Piecewise Linear Approximation of Rational Triangular Surfaces

Piecewise linear approximation of rational triangular surfaces is useful in surfaces intersection, surfaces rendering and mesh generation. The approximation error bound is usually estimated based on the information about second-order derivative bounds of the rational triangular surfaces. But the derivative bounds of rational triangular surfaces are difficult and less effective to be estimated. To solve this problem, using homogeneous coordinates and inequality method, we present an algorithm to estimate subdivision depths for rational triangular surfaces which are defined in any arbitrary triangle. The estimation is performed on the polynomial surfaces, of which the given rational surfaces are the images under the standard perspective projection. It is more efficient than evaluating the derivative bounds of the given surfaces directly. The subdivison depth is obtained in advance, however, it guarantees the required flatness of the given surface after the subdivision. Moreover, using Mbius reparameterization technique, the variance of the log weights of rational triangular Bézier surfaces is minimized, which can obviously improve the efficiency of the algorithm. In particular, the optimal reparameterization is solved explicitily, so reparameterization hardly increases operating times. Numerical examples suggest that this algorithm not only possesses more powerful properties, but also is more effective compared with any other old methods.