Ray tracing general parametric surfaces using interval arithmetic

Abstract

This paper describes an algorithm for ray tracing general parametric surfaces. After dividing the surface adaptively into small parts, a binary tree of these parts is built. For each part a bounding volume is calculated with interval arithmetic. From linear approximations and intervals for the partial derivatives it is possible to construct parallelepipds that adapt the orientation and shape of the surface parts very well and form very tight enclosures. Therefore we can develop an algorithm for rendering that is similar to that used with Bezier and B-spline surfaces, where the bounding volumes are derived from the convex hull property. The tree of enclosures (generated once in a preprocessing step) guarantees that each ray that hits the surface leads to an iteration on a very small surface part; this iteration can be robustly (and very quickly) performed in real arithmetic.