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# de Casteljau Algorithm for a Tensor-Product Bézier Surface

This Demonstration shows three different ways of applying the de Casteljau algorithm to a tensor-product Bézier surface.

### DETAILS

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.

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