Efficient and accurate B-rep generation of low degree sculptured solids using exact arithmetic

Abstract

We present efficient representations and algorithms for exact boundary computation on low degree sculptured CSG solids using exact arithmetic. Most of the previous work using exact arithmetic has been restricted to polyhedral models. In this paper, we generalize it to higher order objects, whose boundaries are composed of rational parametric surfaces. The use of exact arithmetic and representation guarantees that a geometric algorithm is numerically accurate and is likely to be required for perturbation techniques which handle degeneracies. We present efficient algorithms for computing the intersection curves of trimmed parametric surfaces, decomposing them into multiple components for efficient point location queries inside the trimmed regions, and computing the boundary of the resulting solid using topological information and component classification tests. We also employ a number of previously developed algorithms like algebraic curve classification, multivariate Sturm sequences, and multivariate resultants. We have implemented key parts of these algorithms and preliminary implementations indicate the performance of our algorithm to be about one order of magnitude slower than similar algorithms using IEEE floating-point arithmetic.