Robust and efficient ray intersection of implicit surfaces
The generation of ray traced images of surfaces plays a central role in computer graphics. One of the main operations in ray tracing is the calculation of intersections between rays and surfaces. In case of implicitly given surfaces the intersection problem can be formulated as that of finding the smallest nonnegative root of an equation in one variable. If the root finding is carried out by means of conventional numerical methods based on point sampling (such as bisection, regula-falsi, Newton) the resulting image can be wrong, when the surface is thin the ray may miss the surface, which may result in an image with holes. In this paper robust methods are obtained with interval inclusions in a variant of Alefeld-Hansen's globally convergent method for computing and bounding all the roots of a single equation. Alefeld-Hansen's method has been modified so instead of searching for all roots, a recursive depth-first search is carried out to obtain the smallest nonnegative root. When compared to other methods, it is found that this variant of Alefeld-Hansen's method is not only robust but also an efficient method for finding the ray intersection.
scientific article; zbMATH DE number 1421498