Toward Precise Curve Offsetting Constrained to Parametric Surfaces

2025-09-11

AI Summary

Exact offset curve computation on parametric surfaces remains a fundamental challenge in CAD/CAM. Conventional analytical approaches rely on expensive geodesic distance queries and struggle with self-intersections, while discrete methods suffer from limited accuracy. This paper introduces a novel Voronoi decomposition method based on intrinsic triangulation of the parameter domain and planar cutting operations. The input curve is discretized into line segments; an intrinsic triangulation approximating geodesic distances is constructed in the parameter space; and a sequence of planar cuts—guided by this triangulation—indirectly yields a surface-restricted Voronoi diagram. This enables localized, geodesic-free offset tracing without explicit geodesic computation or global distance solving. Crucially, the method avoids traditional self-intersection detection and global optimization, thereby significantly improving both accuracy and computational efficiency. Experimental results demonstrate superior performance over state-of-the-art methods across diverse complex surfaces, confirming its practicality and extensibility.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Computing precise curve offsets on parametric surfaces
• Overcoming traditional geodesic distance query limitations
• Addressing precision and efficiency in offset algorithms

Innovation

• Voronoi decomposition on parametric surfaces
• Intrinsic triangulation in parameter space
• Plane cutting operations for decomposition