bsplinespace.md

B-spline space

Setup

using BasicBSpline
using StaticArrays
using Plots

Defnition

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

!!! tip "Def. Polynomial space" Polynomial space with degree p. math \mathcal{P}[p] =\left\{f:\mathbb{R}\to\mathbb{R}\ ;\ t\mapsto a_0+a_1t^1+\cdots+a_pt^p \ \left| \ % a_i\in \mathbb{R} \right. \right\} This space \mathcal{P}[p] is a (p+1)-dimensional linear space.

Note that \{t\mapsto t^i\}_{0 \le i \le p} is a basis of \mathcal{P}[p], and also the set of Bernstein polynomial \{B_{(i,p)}\}_i is a basis of \mathcal{P}[p].

$$\begin{aligned} B_{(i,p)}(t) &=\binom{p}{i-1}t^{i-1}(1-t)^{p-i+1} &(i=1, \dots, p+1) \end{aligned}$$

Where \binom{p}{i-1} is a binomial coefficient. You can try Bernstein polynomial on desmos graphing calculator!

!!! tip "Def. B-spline space" For given polynomial degree p\ge 0 and knot vector k=(k_1,\dots,k_l), B-spline space \mathcal{P}[p,k] is defined as follows: math \mathcal{P}[p,k] =\left\{f:\mathbb{R}\to\mathbb{R} \ \left| \ % \begin{gathered} \operatorname{supp}(f)\subseteq [k_1, k_l] \\ \exists \tilde{f}\in\mathcal{P}[p], f|_{[k_{i}, k_{i+1})} = \tilde{f}|_{[k_{i}, k_{i+1})} \\ \forall t \in \mathbb{R}, \exists \delta > 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]

Degeneration

!!! tip "Def. Degeneration" 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) \end{aligned}

Dimensions

!!! info "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: math \dim(\mathcal{P}[p,k])=\# k - p -1

P1 = BSplineSpace{1}(KnotVector([1,2,3,4,5,6,7]))
P2 = BSplineSpace{1}(KnotVector([1,2,4,4,4,6,7]))
P3 = BSplineSpace{1}(KnotVector([1,2,3,5,5,5,7]))
dim(P1), exactdim_R(P1), exactdim_I(P1)
dim(P2), exactdim_R(P2), exactdim_I(P2)
dim(P3), exactdim_R(P3), exactdim_I(P3)

Visualization:

gr()
pl1 = plot(P1); plot!(pl1, knotvector(P1))
pl2 = plot(P2); plot!(pl2, knotvector(P2))
pl3 = plot(P3); plot!(pl3, knotvector(P3))
plot(pl1, pl2, pl3; layout=(1,3), legend=false)
savefig("dimension-degeneration.png") # hide
nothing # hide