Derivative of B-spline

using BasicBSpline
using StaticArrays
using Plots

Thm. Derivative of B-spline basis function

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

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

Note that $\dot{B}_{(i,p,k)}\in\mathcal{P}[p-1,k]$.

k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
P = BSplineSpace{3}(k)
plotly()
plot(
    plot(BSplineDerivativeSpace{0}(P), label="0th derivative", color=:black),
    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),
)