We present a fast and robust algorithm for computing the minimum distance from a point in to a toroidal patch. Our approach replaces iterative numerical optimization with a topologically complete analytic classification that exhaustively enumerates all configurations on a toroidal patch. By reducing the problem to planar arc tests on circles obtained through simple radial normalization, the method identifies 4 regular regimes, which extend to 16 via Boolean combinations of mirrored-domain inclusion bits when the patch’s parameter domain overlaps the self-intersection interval. Binary decisions on these bits determine the nearest feature (surface, boundary curve, or corner) at constant cost and without trigonometric evaluations at query time. Each query is processed in about 2 microseconds on average, achieving over 99.99% agreement with a high-precision reference within a tolerance of , while significantly outperforming global optimizers. The algorithm also robustly handles patches derived from spindle tori with self-intersections, providing a consistent and efficient building block for geometry processing and rendering tasks, such as path tracing of surfaces of revolution represented as collections of toroidal patches. • Non-iterative algorithm for point–toroidal patch minimum distance computation. • Problem reformulated via topological relations between torus arcs and the point. • Closed-form solutions per topological case, enabling robust, efficient evaluation. • Demonstrates robust and efficient performance in large-scale rendering experiments.
Kim et al. (Fri,) studied this question.