Satisfying coplanarity constraints on points sets in three dimensions with finite precision arithmetic

Abstract

Geometric operations that form the very heart of many commercial solid modelers are susceptible to failures without any prior warning due to round-off errors in their computations. The consequences of these unwarranted failures can sometimes be really severe. Collinearity and coplanarity of point sets are important properties that need to be preserved in many applications. We present algorithms that preserve the collinearity and coplanarity relationships among point sets in three dimensions. We further extend these algorithms for approximating three-dimensional cellular objects with planar surfaces while maintaining the planarity of component faces. Partial expansions of continued fraction expansions of rational numbers underlies the basic strategy of our approximation paradigm. The solutions are guaranteed to be correct and they work for a large class of engineering objects.