# 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) \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)
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),
)
BasicBSpline.BSplineDerivativeSpaceType
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]))

julia> dP = BSplineDerivativeSpace{1}(P)
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)
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BasicBSpline.derivativeFunction
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]))

julia> BasicBSpline.derivative(ans)
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′₊₀Function
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′₋₀Function
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).

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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).

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