de Casteljau Algorithm for a Tensor-Product Bézier Surface

A polynomial Bézier surface in tensor-product form is described by the formula

, where are the control points of ; they form the control net of .

Recall that is the Bernstein basis polynomial of degree in , .

In order to determine the point (for and ), the de Casteljau algorithm can be used. For a polynomial Bézier curve

, (for ),

where are the control points, this algorithm calculates a current point ) by applying the following recurrence formula:

, for to ,

, for to .

Finally, we obtain

.

In order to determine the point (for and ), the de Casteljau algorithm can be applied to the surface in the following three ways:

First in the direction: by de Casteljau, we first determine the points

for to .

Next, the algorithm is used to compute

.

Second in the direction: analogously, we first determine the points

for to .

Then we compute

.

Simultaneously in the and directions: this method is easier to use in the case where :

for to and to ,

, for to , to , and to ,

yielding

.

If , the above procedure is applied in order to calculate for to , and next the direction method is used to compute the remaining iteration levels. The case is treated analogously.