Toward precise curve offsetting constrained to parametric surfaces

Abstract

Computing offsets of curves on parametric surfaces is a fundamental yet challenging operation in computer-aided design and manufacturing. Traditional analytical approaches suffer from time-consuming geodesic distance queries and complex self-intersection handling, while discrete methods often struggle with precision. In this paper, we propose a totally different algorithm paradigm. Our key insight is that by representing the source curve as a sequence of line-segment primitives, the Voronoi decomposition constrained to the parametric surface enables localized offset computation. Specifically, the offsetting process can be efficiently traced by independently visiting the corresponding Voronoi cells. To address the challenge of computing the Voronoi decomposition on parametric surfaces, we introduce two key techniques. First, we employ intrinsic triangulation in the parameter space to accurately capture geodesic distances. Second, instead of directly computing the surface-constrained Voronoi decomposition, we decompose the triangulated parameter plane using a series of plane-cutting operations. Experimental results demonstrate that our algorithm achieves superior accuracy and runtime performance compared to existing methods. We also present several practical applications enabled by our approach.