Some estimates of the height of rational Bernstein-Bezier triangular surfaces

Subdivision, intersection and triangulation of a surface are the basic and common operations in computer aided design and computer graphics. A complicated surface can be approximated by some triangular patches. The key technique of this problem is the estimation of approximation error. This work has been done in some papers by estimating second derivatives of the surface, thus very complicated computation must be executed and hence time is overly consumed. Avoiding this estimation and exerting the influence of the control points of the surfaces, this paper gives a formula for computing the exact distance between a degree 2 Bernstein-Bezier triangular surface and its base triangle. Also, this paper gives estimates of maximal distance between a degree 2 or 3 rational Bernstein-Bezier triangular surface and its base triangle with a high precision. These results are easy for programming, simple for computing, and effective for improving the subdivision/triangulation algorithms of common rational Bezier surfaces in a design system.