Abstract: This paper presents a new algorithm for computing the points at which a ray intersects a rational Bézier surface patch, and also an algorithm for determining if an intersection point lies within a region trimmed by piecewise Bézier curves. Both algorithms are based on a recent innovation known as Bézier clipping, described herein. The intersection algorithm is faster than previous methods for which published performance data allow reliable comparison. It robustly finds all intersections without requiring special preprocessing.