Computing values and derivatives of Bézier and B-spline tensor products

We give an eecient algorithm for evaluating B ezier and B-spline tensor products for both positions and normals. The algorithm is an extension of a method for computing the position and tangent to a B ezier curve, and is asymptotically twice as fast as the standard bilinear algorithm. Many applications, such as rendering a surface using Phong shading, require evaluating both the value and derivatives of a surface. Repeated bilinear interpolation can be used to compute values and derivatives of n n tensor product B ezier surfaces (see Figure 1a and Farin Far93]). The nal computed point is a point on the surface, and the points in the next to last step can be used to calculate the derivatives of the surface. However, this algorithm cannot be used for an n m tensor product surface with m 6 = n. In this paper we develop an algorithm to handle the general case of arbitrary m and n. Our approach is to run the univariate version of de Casteljau's algorithm successively in each parametric direction. Each time, we stop the evaluation \one short" of completion. The resulting multi-linear function is then evaluated to compute both the value and the derivatives. The algorithm is illustrated for a tensor product B ezier surface by the following pseudo-Pascal code: (* Given the control points P i;j for a n m tensor product Bezier patch F and a pair of parameter values u and v, return the value and partial derivatives of F at u; v *)