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The de Casteljau Algorithm

# The de Casteljau Algorithm

This is a follow up to Blender and Blossoms.

The de Casteljau Algorithm is a recursive way to calculate the Point on a Bézier curve given its control polygon $P = \{ \vec{b_i} \}$ consisting of the Points $b_i$. It is defined as follows in the Book:

Given the points $\vec{b}_0 ... \vec{b}_n$, $t \in \mathbb{R}$ and a function $f(t): \mathbb{R} \to \mathbb{R}$. Set

\begin{eqnarray*}\vec{b}_i^0 & := & \vec{b_i} \\ \vec{b}_i^r & := & (1-f(t)) \vec{b}_i^{r-1} + f(t)\,\vec{b}_{i+1}^{r-1}(t)\end{eqnarray*}

Then $\vec{b}_0^n$ is the point with the parameter $t$ on the corresponding Beziér curve.

So this is essentially a recursive definition: Starting from $n$ points, we always combine two of them and remain with $n-1$ points. We continue this scheme until only one point is left. This is the point on the curve for the value $t$.

# Implementation

The de Casteljau Algorithm is short and beautifully implemented:

import numpy as np

class DeCasteljau:
def __init__(self, *pts, func = lambda t: t):
self._b = np.asarray(pts, dtype=np.float64)
self._func = func

def __call__(self, t):
b, n = self._b, len(self._b)
for r in range(1,n):
for i in range(n-r):
b[i] = (1-self._func(t))*b[i] + self._func(t)*b[i+1]

return b[0]


And this is how it looks.

The orange curve is the defining polygon and the dots are the corresponding beziér curve evaluated for 100 values of $t \in [0,1]$ and $f$ being the identity.

You can find the complete code in the GitHub repository under chap04.