Update docs (#378)

* update geometric modeling

* update

* update

* update

* update

* update docs around bsplinemanifold

* fix typo

docs/src/api.md

@@ -27,6 +27,8 @@ exactdim_I(P::BSplineSpace)
 BSplineSpace
 isnondegenerate
 isdegenerate(P::BSplineSpace)
+BSplineManifold
+unbounded_mapping
 ```
 
 ```@docs

docs/src/geometricmodeling.md

@@ -1,13 +1,13 @@
 # Geometric modeling
 
-## Load packages
+## Setup
 
 ```@example geometricmodeling
 using BasicBSpline
 using StaticArrays
 using Plots
 using LinearAlgebra
-plotly()
+gr()
 ```
 
 ## Arc
@@ -20,69 +20,45 @@ 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.html") # hide
+savefig("geometricmodeling-arc.png") # hide
 nothing # hide
 ```
 
-```@raw html
-<object type="text/html" data="../geometricmodeling-arc.html" style="width:100%;height:420px;"></object>
-```
+![](geometricmodeling-arc.png)
 
 ## Circle
 ```@example geometricmodeling
 p = 2
 k = KnotVector([0,0,0,1,1,2,2,3,3,4,4,4])
 P = BSplineSpace{p}(k)
-a = [
-    SVector( 1, 0),
-    SVector( 1, 1),
-    SVector( 0, 1),
-    SVector(-1, 1),
-    SVector(-1, 0),
-    SVector(-1,-1),
-    SVector( 0,-1),
-    SVector( 1,-1),
-    SVector( 1, 0)
-]
-w = [1,1/√2,1,1/√2,1,1/√2,1,1/√2,1]
+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.html") # hide
+savefig("geometricmodeling-circle.png") # hide
 nothing # hide
 ```
 
-```@raw html
-<object type="text/html" data="../geometricmodeling-circle.html" style="width:100%;height:420px;"></object>
-```
+![](geometricmodeling-circle.png)
 
 ## Torus
 ```@example geometricmodeling
-R = 3
-r = 1
+plotly()
+R1 = 3
+R2 = 1
 
-a0 = [
-    SVector( 1, 0, 0),
-    SVector( 1, 1, 0),
-    SVector( 0, 1, 0),
-    SVector(-1, 1, 0),
-    SVector(-1, 0, 0),
-    SVector(-1,-1, 0),
-    SVector( 0,-1, 0),
-    SVector( 1,-1, 0),
-    SVector( 1, 0, 0)
-]
+A = push.(a, 0)
 
-a1 = (R+r)*a0
-a5 = (R-r)*a0
-a2 = [p+r*SVector(0,0,1) for p in a1]
-a3 = [p+r*SVector(0,0,1) for p in R*a0]
-a4 = [p+r*SVector(0,0,1) for p in a5]
-a6 = [p-r*SVector(0,0,1) for p in a5]
-a7 = [p-r*SVector(0,0,1) for p in R*a0]
-a8 = [p-r*SVector(0,0,1) for p in a1]
-a9 = a1
+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,a9)
+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
@@ -95,6 +71,7 @@ nothing # hide
 
 ## Paraboloid
 ```@example geometricmodeling
+plotly()
 p = 2
 k = KnotVector([-1,-1,-1,1,1,1])
 P = BSplineSpace{p}(k)
@@ -111,6 +88,7 @@ nothing # hide
 
 ## Hyperbolic paraboloid
 ```@example geometricmodeling
+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)

docs/src/math/bsplinebasis.md

@@ -38,7 +38,7 @@ You can manipulate these plots on [desmos graphing calculator](https://www.desmo
 
 These B-spline basis functions can be calculated with [`bsplinebasis₊₀`](@ref).
 
-```@repl math_bsplinebasis
+```@example math_bsplinebasis
 p = 2
 k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
 P = BSplineSpace{p}(k)
@@ -62,7 +62,7 @@ The first terms can be defined in different ways ([`bsplinebasis₋₀`](@ref)).
 \end{aligned}
 ```
 
-```@repl math_bsplinebasis
+```@example math_bsplinebasis
 p = 2
 k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
 P = BSplineSpace{p}(k)
@@ -94,11 +94,12 @@ In these cases, each B-spline basis function ``B_{(i,2,k)}`` is coninuous, so [`
     \end{aligned}
     ```
 
-```@repl math_bsplinebasis
+```@example math_bsplinebasis
 p = 2
 k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
 P = BSplineSpace{p}(k)
-plot(t->sum(bsplinebasis₊₀(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)
+plot(t->sum(bsplinebasis₊₀(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(-0.1, 1.1), label="sum of bspline basis functions")
+plot!(k, label="knot vector", legend=:inside)
 savefig("sumofbsplineplot.png") # hide
 nothing # hide
 ```
@@ -107,11 +108,12 @@ nothing # hide
 
 To satisfy the partition of unity on whole interval ``[0,10]``, sometimes more knots will be inserted to the endpoints of the interval.
 
-```@repl math_bsplinebasis
+```@example math_bsplinebasis
 p = 2
 k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]) + p * KnotVector([0,10])
 P = BSplineSpace{p}(k)
-plot(t->sum(bsplinebasis₊₀(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)
+plot(t->sum(bsplinebasis₊₀(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(-0.1, 1.1), label="sum of bspline basis functions")
+plot!(k, label="knot vector", legend=:inside)
 savefig("sumofbsplineplot2.png") # hide
 nothing # hide
 ```
@@ -133,11 +135,12 @@ Therefore, to satisfy partition of unity on closed interval ``[k_{p+1}, k_{l-p}]
 \end{aligned}
 ```
 
-```@repl math_bsplinebasis
+```@example math_bsplinebasis
 p = 2
 k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]) + p * KnotVector([0,10])
 P = BSplineSpace{p}(k)
-plot(t->sum(bsplinebasis(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(0,1.1), label=false)
+plot(t->sum(bsplinebasis(P,i,t) for i in 1:dim(P)), 0, 10, ylims=(-0.1, 1.1), label="sum of bspline basis functions")
+plot!(k, label="knot vector", legend=:inside)
 savefig("sumofbsplineplot3.png") # hide
 nothing # hide
 ```
@@ -215,8 +218,8 @@ The [`bsplinebasisall`](@ref) function is much more efficient than evaluating B-
 P = BSplineSpace{2}(KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0]))
 t = 6.3
 plot(P; label="P", ylims=(0,1))
-scatter!(fill(t,3), [bsplinebasis(P,2,t), bsplinebasis(P,3,t), bsplinebasis(P,4,t)]; markershape=:hline, label="bsplinebasis")
-scatter!(fill(t,3), bsplinebasisall(P, 2, t); markershape=:vline, label="bsplinebasisall")
+scatter!(fill(t,3), [bsplinebasis(P,2,t), bsplinebasis(P,3,t), bsplinebasis(P,4,t)]; markershape=:circle, markersize=10, label="bsplinebasis")
+scatter!(fill(t,3), bsplinebasisall(P, 2, t); markershape=:star7, markersize=10, label="bsplinebasisall")
 savefig("bsplinebasis_vs_bsplinebasisall.png") # hide
 nothing # hide
 ```
@@ -247,7 +250,7 @@ for p in 1:3
     plot(P, legend=:topleft, label="B-spline basis (p=1)")
     plot!(t->intervalindex(P,t),0,10, label="Interval index")
     plot!(t->sum(bsplinebasis(P,i,t) for i in 1:dim(P)),0,10, label="Sum of B-spline basis")
-    plot!(k, label="knot vector")
+    plot!(k, label="knot vector", legend=:inside)
     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))
     savefig("bsplinebasisall-$(p).html") # hide
     nothing # hide

docs/src/math/bsplinemanifold.md

@@ -4,6 +4,7 @@
 ```@example math_bsplinemanifold
 using BasicBSpline
 using StaticArrays
+using StaticArrays
 using Plots; plotly()
 ```
 
@@ -24,10 +25,6 @@ using Plots; plotly()
 The next plot shows ``B_{(3,p^1,k^1)} \otimes B_{(4,p^2,k^2)}`` basis function.
 
 ```@example math_bsplinemanifold
-using BasicBSpline
-using Plots
-plotly()
-
 # Define shape
 k1 = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
 k2 = knotvector"31 2  121"
@@ -60,7 +57,7 @@ Higher dimensional tensor products ``\mathcal{P}[p^1,k^1]\otimes\cdots\otimes\ma
 B-spline manifold is a parametric representation of a shape.
 
 !!! tip "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:
+    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 parametrization:
     ```math
     \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}
@@ -69,16 +66,22 @@ B-spline manifold is a parametric representation of a shape.
 
 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`.
+Note that the [`BSplineManifold`](@ref) objects are callable, and the arguments will be checked if it fits in the domain of [`BSplineSpace`](@ref).
 
-```@docs
-BSplineManifold
+```@example math_bsplinemanifold
+P = BSplineSpace{2}(KnotVector([0,0,0,1,1,1]))
+a = [SVector(1,0), SVector(1,1), SVector(0,1)]  # `length(a) == dim(P)`
+M = BSplineManifold(a, P)
+M(0.4)  # Calculate `sum(a[i]*bsplinebasis(P, i, 0.4) for i in 1:dim(P))`
 ```
 
-If you need extension of `BSplineManifold` or don't need the arguments check, you can call `unbounded_mapping`.
+If you need extension of [`BSplineManifold`](@ref) or don't need the arguments check, you can call [`unbounded_mapping`](@ref).
 
-```@docs
-unbounded_mapping
+```@repl math_bsplinemanifold
+M(0.4)
+unbounded_mapping(M, 0.4)
+M(1.2)
+unbounded_mapping(M, 1.2)
 ```
 
 `unbounded_mapping(M,t...)` is a little bit faster than `M(t...)` because it does not check the domain.