsplinebasisall(P, 2, t)3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
  0.21018181818181822
  0.7166753246753247
- 0.0731428571428571
julia> @benchmark (bsplinebasis($P, 2, $t), bsplinebasis($P, 3, $t), bsplinebasis($P, 4, $t))BenchmarkTools.Trial: 10000 samples with 996 evaluations.
- Range (min … max):  30.020 ns … 78.916 ns  ┊ GC (min … max): 0.00% … 0.00%
- Time  (median):     30.522 ns              ┊ GC (median):    0.00%
- Time  (mean ± σ):   30.747 ns ±  0.984 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
+ 0.0731428571428571
julia> @benchmark (bsplinebasis($P, 2, $t), bsplinebasis($P, 3, $t), bsplinebasis($P, 4, $t))BenchmarkTools.Trial: 10000 samples with 994 evaluations.
+ Range (min … max):  32.093 ns … 62.778 ns  ┊ GC (min … max): 0.00% … 0.00%
+ Time  (median):     32.294 ns              ┊ GC (median):    0.00%
+ Time  (mean ± σ):   32.392 ns ±  0.774 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
 
-     █ ▆ ▄ ▂                                                  
-  ▂▇▁█▁█▁█▁█▁█▁█▁▇▁▆▁▆▁▅▁▅▁▄▁▄▁▄▁▃▁▃▁▃▁▃▁▂▁▂▁▂▁▂▁▂▁▂▁▂▁▂▁▂▁▂▂ ▃
-  30 ns           Histogram: frequency by time          33 ns <
+     ▇  █  ▂                                                  
+  ▂▁▁█▁▁█▁▁█▁▁▁▄▁▁▂▁▁▂▁▁▅▁▁▁▂▁▁▁▁▁▁▁▁▁▁▁▁▂▁▁▁▁▁▁▁▁▁▂▁▁▂▁▁▂▁▁▂ ▂
+  32.1 ns         Histogram: frequency by time        33.9 ns <
 
  Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark bsplinebasisall($P, 2, $t)BenchmarkTools.Trial: 10000 samples with 1000 evaluations.
- Range (min … max):  9.599 ns … 24.000 ns  ┊ GC (min … max): 0.00% … 0.00%
- Time  (median):     9.700 ns              ┊ GC (median):    0.00%
- Time  (mean ± σ):   9.728 ns ±  0.417 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
+ Range (min … max):  8.000 ns … 26.000 ns  ┊ GC (min … max): 0.00% … 0.00%
+ Time  (median):     8.100 ns              ┊ GC (median):    0.00%
+ Time  (mean ± σ):   8.069 ns ±  0.339 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
 
-                     █                                       
-  ▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▅▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂ ▂
-  9.6 ns         Histogram: frequency by time         9.9 ns <
+  ▆                                                        █ 
+  █▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▂
+  8 ns           Histogram: frequency by time         8.1 ns <
+
+ Memory estimate: 0 bytes, allocs estimate: 0.
BasicBSpline.intervalindex — Function

Return an index of a interval in the domain of B-spline space

Examples

julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);
 
- Memory estimate: 0 bytes, allocs estimate: 0.
BasicBSpline.intervalindex — Function

Return an index of a interval in the domain of B-spline space

Examples

julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);
 
 julia> P = BSplineSpace{2}(k);
 
+
 julia> domain(P)
-2.5..9.0
+2.5 .. 9.0
 
 julia> intervalindex(P,2.6)
 1
@@ -137,7 +160,7 @@
 3
 
 julia> intervalindex(P,9.5)
-3
source
BasicBSpline.bsplinebasisall — Function

B-spline basis functions at point t on i-th interval.

Examples

julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+3
source
BasicBSpline.bsplinebasisall — Function

B-spline basis functions at point t on i-th interval.

Examples

julia> 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> p = 2
@@ -162,7 +185,7 @@
 3-element Vector{Float64}:
  0.38472727272727264
  0.6107012987012989
- 0.00457142857142858
source

The next figures illustlates the relation between domain(P), intervalindex(P,t) and bsplinebasisall(P,i,t).

k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+ 0.00457142857142858
source

The next figures illustlates the relation between domain(P), intervalindex(P,t) and bsplinebasisall(P,i,t).

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)
@@ -185,4 +208,4 @@
 plot4 = histogram([rand()+rand()+rand()+rand() for _ in 1:N], normalize=true, label=false)
 plot!(t->BasicBSpline.uniform_bsplinebasis_kernel(Val(3),t), label=false)
 # plot all
-plot(plot1,plot2,plot3,plot4)
+plot(plot1,plot2,plot3,plot4)
diff --git a/dev/math-bsplinemanifold/index.html b/dev/math-bsplinemanifold/index.html index 0c795e118..ec9257fd6 100644 --- a/dev/math-bsplinemanifold/index.html +++ b/dev/math-bsplinemanifold/index.html @@ -1,8 +1,8 @@ -B-spline manifold · BasicBSpline.jl
Edit on GitHub

B-spline manifold

Multi-dimensional B-spline

Thm. Basis of tensor product of B-spline spaces

The tensor product of B-spline spaces $\mathcal{P}[p^1,k^1]\otimes\mathcal{P}[p^2,k^2]$ is a linear space with the following basis.

\[\mathcal{P}[p^1,k^1]\otimes\mathcal{P}[p^2,k^2] +B-spline manifold · BasicBSpline.jl

GitHub

B-spline manifold

Multi-dimensional B-spline

Thm. Basis of tensor product of B-spline spaces

The tensor product of B-spline spaces $\mathcal{P}[p^1,k^1]\otimes\mathcal{P}[p^2,k^2]$ is a linear space with the following basis.

\[\mathcal{P}[p^1,k^1]\otimes\mathcal{P}[p^2,k^2] = \operatorname*{span}_{i,j} (B_{(i,p^1,k^1)} \otimes B_{(j,p^2,k^2)})\]

where the basis are defined as

\[(B_{(i,p^1,k^1)} \otimes B_{(j,p^2,k^2)})(t^1, t^2) = B_{(i,p^1,k^1)}(t^1) \cdot B_{(j,p^2,k^2)}(t^2)\]

Higher dimensional tensor products $\mathcal{P}[p^1,k^1]\otimes\cdots\otimes\mathcal{P}[p^d,k^d]$ are defined similarly.

B-spline manifold

B-spline manifold is a parametric representation of a shape.

Def. B-spline manifold

For given $d$-dimensional B-spline basis functions $B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)}$ and given points $\bm{a}_{i^1 \dots i^d} \in V$, B-spline manifold is defined by the following equality:

\[\bm{p}(t^1,\dots,t^d;\bm{a}_{i^1 \dots i^d}) -=\sum_{i^1,\dots,i^d}(B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)})(t^1,\dots,t^d) \bm{a}_{i^1 \dots i^d}\]

Where $\bm{a}_{i^1 \dots i^d}$ are called control points.

We will also write $\bm{p}(t^1,\dots,t^d; \bm{a})$, $\bm{p}(t^1,\dots,t^d)$, $\bm{p}(t; \bm{a})$ or $\bm{p}(t)$ for simplicity.

Note that the BSplineManifold objects are callable, and the arguments will be checked if it fits in the domain of BSplineSpace.

BasicBSpline.BSplineManifold — Type

Construct B-spline manifold from given control points and B-spline spaces.

Examples

julia> using StaticArrays
+=\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}\]
Where $\bm{a}_{i^1 \dots i^d}$ are called control points.

We will also write $\bm{p}(t^1,\dots,t^d; \bm{a})$, $\bm{p}(t^1,\dots,t^d)$, $\bm{p}(t; \bm{a})$ or $\bm{p}(t)$ for simplicity.

Note that the BSplineManifold objects are callable, and the arguments will be checked if it fits in the domain of BSplineSpace.

BasicBSpline.BSplineManifold — Type

Construct B-spline manifold from given control points and B-spline spaces.

Examples

julia> using StaticArrays
 
 julia> P = BSplineSpace{2}(KnotVector([0,0,0,1,1,1]))
 BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 0, 0, 1, 1, 1]))
@@ -15,6 +15,7 @@
 
 julia> M = BSplineManifold(a, P);
 
+
 julia> M(0.4)
 2-element SVector{2, Float64} with indices SOneTo(2):
  0.84
@@ -22,14 +23,16 @@
 
 julia> M(1.2)
 ERROR: DomainError with 1.2:
-The input 1.2 is out of range.
source

If you need extension of BSplineManifold or don't need the arguments check, you can call unbounded_mapping.

BasicBSpline.unbounded_mapping — Function
unbounded_mapping(M::BSplineManifold{Dim}, t::Vararg{Real,Dim})

Examples

julia> P = BSplineSpace{1}(KnotVector([0,0,1,1]))
+The input 1.2 is out of range.
+[...]
source

If you need extension of BSplineManifold or don't need the arguments check, you can call unbounded_mapping.

BasicBSpline.unbounded_mapping — Function
unbounded_mapping(M::BSplineManifold{Dim}, t::Vararg{Real,Dim})

Examples

julia> P = BSplineSpace{1}(KnotVector([0,0,1,1]))
 BSplineSpace{1, Int64, KnotVector{Int64}}(KnotVector([0, 0, 1, 1]))
 
 julia> domain(P)
-0..1
+0 .. 1
 
 julia> M = BSplineManifold([0,1], P);
 
+
 julia> unbounded_mapping(M, 0.1)
 0.1
 
@@ -41,7 +44,8 @@
 
 julia> M(1.2)
 ERROR: DomainError with 1.2:
-The input 1.2 is out of range.
source

unbounded_mapping(M,t...) is a little bit faster than M(t...) because it does not check the domain.

B-spline curve

## 1-dim B-spline manifold
+The input 1.2 is out of range.
+[...]
source

unbounded_mapping(M,t...) is a little bit faster than M(t...) because it does not check the domain.

B-spline curve

## 1-dim B-spline manifold
 p = 2 # degree of polynomial
 k = KnotVector(1:12) # knot vector
 P = BSplineSpace{p}(k) # B-spline space
@@ -58,4 +62,4 @@
 =\bm{p}(t; T(\bm{a}))\]

Fixing arguments (currying)

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}.

julia> p = 2;
julia> k = KnotVector(1:8);
julia> P = BSplineSpace{p}(k);
julia> a = [SVector(5i+5j+rand(), 5i-5j+rand(), rand()) for i in 1:dim(P), j in 1:dim(P)];
julia> M = BSplineManifold(a,(P,P));
julia> M isa BSplineManifold{2}true
julia> M(:,:) isa BSplineManifold{2}true
julia> M(4.3,:) isa BSplineManifold{1}  # Fix first argumenttrue
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)
+plot!(M(:,5.2), linewidth = 5, color=:green) diff --git a/dev/math-bsplinespace/index.html b/dev/math-bsplinespace/index.html index 88eeaa951..e61626053 100644 --- a/dev/math-bsplinespace/index.html +++ b/dev/math-bsplinespace/index.html @@ -1,5 +1,5 @@ -B-spline space · BasicBSpline.jl
Edit on GitHub

B-spline space

Defnition

Before defining B-spline space, we'll define polynomial space with degree $p$.

Def. Polynomial space

Polynomial space with degree $p$.

\[\mathcal{P}[p] +B-spline space · BasicBSpline.jl

GitHub

B-spline space

Defnition

Before defining B-spline space, we'll define polynomial space with degree $p$.

Def. Polynomial space

Polynomial space with degree $p$.

\[\mathcal{P}[p] =\left\{f:\mathbb{R}\to\mathbb{R}\ ;\ t\mapsto a_0+a_1t^1+\cdots+a_pt^p \ \left| \ % a_i\in \mathbb{R} \right. @@ -14,24 +14,24 @@ \exists \tilde{f}\in\mathcal{P}[p], f|_{[k_{i}, k_{i+1})} = \tilde{f}|_{[k_{i}, k_{i+1})} \\ \forall t \in \mathbb{R}, \exists \delta > 0, f|_{(t-\delta,t+\delta)}\in C^{p-\mathfrak{n}_k(t)} \end{gathered} \right. -\right\}\]

Note that each element of the space $\mathcal{P}[p,k]$ is a piecewise polynomial.

[TODO: fig]

BasicBSpline.BSplineSpace — Type

Construct B-spline space from given polynominal degree and knot vector.

\[\mathcal{P}[p,k]\]

Examples

julia> p = 2
+\right\}\]

Note that each element of the space $\mathcal{P}[p,k]$ is a piecewise polynomial.

[TODO: fig]

BasicBSpline.BSplineSpace — Type

Construct B-spline space from given polynominal degree and knot vector.

\[\mathcal{P}[p,k]\]

Examples

julia> p = 2
 2
 
 julia> k = KnotVector([1,3,5,6,8,9])
 KnotVector([1, 3, 5, 6, 8, 9])
 
 julia> BSplineSpace{p}(k)
-BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 3, 5, 6, 8, 9]))
source

Degeneration

Def. Degeneration

A B-spline space is said to be non-degenerate if its degree and knot vector satisfies following property:

\[\begin{aligned} +BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 3, 5, 6, 8, 9]))

source

Degeneration

Def. Degeneration

A B-spline space is said to be non-degenerate if its degree and knot vector satisfies following property:

\[\begin{aligned} k_{i}&<k_{i+p+1} & (1 \le i \le l-p-1) -\end{aligned}\]

BasicBSpline.isnondegenerate — Function

Check if given B-spline space is non-degenerate.

Examples

julia> isnondegenerate(BSplineSpace{2}(KnotVector([1,3,5,6,8,9])))
+\end{aligned}\]
BasicBSpline.isnondegenerate — Function

Check if given B-spline space is non-degenerate.

Examples

julia> isnondegenerate(BSplineSpace{2}(KnotVector([1,3,5,6,8,9])))
 true
 
 julia> isnondegenerate(BSplineSpace{1}(KnotVector([1,3,3,3,8,9])))
-false
source
BasicBSpline.isdegenerate — Method

Check if given B-spline space is degenerate.

Examples

julia> isdegenerate(BSplineSpace{2}(KnotVector([1,3,5,6,8,9])))
+false
source
BasicBSpline.isdegenerate — Method

Check if given B-spline space is degenerate.

Examples

julia> isdegenerate(BSplineSpace{2}(KnotVector([1,3,5,6,8,9])))
 false
 
 julia> isdegenerate(BSplineSpace{1}(KnotVector([1,3,3,3,8,9])))
-true
source

Dimensions

Thm. Dimension of B-spline space

The B-spline space is a linear space, and if a B-spline space is non-degenerate, its dimension is calculated by:

\[\dim(\mathcal{P}[p,k])=\# k - p -1\]

Dimensions

Thm. Dimension of B-spline space

The B-spline space is a linear space, and if a B-spline space is non-degenerate, its dimension is calculated by:

\[\dim(\mathcal{P}[p,k])=\# k - p -1\]

BasicBSpline.dim — Function

Return dimention of a B-spline space.

\[\dim(\mathcal{P}[p,k]) =\# k - p -1\]

Examples

julia> dim(BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7])))
 5
 
@@ -39,11 +39,18 @@
 5
 
 julia> dim(BSplineSpace{1}(KnotVector([1,2,3,5,5,5,7])))
-5
source
BasicBSpline.exactdim_R — Method

Exact dimension of a B-spline space.

Examples

julia> exactdim_R(BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7])))
+5
source
BasicBSpline.exactdim_R — Method

Exact dimension of a B-spline space.

Examples

julia> exactdim_R(BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7])))
 5
 
 julia> exactdim_R(BSplineSpace{1}(KnotVector([1,2,4,4,4,6,7])))
 4
 
 julia> exactdim_R(BSplineSpace{1}(KnotVector([1,2,3,5,5,5,7])))
-4
source
+4source
BasicBSpline.exactdim_I — Method

Exact dimension of a B-spline space.

Examples

julia> exactdim_I(BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7])))
+5
+
+julia> exactdim_I(BSplineSpace{1}(KnotVector([1,2,4,4,4,6,7])))
+4
+
+julia> exactdim_I(BSplineSpace{1}(KnotVector([1,2,3,5,5,5,7])))
+3
source
diff --git a/dev/math-derivative/index.html b/dev/math-derivative/index.html index 6509c645e..9b40dae1a 100644 --- a/dev/math-derivative/index.html +++ b/dev/math-derivative/index.html @@ -1,5 +1,5 @@ -Derivative · BasicBSpline.jl
Edit on GitHub

Derivative of B-spline

Thm. Derivative of B-spline basis function

The derivative of B-spline basis function can be expressed as follows:

\[\begin{aligned} +Derivative · BasicBSpline.jl

GitHub

Derivative of B-spline

Thm. Derivative of B-spline basis function

The derivative of B-spline basis function can be expressed as follows:

\[\begin{aligned} \dot{B}_{(i,p,k)}(t) &=\frac{d}{dt}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) @@ -10,7 +10,7 @@ plot(BSplineDerivativeSpace{1}(P), label="1st derivative", color=:red), plot(BSplineDerivativeSpace{2}(P), label="2nd derivative", color=:green), plot(BSplineDerivativeSpace{3}(P), label="3rd derivative", color=:blue), -)

BasicBSpline.BSplineDerivativeSpace — Type
BSplineDerivativeSpace{r}(P::BSplineSpace)

Construct derivative of B-spline space from given differential order and B-spline space.

\[D^{r}(\mathcal{P}[p,k]) +)

BasicBSpline.BSplineDerivativeSpace — Type
BSplineDerivativeSpace{r}(P::BSplineSpace)

Construct derivative of B-spline space from given differential order and B-spline space.

\[D^{r}(\mathcal{P}[p,k]) =\left\{t \mapsto \left. \frac{d^r f}{dt^r}(t) \ \right| \ f \in \mathcal{P}[p,k] \right\}\]

Examples

julia> P = BSplineSpace{2}(KnotVector([1,2,3,4,5,6]))
 BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6]))
 
@@ -18,7 +18,7 @@
 BSplineDerivativeSpace{1, BSplineSpace{2, Int64, KnotVector{Int64}}, Int64}(BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6])))
 
 julia> degree(P), degree(dP)
-(2, 1)
source
BasicBSpline.derivative — Function
derivative(::BSplineDerivativeSpace{r}) -> BSplineDerivativeSpace{r+1}
 derivative(::BSplineSpace) -> BSplineDerivativeSpace{1}

Derivative of B-spline related space.

Examples

julia> BSplineSpace{2}(KnotVector(0:5))
 BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 1, 2, 3, 4, 5]))
 
@@ -26,7 +26,7 @@
 BSplineDerivativeSpace{1, BSplineSpace{2, Int64, KnotVector{Int64}}, Int64}(BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 1, 2, 3, 4, 5])))
 
 julia> BasicBSpline.derivative(ans)
-BSplineDerivativeSpace{2, BSplineSpace{2, Int64, KnotVector{Int64}}, Int64}(BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 1, 2, 3, 4, 5])))
source
BasicBSpline.bsplinebasis′₊₀ — Function
bsplinebasis′₊₀(::AbstractFunctionSpace, ::Integer, ::Real) -> Real

1st derivative of B-spline basis function. Right-sided limit version.

\[\dot{B}_{(i,p,k)}(t) -=p\left(\frac{1}{k_{i+p}-k_{i}}B_{(i,p-1,k)}(t)-\frac{1}{k_{i+p+1}-k_{i+1}}B_{(i+1,p-1,k)}(t)\right)\]

bsplinebasis′₊₀(P, i, t) is equivalent to bsplinebasis₊₀(derivative(P), i, t).

source
BasicBSpline.bsplinebasis′₋₀ — Function
bsplinebasis′₋₀(::AbstractFunctionSpace, ::Integer, ::Real) -> Real

1st derivative of B-spline basis function. Left-sided limit version.

\[\dot{B}_{(i,p,k)}(t) -=p\left(\frac{1}{k_{i+p}-k_{i}}B_{(i,p-1,k)}(t)-\frac{1}{k_{i+p+1}-k_{i+1}}B_{(i+1,p-1,k)}(t)\right)\]

bsplinebasis′₋₀(P, i, t) is equivalent to bsplinebasis₋₀(derivative(P), i, t).

source
BasicBSpline.bsplinebasis′ — Function
bsplinebasis′(::AbstractFunctionSpace, ::Integer, ::Real) -> Real

1st derivative of B-spline basis function. Modified version.

\[\dot{B}_{(i,p,k)}(t) -=p\left(\frac{1}{k_{i+p}-k_{i}}B_{(i,p-1,k)}(t)-\frac{1}{k_{i+p+1}-k_{i+1}}B_{(i+1,p-1,k)}(t)\right)\]

bsplinebasis′(P, i, t) is equivalent to bsplinebasis(derivative(P), i, t).

source
+BSplineDerivativeSpace{2, BSplineSpace{2, Int64, KnotVector{Int64}}, Int64}(BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 1, 2, 3, 4, 5])))
source
BasicBSpline.bsplinebasis′₊₀ — Function
bsplinebasis′₊₀(::AbstractFunctionSpace, ::Integer, ::Real) -> Real

1st derivative of B-spline basis function. Right-sided limit version.

\[\dot{B}_{(i,p,k)}(t) +=p\left(\frac{1}{k_{i+p}-k_{i}}B_{(i,p-1,k)}(t)-\frac{1}{k_{i+p+1}-k_{i+1}}B_{(i+1,p-1,k)}(t)\right)\]

bsplinebasis′₊₀(P, i, t) is equivalent to bsplinebasis₊₀(derivative(P), i, t).

source
BasicBSpline.bsplinebasis′₋₀ — Function
bsplinebasis′₋₀(::AbstractFunctionSpace, ::Integer, ::Real) -> Real

1st derivative of B-spline basis function. Left-sided limit version.

\[\dot{B}_{(i,p,k)}(t) +=p\left(\frac{1}{k_{i+p}-k_{i}}B_{(i,p-1,k)}(t)-\frac{1}{k_{i+p+1}-k_{i+1}}B_{(i+1,p-1,k)}(t)\right)\]

bsplinebasis′₋₀(P, i, t) is equivalent to bsplinebasis₋₀(derivative(P), i, t).

source
BasicBSpline.bsplinebasis′ — Function
bsplinebasis′(::AbstractFunctionSpace, ::Integer, ::Real) -> Real

1st derivative of B-spline basis function. Modified version.

\[\dot{B}_{(i,p,k)}(t) +=p\left(\frac{1}{k_{i+p}-k_{i}}B_{(i,p-1,k)}(t)-\frac{1}{k_{i+p+1}-k_{i+1}}B_{(i+1,p-1,k)}(t)\right)\]

bsplinebasis′(P, i, t) is equivalent to bsplinebasis(derivative(P), i, t).

source
diff --git a/dev/math-fitting/index.html b/dev/math-fitting/index.html index 73ceddcb0..3131c34fb 100644 --- a/dev/math-fitting/index.html +++ b/dev/math-fitting/index.html @@ -1,5 +1,5 @@ -Fitting · BasicBSpline.jl
Edit on GitHub

Fitting with B-spline manifold

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.

Fitting with least squares method.

BasicBSplineFitting.fittingcontrolpoints — Function

Fitting controlpoints with least squares method.

fittingcontrolpoints(func, Ps::Tuple)

This function will calculate $\bm{a}_i$ to minimize the following integral.

\[\int_I \left\|f(t)-\sum_i B_{(i,p,k)}(t) \bm{a}_i\right\|^2 dt\]

Similarly, for the two-dimensional case, minimize the following integral.

\[\int_{I^1 \times I^2} \left\|f(t^1, t^2)-\sum_{i,j} B_{(i,p^1,k^1)}(t^1)B_{(j,p^2,k^2)}(t^2) \bm{a}_{ij}\right\|^2 dt^1dt^2\]

Currently, this function supports up to three dimensions.

Examples

julia> f(t) = SVector(cos(t),sin(t),t);
+Fitting · BasicBSpline.jl
GitHub
Fitting with B-spline manifold
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.
Fitting with least squares method.
BasicBSplineFitting.fittingcontrolpoints — Function
Fitting controlpoints with least squares method.
fittingcontrolpoints(func, Ps::Tuple)
This function will calculate $\bm{a}_i$ to minimize the following integral.
\[\int_I \left\|f(t)-\sum_i B_{(i,p,k)}(t) \bm{a}_i\right\|^2 dt\]
Similarly, for the two-dimensional case, minimize the following integral.
\[\int_{I^1 \times I^2} \left\|f(t^1, t^2)-\sum_{i,j} B_{(i,p^1,k^1)}(t^1)B_{(j,p^2,k^2)}(t^2) \bm{a}_{ij}\right\|^2 dt^1dt^2\]
Currently, this function supports up to three dimensions.
Examples
julia> f(t) = SVector(cos(t),sin(t),t);
 
 julia> P = BSplineSpace{3}(KnotVector(range(0,2π,30)) + 3*KnotVector([0,2π]));
 
@@ -8,7 +8,7 @@
 julia> M = BSplineManifold(a, P);
 
 julia> norm(M(1) - f(1)) < 1e-5
-true
p1 = 2
+true
source
p1 = 2
 p2 = 2
 k1 = KnotVector(-10:10)+p1*KnotVector([-10,10])
 k2 = KnotVector(-10:10)+p2*KnotVector([-10,10])
@@ -19,4 +19,4 @@
 
 a = fittingcontrolpoints(f, (P1, P2))
 M = BSplineManifold(a, (P1, P2))
-save_png("fitting.png", M, unitlength=50, xlims=(-10,10), ylims=(-10,10))

Try on Desmos graphing graphing calculator!

+save_png("fitting.png", M, unitlength=50, xlims=(-10,10), ylims=(-10,10))

Try on Desmos graphing graphing calculator!

diff --git a/dev/math-inclusive/index.html b/dev/math-inclusive/index.html index 57a5b529b..8544c5f09 100644 --- a/dev/math-inclusive/index.html +++ b/dev/math-inclusive/index.html @@ -1,13 +1,16 @@ -Inclusive relationship · BasicBSpline.jl
Edit on GitHub

Inclusive relation between B-spline spaces

Thm. Inclusive relation between B-spline spaces

For non-degenerate B-spline spaces, the following relationship holds.

\[\mathcal{P}[p,k] +Inclusive relationship · BasicBSpline.jl

GitHub

Inclusive relation between B-spline spaces

Thm. Inclusive relation between B-spline spaces

For non-degenerate B-spline spaces, the following relationship holds.

\[\mathcal{P}[p,k] \subseteq \mathcal{P}[p',k'] -\Leftrightarrow (m=p'-p \ge 0 \ \text{and} \ k+m\widehat{k}\subseteq k')\]

Base.issubset — Method

Check inclusive relationship between B-spline spaces.

\[\mathcal{P}[p,k] +\Leftrightarrow (m=p'-p \ge 0 \ \text{and} \ k+m\widehat{k}\subseteq k')\]

Base.issubset — Method

Check inclusive relationship between B-spline spaces.

\[\mathcal{P}[p,k] \subseteq\mathcal{P}[p',k']\]

Examples

julia> P1 = BSplineSpace{1}(KnotVector([1,3,5,8]));
 
+
 julia> P2 = BSplineSpace{1}(KnotVector([1,3,5,6,8,9]));
 
+
 julia> P3 = BSplineSpace{2}(KnotVector([1,1,3,3,5,5,8,8]));
 
+
 julia> P1 ⊆ P2
 true
 
@@ -18,7 +21,7 @@
 false
 
 julia> P2 ⊈ P3
-true
source

Here are plots of the B-spline basis functions of the spaces P1, P2, P3.

julia> P1 = BSplineSpace{1}(KnotVector([1,3,5,8]))BSplineSpace{1, Int64, KnotVector{Int64}}(KnotVector([1, 3, 5, 8]))
julia> P2 = BSplineSpace{1}(KnotVector([1,3,5,6,8,9]))BSplineSpace{1, Int64, KnotVector{Int64}}(KnotVector([1, 3, 5, 6, 8, 9]))
julia> P3 = BSplineSpace{2}(KnotVector([1,1,3,3,5,5,8,8]))BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 1, 3, 3, 5, 5, 8, 8]))
julia> plot(
+true
source

Here are plots of the B-spline basis functions of the spaces P1, P2, P3.

julia> P1 = BSplineSpace{1}(KnotVector([1,3,5,8]))BSplineSpace{1, Int64, KnotVector{Int64}}(KnotVector([1, 3, 5, 8]))
julia> P2 = BSplineSpace{1}(KnotVector([1,3,5,6,8,9]))BSplineSpace{1, Int64, KnotVector{Int64}}(KnotVector([1, 3, 5, 6, 8, 9]))
julia> P3 = BSplineSpace{2}(KnotVector([1,1,3,3,5,5,8,8]))BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([1, 1, 3, 3, 5, 5, 8, 8]))
julia> plot(
            plot([t->bsplinebasis₊₀(P1,i,t) for i in 1:dim(P1)], 1, 9, ylims=(0,1), legend=false),
            plot([t->bsplinebasis₊₀(P2,i,t) for i in 1:dim(P2)], 1, 9, ylims=(0,1), legend=false),
            plot([t->bsplinebasis₊₀(P3,i,t) for i in 1:dim(P3)], 1, 9, ylims=(0,1), legend=false),
@@ -39,14 +42,16 @@
            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),
            layout=(3,1),
            link=:x
-       )Plot{Plots.PlotlyBackend() n=6}
BasicBSpline.issqsubset — Function

Check inclusive relationship between B-spline spaces.

\[\mathcal{P}[p,k] + )Plot{Plots.PlotlyBackend() n=6}

BasicBSpline.changebasis_R — Function

Return a coefficient matrix $A$ which satisfy

\[B_{(i,p,k)} = \sum_{j}A_{i,j}B_{(j,p',k')}\]

Assumption:

  • $P ⊆ P^{\prime}$
source
BasicBSpline.changebasis_I — Function

Return a coefficient matrix $A$ which satisfy

\[B_{(i,p,k)} = \sum_{j}A_{i,j}B_{(j,p',k')}\]

Assumption:

  • $P ⊑ P^{\prime}$
source
BasicBSpline.issqsubset — Function

Check inclusive relationship between B-spline spaces.

\[\mathcal{P}[p,k] \sqsubseteq\mathcal{P}[p',k'] \Leftrightarrow \mathcal{P}[p,k]|_{[k_{p+1},k_{l-p}]} -\subseteq\mathcal{P}[p',k']|_{[k'_{p'+1},k'_{l'-p'}]}\]

source
BasicBSpline.expandspace — Function

Expand B-spline space with given additional degree and knotvector. The behavior of expandspace is same as expandspace_I.

source
BasicBSpline.expandspace_R — Function

Expand B-spline space with given additional degree and knotvector. This function is compatible with issubset (⊆).

Examples

julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);
+\subseteq\mathcal{P}[p',k']|_{[k'_{p'+1},k'_{l'-p'}]}\]
source
BasicBSpline.expandspace — Function

Expand B-spline space with given additional degree and knotvector. The behavior of expandspace is same as expandspace_I.

source
BasicBSpline.expandspace_R — Function

Expand B-spline space with given additional degree and knotvector. This function is compatible with issubset (⊆).

Examples

julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);
+
 
 julia> P = BSplineSpace{2}(k);
 
+
 julia> 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]))
 
@@ -57,13 +62,15 @@
 false
 
 julia> domain(P)
-2.5..9.0
+2.5 .. 9.0
 
 julia> domain(P′)
-1.5..9.5
source
BasicBSpline.expandspace_I — Function

Expand B-spline space with given additional degree and knotvector. This function is compatible with issqsubset (⊑)

Examples

julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);
+1.5 .. 9.5
source
BasicBSpline.expandspace_I — Function

Expand B-spline space with given additional degree and knotvector. This function is compatible with issqsubset (⊑)

Examples

julia> k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]);
+
 
 julia> P = BSplineSpace{2}(k);
 
+
 julia> 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]))
 
@@ -74,7 +81,7 @@
 true
 
 julia> domain(P)
-2.5..9.0
+2.5 .. 9.0
 
 julia> domain(P′)
-2.5..9.0
source
+2.5 .. 9.0source diff --git a/dev/math-knotvector/index.html b/dev/math-knotvector/index.html index f7baf6232..37f19ec89 100644 --- a/dev/math-knotvector/index.html +++ b/dev/math-knotvector/index.html @@ -1,37 +1,45 @@ -Knot vector · BasicBSpline.jl
Edit on GitHub

Knot vector

Definition

Def. Knot vector

A finite sequence

\[k = (k_1, \dots, k_l)\]

is called knot vector if the sequence is broad monotonic increase, i.e. $k_{i} \le k_{i+1}$.

[TODO: fig]

BasicBSpline.KnotVector — Type

Construct knot vector from given array.

\[k=(k_1,\dots,k_l)\]

Examples

julia> k = KnotVector([1,2,3])
+Knot vector · BasicBSpline.jl
GitHub
Knot vector
Definition
Def.  Knot vector
A finite sequence
\[k = (k_1, \dots, k_l)\]
is called knot vector if the sequence is broad monotonic increase, i.e. $k_{i} \le k_{i+1}$.
[TODO: fig]
BasicBSpline.KnotVector — Type
Construct knot vector from given array.
\[k=(k_1,\dots,k_l)\]
Examples
julia> k = KnotVector([1,2,3])
 KnotVector([1, 2, 3])
 
 julia> k = KnotVector(1:3)
-KnotVector([1, 2, 3])
source
BasicBSpline.UniformKnotVector — Type
Construct uniform knot vector from given range.
\[k=(k_1,\dots,k_l)\]
Examples
julia> k = UniformKnotVector(1:8)
 UniformKnotVector(1:8)
 
 julia> UniformKnotVector(8:-1:3)
-UniformKnotVector(3:1:8)
source
BasicBSpline.EmptyKnotVector — Type
Knot vector with zero-element.
\[k=()\]
This struct is intended for internal use.
Examples
julia> EmptyKnotVector()
+UniformKnotVector(3:1:8)
source
BasicBSpline.SubKnotVector — Type
A type to represetnt sub knot vector.
\[k=(k_1,\dots,k_l)\]
Examples
julia> k = knotvector"1 11 211"
+KnotVector([1, 3, 4, 6, 6, 7, 8])
+
+julia> view(k, 2:5)
+SubKnotVector([3, 4, 6, 6])
source
BasicBSpline.EmptyKnotVector — Type
Knot vector with zero-element.
\[k=()\]
This struct is intended for internal use.
Examples
julia> EmptyKnotVector()
 EmptyKnotVector{Bool}()
 
 julia> EmptyKnotVector{Float64}()
-EmptyKnotVector{Float64}()
source
Operations for knot vectors
Base.length — Method
Length of knot vector
\[\begin{aligned}
+EmptyKnotVector{Float64}()
source

Operations for knot vectors

Base.length — Method

Length of knot vector

\[\begin{aligned} \#{k} &=(\text{number of knot elements of} \ k) \\ \end{aligned}\]

For example, $\#{(1,2,2,3)}=4$.

Examples

julia> k = KnotVector([1,2,2,3]);
 
+
 julia> length(k)
-4
source

Although a knot vector is not a vector in linear algebra, but we introduce additional operator $+$.

Although a knot vector is not a vector in linear algebra, but we introduce additional operator $+$.

Base.:+ — Method

Sum of knot vectors

\[\begin{aligned} k^{(1)}+k^{(2)} &=(k^{(1)}_1, \dots, k^{(1)}_{l^{(1)}}) + (k^{(2)}_1, \dots, k^{(2)}_{l^{(2)}}) \\ &=(\text{sort of union of} \ k^{(1)} \ \text{and} \ k^{(2)} \text{)} \end{aligned}\]

For example, $(1,2,3,5)+(4,5,8)=(1,2,3,4,5,5,8)$.

Examples

julia> k1 = KnotVector([1,2,3,5]);
 
+
 julia> k2 = KnotVector([4,5,8]);
 
+
 julia> k1 + k2
-KnotVector([1, 2, 3, 4, 5, 5, 8])
source

Note that the operator +(::KnotVector, ::KnotVector) is commutative. This is why we choose the $+$ sign. We also introduce product operator $\cdot$ for knot vector.

Base.:* — Method

Product of integer and knot vector

\[\begin{aligned} +KnotVector([1, 2, 3, 4, 5, 5, 8])

source

Note that the operator +(::KnotVector, ::KnotVector) is commutative. This is why we choose the $+$ sign. We also introduce product operator $\cdot$ for knot vector.

Base.:* — Method

Product of integer and knot vector

\[\begin{aligned} m\cdot k&=\underbrace{k+\cdots+k}_{m} \end{aligned}\]

For example, $2\cdot (1,2,2,5)=(1,1,2,2,2,2,5,5)$.

Examples

julia> k = KnotVector([1,2,2,5]);
 
+
 julia> 2 * k
-KnotVector([1, 1, 2, 2, 2, 2, 5, 5])
source

Inclusive relationship between knot vectors.

Base.issubset — Method

Check a inclusive relationship $k\subseteq k'$, for example:

\[\begin{aligned} +KnotVector([1, 1, 2, 2, 2, 2, 5, 5])

source

Inclusive relationship between knot vectors.

Base.issubset — Method

Check a inclusive relationship $k\subseteq k'$, for example:

\[\begin{aligned} (1,2) &\subseteq (1,2,3) \\ (1,2,2) &\not\subseteq (1,2,3) \\ (1,2,3) &\subseteq (1,2,3) \\ @@ -42,13 +50,15 @@ false julia> KnotVector([1,2,3]) ⊆ KnotVector([1,2,3]) -true

source
Base.unique — Method

Unique elements of knot vector.

\[\begin{aligned} \widehat{k} &=(\text{unique knot elements of} \ k) \\ \end{aligned}\]

For example, $\widehat{(1,2,2,3)}=(1,2,3)$.

Examples

julia> k = KnotVector([1,2,2,3]);
 
+
 julia> unique(k)
-KnotVector([1, 2, 3])
source
BasicBSpline.countknots — Method

For given knot vector $k$, the following function $\mathfrak{n}_k:\mathbb{R}\to\mathbb{Z}$ represents the number of knots that duplicate the knot vector $k$.

\[\mathfrak{n}_k(t) = \#\{i \mid k_i=t \}\]

For example, if $k=(1,2,2,3)$, then $\mathfrak{n}_k(0.3)=0$, $\mathfrak{n}_k(1)=1$, $\mathfrak{n}_k(2)=2$.

julia> k = KnotVector([1,2,2,3]);
+KnotVector([1, 2, 3])
source
BasicBSpline.countknots — Method

For given knot vector $k$, the following function $\mathfrak{n}_k:\mathbb{R}\to\mathbb{Z}$ represents the number of knots that duplicate the knot vector $k$.

\[\mathfrak{n}_k(t) = \#\{i \mid k_i=t \}\]

For example, if $k=(1,2,2,3)$, then $\mathfrak{n}_k(0.3)=0$, $\mathfrak{n}_k(1)=1$, $\mathfrak{n}_k(2)=2$.

julia> k = KnotVector([1,2,2,3]);
+
 
 julia> countknots(k,0.3)
 0
@@ -57,4 +67,14 @@
 1
 
 julia> countknots(k,2.0)
-2
source
+2source

KnotVector with string macro

BasicBSpline.@knotvector_str — Macro
@knotvector_str -> KnotVector

Construct a knotvector by specifying the numbers of duplicates of knots.

Examples

julia> knotvector"11111"
+KnotVector([1, 2, 3, 4, 5])
+
+julia> knotvector"123"
+KnotVector([1, 2, 2, 3, 3, 3])
+
+julia> knotvector" 2 2 2"
+KnotVector([2, 2, 4, 4, 6, 6])
+
+julia> knotvector"     1"
+KnotVector([6])
source
diff --git a/dev/math-rationalbsplinemanifold/index.html b/dev/math-rationalbsplinemanifold/index.html index 65b3349ce..7b600ae5f 100644 --- a/dev/math-rationalbsplinemanifold/index.html +++ b/dev/math-rationalbsplinemanifold/index.html @@ -1,9 +1,9 @@ -Rational B-spline manifold · BasicBSpline.jl
Edit on GitHub

Rational B-spline manifold

Non-uniform rational basis spline (NURBS) is also supported in BasicBSpline.jl package.

Rational B-spline manifold

Rational B-spline manifold is a parametric representation of a shape.

Def. Rational B-spline manifold

For given $d$-dimensional B-spline basis functions $B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)}$, given points $\bm{a}_{i^1 \dots i^d} \in V$ and real numbers $w_{i^1 \dots i^d} > 0$, rational B-spline manifold is defined by the following equality:

\[\bm{p}(t^1,\dots,t^d; \bm{a}_{i^1 \dots i^d}, w_{i^1 \dots i^d}) +Rational B-spline manifold · BasicBSpline.jl

GitHub

Rational B-spline manifold

Non-uniform rational basis spline (NURBS) is also supported in BasicBSpline.jl package.

Rational B-spline manifold

Rational B-spline manifold is a parametric representation of a shape.

Def. Rational B-spline manifold

For given $d$-dimensional B-spline basis functions $B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)}$, given points $\bm{a}_{i^1 \dots i^d} \in V$ and real numbers $w_{i^1 \dots i^d} > 0$, rational B-spline manifold is defined by the following equality:

\[\bm{p}(t^1,\dots,t^d; \bm{a}_{i^1 \dots i^d}, w_{i^1 \dots i^d}) =\sum_{i^1,\dots,i^d} \frac{(B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)})(t^1,\dots,t^d) w_{i^1 \dots i^d}} {\sum\limits_{j^1,\dots,j^d}(B_{(j^1,p^1,k^1)} \otimes \cdots \otimes B_{(j^d,p^d,k^d)})(t^1,\dots,t^d) w_{j^1 \dots j^d}} -\bm{a}_{i^1 \dots i^d}\]

Where $\bm{a}_{i^1,\dots,i^d}$ are called control points, and $w_{i^1 \dots i^d}$ are called weights.

BasicBSpline.RationalBSplineManifold — Type

Construct Rational B-spline manifold from given control points, weights and B-spline spaces.

Examples

julia> using StaticArrays, LinearAlgebra
+\bm{a}_{i^1 \dots i^d}\]
Where $\bm{a}_{i^1,\dots,i^d}$ are called control points, and $w_{i^1 \dots i^d}$ are called weights.
BasicBSpline.RationalBSplineManifold — Type

Construct Rational B-spline manifold from given control points, weights and B-spline spaces.

Examples

julia> using StaticArrays, LinearAlgebra
 
 julia> P = BSplineSpace{2}(KnotVector([0,0,0,1,1,1]))
 BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([0, 0, 0, 1, 1, 1]))
@@ -22,10 +22,11 @@
 
 julia> M = RationalBSplineManifold(a,w,P);  # 1/4 arc
 
+
 julia> M(0.3)
 2-element SVector{2, Float64} with indices SOneTo(2):
  0.8973756499953727
  0.4412674277525845
 
 julia> norm(M(0.3))
-1.0
source

Properties

Similar to BSplineManifold, RationalBSplineManifold supports the following methods and properties.

  • currying
  • refinement
  • Affine commutativity
+1.0source

Properties

Similar to BSplineManifold, RationalBSplineManifold supports the following methods and properties.

  • currying
  • refinement
  • Affine commutativity
diff --git a/dev/math-refinement/index.html b/dev/math-refinement/index.html index 7e88c5a0f..a4fe60525 100644 --- a/dev/math-refinement/index.html +++ b/dev/math-refinement/index.html @@ -1,7 +1,7 @@ -Refinement · BasicBSpline.jl
Edit on GitHub

Refinement

Documentation

Example

Define original manifold

p = 2 # degree of polynomial
+Refinement · BasicBSpline.jl
GitHub
Refinement
Documentation
Example
Define original manifold
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
h-refinement
Insert additional knots to knot vector.
julia> k₊ = (KnotVector([3.3,4.2]),KnotVector([3.8,3.2,5.3])) # additional knot vectors(KnotVector([3.3, 4.2]), KnotVector([3.2, 3.8, 5.3]))
julia> M_h = refinement(M, k₊) # refinement of B-spline manifoldBSplineManifold{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]])
julia> save_png("2dim_h-refinement.png", M_h) # save image
Note that this shape and the last shape are equivalent.
p-refinement
Increase the polynomial degree of B-spline manifold.
julia> p₊ = (Val(1), Val(2)) # additional degrees(Val{1}(), Val{2}())
julia> M_p = refinement(M, p₊) # refinement of B-spline manifoldBSplineManifold{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, -3.3810628823825324] [-3.2286534014564956, -2.3898453165277362] … [-3.857369028126979, 2.625613347418197] [-3.750274329305369, 3.444604754180493]; [-1.9190826728891883, -3.6996476508119316] [-1.8954644502285198, -2.596850938310199] … [-2.320089840437407, 2.6673570013295036] [-2.4013083859231266, 3.4176156450587825]; … ; [2.63663136518713, -3.36586722890932] [2.6119919237133846, -2.773961995558151] … [2.5855688772510934, 2.690164285500753] [2.692907338976954, 3.352168201374152]; [3.721273556424996, -3.4913984254174055] [3.554093269596626, -2.7596655006099793] … [3.996017425914788, 2.6027151956081527] [4.087903053338675, 3.724832581624184]])
julia> save_png("2dim_p-refinement.png", M_p) # save image
Note that this shape and the last shape are equivalent.

+M = BSplineManifold(a,(P,P)) # Define B-spline manifold

h-refinement

Insert additional knots to knot vector.

julia> k₊ = (KnotVector([3.3,4.2]),KnotVector([3.8,3.2,5.3])) # additional knot vectors(KnotVector([3.3, 4.2]), KnotVector([3.2, 3.8, 5.3]))
julia> M_h = refinement(M, k₊) # refinement of B-spline manifoldBSplineManifold{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]])
julia> save_png("2dim_h-refinement.png", M_h) # save image

Note that this shape and the last shape are equivalent.

p-refinement

Increase the polynomial degree of B-spline manifold.

julia> p₊ = (Val(1), Val(2)) # additional degrees(Val{1}(), Val{2}())
julia> M_p = refinement(M, p₊) # refinement of B-spline manifoldBSplineManifold{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]])
julia> save_png("2dim_p-refinement.png", M_p) # save image

Note that this shape and the last shape are equivalent.

diff --git a/dev/math/index.html b/dev/math/index.html index 8228927fc..af67c4d1d 100644 --- a/dev/math/index.html +++ b/dev/math/index.html @@ -1,3 +1,3 @@ -Introduction · BasicBSpline.jl
Edit on GitHub

Mathematical properties of B-spline

Introduction

B-spline is a mathematical object, and it has a lot of application. (e.g. Geometric representation: NURBS, Interpolation, Numerical analysis: IGA)

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:

using BasicBSpline
-using Plots; plotly()
Plots.PlotlyBackend()

Notice

Some of notations in this page are our original, but these are well-considered results.

References

Most of this documentation around B-spline is self-contained. If you want to learn more, the following resources are recommended.

日本語の文献では以下がおすすめです。

+Introduction · BasicBSpline.jl
GitHub

Mathematical properties of B-spline

Introduction

B-spline is a mathematical object, and it has a lot of application. (e.g. Geometric representation: NURBS, Interpolation, Numerical analysis: IGA)

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:

using BasicBSpline
+using Plots; plotly()
Plots.PlotlyBackend()

Notice

Some of notations in this page are our original, but these are well-considered results.

References

Most of this documentation around B-spline is self-contained. If you want to learn more, the following resources are recommended.

日本語の文献では以下がおすすめです。

diff --git a/dev/plotlyjs/index.html b/dev/plotlyjs/index.html index e0046d609..b1d7635d3 100644 --- a/dev/plotlyjs/index.html +++ b/dev/plotlyjs/index.html @@ -1,5 +1,5 @@ -PlotlyJS.jl · BasicBSpline.jl
Edit on GitHub

PlotlyJS.jl

Cardioid

using BasicBSpline
+PlotlyJS.jl · BasicBSpline.jl
GitHub
PlotlyJS.jl
Cardioid
using BasicBSpline
 using BasicBSplineFitting
 using StaticArrays
 using PlotlyJS
@@ -37,4 +37,4 @@
 zs_f = getindex.(M.(ts),3)
 fig = Plot(scatter3d(x=xs_a, y=ys_a, z=zs_a, name="control points", line_color="blue", marker_size=8))
 addtraces!(fig, scatter3d(x=xs_f, y=ys_f, z=zs_f, name="B-spline curve", mode="lines", line_color="red"))
-relayout!(fig, width=500, height=500)

+relayout!(fig, width=500, height=500)
diff --git a/dev/plots-arc.html b/dev/plots-arc.html index 5094e08b6..89fd2078b 100644 --- a/dev/plots-arc.html +++ b/dev/plots-arc.html @@ -1,7 +1,7 @@ -
+
+
+
+
+
+
+
Edit on GitHub

Plots.jl

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.

BSplineSpace

k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
+Plots.jl · BasicBSpline.jl
GitHub
Plots.jl
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.
BSplineSpace
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
@@ -58,4 +58,4 @@
 ts = 0:0.01:2
 plot(cospi.(ts),sinpi.(ts), label="circle")
 plot!(M, label="B-spline curve")
-plot!(R, label="Rational B-spline curve")

+plot!(R, label="Rational B-spline curve")
diff --git a/dev/search/index.html b/dev/search/index.html deleted file mode 100644 index 734b1ad94..000000000 --- a/dev/search/index.html +++ /dev/null @@ -1,2 +0,0 @@ - -Search · BasicBSpline.jl

Loading search...

    diff --git a/dev/search_index.js b/dev/search_index.js index afd360df0..e9b2db535 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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 tk_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/#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\njulia> P = BSplineSpace{2}(k);\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":"","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"","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":"","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":"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\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\njulia> k2 = KnotVector([4,5,8]);\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\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\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\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":"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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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_R","category":"page"},{"location":"internal/","page":"Private API","title":"Private API","text":"BasicBSpline._changebasis_I","category":"page"},{"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\njulia> P2 = BSplineSpace{1}(KnotVector([1,3,5,6,8,9]));\n\njulia> P3 = BSplineSpace{2}(KnotVector([1,1,3,3,5,5,8,8]));\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":"","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":"","category":"page"},{"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\njulia> P = BSplineSpace{2}(k);\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\njulia> P = BSplineSpace{2}(k);\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\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","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\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","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":"","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":"","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":"","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-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\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":"","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":"","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":"","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":"","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":"","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"}] +[{"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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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 tk_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_i le tk_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> 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":"","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"","category":"page"},{"location":"math-bsplinebasis/","page":"B-spline basis function","title":"B-spline basis function","text":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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":"","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"}] } diff --git a/dev/subbsplineplot.html b/dev/subbsplineplot.html index c7d37f26e..8f759ac22 100644 --- a/dev/subbsplineplot.html +++ b/dev/subbsplineplot.html @@ -1,7 +1,7 @@ -
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    Edit on GitHub

    Geometric modeling

    Load packages

    using BasicBSpline
+Geometric modeling · BasicBSpline.jl
    GitHub
    Geometric modeling
    Load packages
    using BasicBSpline
 using StaticArrays
 using Plots
 using LinearAlgebra
@@ -60,4 +60,4 @@
 M = BSplineManifold(a,P,P)
 plot(M)
    Hyperbolic paraboloid
    a = [SVector(i,j,2i^2-2j^2) for i in -1:1, j in -1:1]
 M = BSplineManifold(a,P,P)
-plot(M)
    
+plot(M)