Update docs

docs/make.jl

@@ -59,10 +59,10 @@ makedocs(;
             "B-spline space" => "math/bsplinespace.md",
             "B-spline basis function" => "math/bsplinebasis.md",
             "B-spline manifold" => "math/bsplinemanifold.md",
-            "Derivative" => "math/derivative.md",
+            "Rational B-spline manifold" => "math/rationalbsplinemanifold.md",
             "Inclusive relationship" => "math/inclusive.md",
             "Refinement" => "math/refinement.md",
-            "Rational B-spline manifold" => "math/rationalbsplinemanifold.md",
+            "Derivative" => "math/derivative.md",
             "Fitting" => "math/fitting.md",
         ],
         # "Differentiation" => [

docs/src/api.md

@@ -29,6 +29,7 @@ isnondegenerate
 isdegenerate(P::BSplineSpace)
 BSplineManifold
 unbounded_mapping
+RationalBSplineManifold
 ```
 
 ```@docs

docs/src/math/bsplinemanifold.md

@@ -53,7 +53,7 @@ nothing # hide
 
 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
+## Definition
 B-spline manifold is a parametric representation of a shape.
 
 !!! tip "Def.  B-spline manifold"
@@ -86,7 +86,7 @@ unbounded_mapping(M, 1.2)
 
 `unbounded_mapping(M,t...)` is a little bit faster than `M(t...)` because it does not check the domain.
 
-### B-spline curve
+## B-spline curve
 ```@example math_bsplinemanifold
 ## 1-dim B-spline manifold
 p = 2 # degree of polynomial
@@ -103,7 +103,7 @@ nothing # hide
 <object type="text/html" data="../1dim-manifold.html" style="width:100%;height:420px;"></object>
 ```
 
-### B-spline surface
+## B-spline surface
 ```@example math_bsplinemanifold
 ## 2-dim B-spline manifold
 p = 2 # degree of polynomial
@@ -121,6 +121,58 @@ nothing # hide
 <object type="text/html" data="../2dim-manifold.html" style="width:100%;height:420px;"></object>
 ```
 
+**Paraboloid**
+```@example math_bsplinemanifold
+plotly()
+p = 2
+k = KnotVector([-1,-1,-1,1,1,1])
+P = BSplineSpace{p}(k)
+a = [SVector(i,j,2i^2+2j^2-2) for i in -1:1, j in -1:1]
+M = BSplineManifold(a,P,P)
+plot(M)
+savefig("paraboloid.html") # hide
+nothing # hide
+```
+
+```@raw html
+<object type="text/html" data="../paraboloid.html" style="width:100%;height:420px;"></object>
+```
+
+**Hyperbolic paraboloid**
+```@example math_bsplinemanifold
+plotly()
+a = [SVector(i,j,2i^2-2j^2) for i in -1:1, j in -1:1]
+M = BSplineManifold(a,P,P)
+plot(M)
+savefig("hyperbolicparaboloid.html") # hide
+nothing # hide
+```
+
+```@raw html
+<object type="text/html" data="../hyperbolicparaboloid.html" style="width:100%;height:420px;"></object>
+```
+
+## B-spline solid
+
+```@example math_bsplinemanifold
+k1 = k2 = KnotVector([0,0,1,1])
+k3 = UniformKnotVector(-6:6)
+P1 = BSplineSpace{1}(k1)
+P2 = BSplineSpace{1}(k2)
+P3 = BSplineSpace{3}(k3)
+e₁(t) = SVector(cos(t),sin(t),0)
+e₂(t) = SVector(-sin(t),cos(t),0)
+a = cat([[e₁(t)*i+e₂(t)*j+SVector(0,0,t) for i in 0:1, j in 0:1] for t in -2:0.5:2]..., dims=3)
+M = BSplineManifold(a,(P1,P2,P3))
+plot(M; colorbar=false)
+savefig("bsplinesolid.html") # hide
+nothing # hide
+```
+
+```@raw html
+<object type="text/html" data="../bsplinesolid.html" style="width:100%;height:420px;"></object>
+```
+
 ## Affine commutativity
 !!! info "Thm.  Affine commutativity"
     Let ``T`` be a affine transform ``V \to W``, then the following equality holds.

docs/src/math/bsplinespace.md

@@ -57,7 +57,7 @@ Note that each element of the space ``\mathcal{P}[p,k]`` is a piecewise polynomi
 ## Degeneration
 
 !!! tip "Def.  Degeneration"
-    A B-spline space is said to be **non-degenerate** if its degree and knot vector satisfies following property:
+    A B-spline space **non-degenerate** if its degree and knot vector satisfies following property:
     ```math
     \begin{aligned}
     k_{i}&<k_{i+p+1} & (1 \le i \le l-p-1)

docs/src/math/introduction.md

@@ -3,12 +3,7 @@
 ## Introduction
 [B-spline](https://en.wikipedia.org/wiki/B-spline) is a mathematical object, and it has a lot of application. (e.g. Geometric representation: [NURBS](https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline), Interpolation, Numerical analysis: [IGA](https://en.wikipedia.org/wiki/Isogeometric_analysis))
 
-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:
-```@example
-using BasicBSpline
-using Plots; plotly()
-```
+We will explain the mathematical definitions and properties of B-spline with Julia code in the following pages.
 
 ## Notice
 Some of notations in this page are our original, but these are well-considered results.

docs/src/math/rationalbsplinemanifold.md

@@ -1,15 +1,17 @@
 # Rational B-spline manifold
 
-```@setup math
+## Setup
+
+```@example math_rationalbsplinemanifold
 using BasicBSpline
-using BasicBSplineExporter
 using StaticArrays
+using LinearAlgebra
 using Plots; plotly()
 ```
 
 [Non-uniform rational basis spline (NURBS)](https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline) is also supported in BasicBSpline.jl package.
 
-## Rational B-spline manifold
+## Definition
 Rational B-spline manifold is a parametric representation of a shape.
 
 !!! tip "Def.  Rational B-spline manifold"
@@ -23,13 +25,115 @@ Rational B-spline manifold is a parametric representation of a shape.
     ```
     Where ``\bm{a}_{i^1,\dots,i^d}`` are called **control points**, and ``w_{i^1 \dots i^d}`` are called **weights**.
 
-```@docs
-RationalBSplineManifold
+## Visualization with projection
+
+A rational B-spline manifold in ``\mathbb{R}^d`` can be understanded as a projected B-spline manifold in ``\mathbb{R}^{d+1}``.
+The next visuallzation this projection.
+
+```@example math_rationalbsplinemanifold
+# Define B-spline space
+k = KnotVector([0.0, 1.5, 2.5, 5.0, 5.5, 8.0, 9.0, 9.5, 10.0])
+P = BSplineSpace{3}(k)
+
+# Define geometric parameters
+a2 = [
+    SVector(-0.65, -0.20),
+    SVector(-0.20, +0.65),
+    SVector(-0.05, -0.10),
+    SVector(+0.75, +0.20),
+    SVector(+0.45, -0.65),
+]
+a3 = [SVector(p...,1) for p in a2]
+w = [2.2, 1.3, 1.9, 2.1, 1.5]
+
+# Define (rational) B-spline manifolds
+R2 = RationalBSplineManifold(a2,w,(P,))
+M3 = BSplineManifold(a3.*w,(P))
+R3 = RationalBSplineManifold(a3,w,(P,))
+
+# Plot
+pl2 = plot(R2, xlims=(-1,1), ylims=(-1,1); color=:blue, linewidth=3, aspectratio=1, label=false)
+pl3 = plot(R3; color=:blue, linewidth=3, controlpoints=(markersize=2,), label="Rational B-spline curve")
+plot!(pl3, M3; color=:red, linewidth=3, controlpoints=(markersize=2,), label="B-spline curve")
+for p in a3.*w
+    plot!(pl3, [0,p[1]], [0,p[2]], [0,p[3]], color=:black, linestyle=:dash, label=false)
+end
+for t in range(domain(P), length=51)
+    p = M3(t)
+    plot!(pl3, [0,p[1]], [0,p[2]], [0,p[3]], color=:red, linestyle=:dash, label=false)
+end
+surface!(pl3, [-1,1], [-1,1], ones(2,2); color=:green, colorbar=false, alpha=0.5)
+plot(pl2, pl3; layout=grid(1,2, widths=[0.35, 0.65]), size=(780,500))
+savefig("rational_bspline_manifold_projection.html") # hide
+nothing # hide
 ```
 
-## Properties
-Similar to `BSplineManifold`, `RationalBSplineManifold` supports the following methods and properties.
+```@raw html
+<object type="text/html" data="../rational_bspline_manifold_projection.html" style="width:100%;height:520px;"></object>
+```
+
+## Conic sections
+
+One of the great aspect of rational B-spline manifolds is exact shape representation of circles.
+This is achieved as conic sections.
+
+### Arc
+```@example math_rationalbsplinemanifold
+gr()
+p = 2
+k = KnotVector([0,0,0,1,1,1])
+P = BSplineSpace{p}(k)
+t = 1  # angle in radians
+a = [SVector(1,0), SVector(1,tan(t/2)), SVector(cos(t),sin(t))]
+w = [1,cos(t/2),1]
+M = RationalBSplineManifold(a,w,P)
+plot(M, xlims=(0,1.1), ylims=(0,1.1), aspectratio=1)
+savefig("geometricmodeling-arc.png") # hide
+nothing # hide
+```
 
-* currying
-* `refinement`
-* Affine commutativity
+![](geometricmodeling-arc.png)
+
+### Circle
+```@example math_rationalbsplinemanifold
+gr()
+p = 2
+k = KnotVector([0,0,0,1,1,2,2,3,3,4,4,4])
+P = BSplineSpace{p}(k)
+a = [normalize(SVector(cosd(t), sind(t)), Inf) for t in 0:45:360]
+w = [ifelse(isodd(i), √2, 1) for i in 1:9]
+M = RationalBSplineManifold(a,w,P)
+plot(M, xlims=(-1.2,1.2), ylims=(-1.2,1.2), aspectratio=1)
+savefig("geometricmodeling-circle.png") # hide
+nothing # hide
+```
+
+![](geometricmodeling-circle.png)
+
+### Torus
+```@example math_rationalbsplinemanifold
+plotly()
+R1 = 3
+R2 = 1
+
+A = push.(a, 0)
+
+a1 = (R1+R2)*A
+a5 = (R1-R2)*A
+a2 = [p+R2*SVector(0,0,1) for p in a1]
+a3 = [p+R2*SVector(0,0,1) for p in R1*A]
+a4 = [p+R2*SVector(0,0,1) for p in a5]
+a6 = [p-R2*SVector(0,0,1) for p in a5]
+a7 = [p-R2*SVector(0,0,1) for p in R1*A]
+a8 = [p-R2*SVector(0,0,1) for p in a1]
+
+a = hcat(a1,a2,a3,a4,a5,a6,a7,a8,a1)
+M = RationalBSplineManifold(a,w*w',P,P)
+plot(M)
+savefig("geometricmodeling-torus.html") # hide
+nothing # hide
+```
+
+```@raw html
+<object type="text/html" data="../geometricmodeling-torus.html" style="width:100%;height:420px;"></object>
+```