Ray tracing of Steiner patches

Abstract

Steiner patches are triangular surface patches for which the Cartesian coordinates of points on the patch are defined parametrically by quadratic polynomial functions of two variables. It has recently been shown that it is possible to express a Steiner patch in an implicit equation which is a degree four polynomial in x,y,z. Furthermore, the parameters of a point known to be on the surface can be computed as rational polynomial functions of x,y,z. These findings lead to a straightforward algorithm for ray tracing Steiner patches in which the ray intersection equation is a degree four polynomial in the parameter of the ray. The algorithm presented represents a major simplification over existing techniques for ray tracing free-form surface patches.