Editing De Casteljau's algorithm (section)

=== Geometric interpretation ===
The geometric interpretation of De Casteljau's algorithm is straightforward. 
*Consider a Bézier curve with control points <math>P_0, \dots, P_n</math>. Connecting the consecutive points we create the control polygon of the curve.
*Subdivide now each line segment of this polygon with the ratio <math>t : (1-t)</math> 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 <math>t</math>.
The following picture shows this process for a cubic Bézier curve:

[[Image:DeCasteljau1.svg]]

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 <math>t</math>, but splits the curve into two pieces at <math>t</math>, 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 <math>\mathbf{R}^n</math>, we may project the point into <math>\mathbf{R}^{n+1}</math>; for example, a curve in three dimensions may have its control points <math>\{(x_i, y_i, z_i)\}</math> and weights <math>\{w_i\}</math> projected to the weighted control points <math>\{(w_ix_i, w_iy_i, w_iz_i, w_i)\}</math>. The algorithm then proceeds as usual, interpolating in <math>\mathbf{R}^4</math>. 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.