De Casteljau's algorithm

In the mathematical field of numerical analysis, De Casteljau's algorithm is a recursive method to evaluate polynomials in Bernstein form or Bézier curves, named after its inventor Paul de Casteljau. De Casteljau's algorithm can also be used to split a single Bézier curve into two Bézier curves at an arbitrary parameter value.

Although the algorithm is slower for most architectures when compared with the direct approach, it is more numerically stable.[citation needed]

Definition edit

A Bézier curve   (of degree  , with control points  ) can be written in Bernstein form as follows


where   is a Bernstein basis polynomial


The curve at point   can be evaluated with the recurrence relation


Then, the evaluation of   at point   can be evaluated in   operations. The result   is given by


Moreover, the Bézier curve   can be split at point   into two curves with respective control points:


Geometric interpretation edit

The geometric interpretation of De Casteljau's algorithm is straightforward.

  • Consider a Bézier curve with control points  . Connecting the consecutive points we create the control polygon of the curve.
  • Subdivide now each line segment of this polygon with the ratio   and connect the points you get. This way you arrive at the new polygon having one fewer segment.
  • Repeat the process until you arrive at the single point – this is the point of the curve corresponding to the parameter  .

The following picture shows this process for a cubic Bézier curve:


Note that the intermediate points that were constructed are in fact the control points for two new Bézier curves, both exactly coincident with the old one. This algorithm not only evaluates the curve at  , but splits the curve into two pieces at  , and provides the equations of the two sub-curves in Bézier form.

The interpretation given above is valid for a nonrational Bézier curve. To evaluate a rational Bézier curve in  , we may project the point into  ; for example, a curve in three dimensions may have its control points   and weights   projected to the weighted control points  . The algorithm then proceeds as usual, interpolating in  . The resulting four-dimensional points may be projected back into three-space with a perspective divide.

In general, operations on a rational curve (or surface) are equivalent to operations on a nonrational curve in a projective space. This representation as the "weighted control points" and weights is often convenient when evaluating rational curves.

Notation edit

When doing the calculation by hand it is useful to write down the coefficients in a triangle scheme as


When choosing a point t0 to evaluate a Bernstein polynomial we can use the two diagonals of the triangle scheme to construct a division of the polynomial






Bézier curve edit

A Bézier curve

When evaluating a Bézier curve of degree n in 3-dimensional space with n + 1 control points Pi




we split the Bézier curve into three separate equations


which we evaluate individually using De Casteljau's algorithm.

Example edit

We want to evaluate the Bernstein polynomial of degree 2 with the Bernstein coefficients


at the point t0.

We start the recursion with


and with the second iteration the recursion stops with


which is the expected Bernstein polynomial of degree 2.

Implementations edit

Here are example implementations of De Casteljau's algorithm in various programming languages.

Haskell edit

deCasteljau :: Double -> [(Double, Double)] -> (Double, Double)
deCasteljau t [b] = b
deCasteljau t coefs = deCasteljau t reduced
    reduced = zipWith (lerpP t) coefs (tail coefs)
    lerpP t (x0, y0) (x1, y1) = (lerp t x0 x1, lerp t y0 y1)
    lerp t a b = t * b + (1 - t) * a

Python edit

def de_casteljau(t: float, coefs: float) -> float:
    """De Casteljau's algorithm."""
    beta = [c for c in coefs]  # values in this list are overridden
    n = len(beta)
    for j in range(1, n):
        for k in range(n - j):
            beta[k] = beta[k] * (1 - t) + beta[k + 1] * t
    return beta[0]

Java edit

public double deCasteljau(double t, double[] coefficients) {
    double[] beta = coefficients;
    int n = beta.length;
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < (n - i); j++) {
            beta[j] = beta[j] * (1 - t) + beta[j + 1] * t;
    return beta[0];

JavaScript edit

The following function applies De Casteljau's algorithm to an array of points, resolving the final midpoint with the additional properties in and out (for the midpoint's "in" and "out" tangents, respectively).

function deCasteljau(points, position = 0.5) {
	let a, b, midpoints = [];
	while (points.length > 1) {
		const num = points.length - 1;
		for (let i = 0; i < num; ++i) {
			a = points[i];
			b = points[i+1];
				a[0] + ((b[0] - a[0]) * position),
				a[1] + ((b[1] - a[1]) * position),
		points = midpoints;
		midpoints = [];
	return Object.assign(points[0], {in: a, out: b});

The following example calls this function with the green points below, exactly halfway along the curve. The resulting coordinates should equal  , or the position of the centremost red point.

Intermediate line segments obtained by recursively applying linear interpolation to adjacent points.
	/* Definition of deCasteljau() function omitted for brevity */
	const nodes = window.document.querySelectorAll("circle.n0-point");
	const points = Array.from(nodes).map(({cx, cy}) => [cx.baseVal.value, cy.baseVal.value]);
	deCasteljau(points); // Result: [192, 32]

See also edit

References edit

  • Farin, Gerald & Hansford, Dianne (2000). The Essentials of CAGD. Natic, MA: A K Peters, Ltd. ISBN 1-56881-123-3

External links edit