De Casteljau's Algorithm:
de Casteljau's algorithm may be extended to handle Bézier surfaces. More exactly, de Casteljau's algorithm may be applied many times to determine the corresponding point on a Bézier surface p(u, v) given (u, v).
Remember that the equation of a Bézier surface
![1531_De Casteljaus Algorithm.png](https://www.expertsmind.com/CMSImages/1531_De%20Casteljaus%20Algorithm.png)
may be rewritten as
![2336_De Casteljaus Algorithm1.png](https://www.expertsmind.com/CMSImages/2336_De%20Casteljaus%20Algorithm1.png)
For i = 0, 1, ..., m, describe qi (v) as follows :
![404_De Casteljaus Algorithm2.png](https://www.expertsmind.com/CMSImages/404_De%20Casteljaus%20Algorithm2.png)
For a fixed v, we have m + 1 points q0 (v), q1 (v), . . . , qm (v). Each qi (v) is a point on the Bézier curve described by control points pi0, pi1, . . . , pin. Plugging these back in the surface equation yields
![619_De Casteljaus Algorithm3.png](https://www.expertsmind.com/CMSImages/619_De%20Casteljaus%20Algorithm3.png)
It means p(u, v) is a point of the Bézier curve described by m + 1 control points q0 (v), q1 (v), . . . , qm(v). Therefore, we have the following conclusion: To determine point p(u, v) on a Bézier surface, we may find m + 1 points q0 (v), q1 (v), . . . , qm (v) and after that from these points determine p(u, v).