Fast and reliable algorithms for boundary evaluation of low-degree sculptured solids

We present e cient representations and algorithms for accurate boundary computation on low-degree sculptured CSG solids using exact arithmetic. Previous approaches to this problem based entirely on oating-point arithmetic su er from severe robustness problems. We propose a general framework for precisely answering geometric queries on algebraic points, through the use of multivariate St urm sequences over \big-rational" numbers. Exact arithmetic guarantees an accurate geometric algorithm, and is likely to be required for perturbation techniques to handle degeneracies. While exact arithmetic is considerably more expensive than xedprecision arithmetic, we show that for many inputs we can improve the performance of overall algorithm by using oating-point arithmetic without sacri cing accuracy. Overall, the approach leads to a boundary evaluation algorithm that is accurate and reasonably e cient.