B-Spline derivative using de Boor's algorithm

Wikipedia gives us a Python implementation for the de Boor's algorithm:

def deBoor(k, x, t, c, p):
    """
    Evaluates S(x).

    Args
    ----
    k: index of knot interval that contains x
    x: position
    t: array of knot positions, needs to be padded as described above
    c: array of control points
    p: degree of B-spline
    """
    d = [c[j + k - p] for j in range(0, p+1)]

    for r in range(1, p+1):
        for j in range(p, r-1, -1):
            alpha = (x - t[j+k-p]) / (t[j+1+k-r] - t[j+k-p])
            d[j] = (1.0 - alpha) * d[j-1] + alpha * d[j]

    return d[p]

Is there a similar algorithm calculating the derivative of the B-Spline interpolated curve (or even n-th derivative)?

I know that mathematically it is reduced to using a spline of the lower order but can't apply it to the de Boor's algorithm.

Solution

I think I found the right way to re-use the de Boor's algorithm for curve derivatives.

First, we consider the definition of the B-Spline curve. It is a linear combination of control points: $C(u) = \sum_{i=0}^{n}N_{i,p}(u)P_i$ (1)

Hence, the derivative is a linear combination of the basis-function derivatives

$C'(u) = \sum_{i=0}^{n}N'_{i,p}(u)P_i$ (2)

The derivative of the basis function is defined as follows:

$N'(u) = \frac{p}{u_{i+p} - u_i} N_{i,p-1} - \frac{p}{u_{i+p+1}-u_{i+1}}N_{i+1,p-1}$ (3)

We plug-in (3) into (2) and after some algebra kung-fu, described here http://public.vrac.iastate.edu/~oliver/courses/me625/week5b.pdf, we obtain:

$C'$u$ = \sum_{i=0}^{n-1}N_{i,p-1}$u$Q_i$ (4), where $Q_i = p\frac{P_{i+1}-P{i}}{u_{i+p+1} - u_{i+1}}$

The derivative of the B-Spline curve is nothing else but a new B-Spline curve of (p-1) degree built on top of the new control points Q. Now, to employ the de Boor's algorithm we compute the new control point set and lower the spline degree p by 1:

def deBoorDerivative(k, x, t, c, p):
    """
    Evaluates S(x).

    Args
    ----
    k: index of knot interval that contains x
    x: position
    t: array of knot positions, needs to be padded as described above
    c: array of control points
    p: degree of B-spline
    """
    q = [p * (c[j+k-p+1] - c[j+k-p]) / (t[j+k+1] - t[j+k-p+1]) for j in range(0, p)]

    for r in range(1, p):
        for j in range(p-1, r-1, -1):
            right = j+1+k-r
            left = j+k-(p-1)
            alpha = (x - t[left]) / (t[right] - t[left])
            q[j] = (1.0 - alpha) * q[j-1] + alpha * q[j]

    return q[p-1]

Test:

import numpy as np
import math as m

points = np.array([[i, m.sin(i / 3.0), m.cos(i / 2)] for i in range(0, 11)])
knots = np.array([0, 0, 0, 0, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 1.0, 1.0, 1.0, 1.0])


def finiteDifferenceDerivative(k, x, t, c, p):
    """ Third order finite difference derivative """

    f = lambda xx : deBoor(k, xx, t, c, p)

    dx = 1e-7

    return (- f(x + 2 * dx) \
            + 8 * f(x + dx) \
            - 8 * f(x - dx) \
            + f(x - 2 * dx)) / ( 12 * dx )


print "Derivatives: "·
print "De Boor:\t", deBoorDerivative(7, 0.44, knots, points, 3)
print "Finite Difference:\t", finiteDifferenceDerivative(7, 0.44, knots, points, 3)

Output:

Derivatives: 
De Boor:              [10. 0.36134438  2.63969004]
Finite Difference:    [9.99999999 0.36134438 2.63969004]