Towards Voronoi Diagrams of Surface Patches

TL;DR

This paper introduces a novel method for accurately computing Voronoi diagrams of surface patches on CAD models, improving medial axis extraction by tetrahedralizing the region and performing hyperplane cuts in 4D.

Contribution

It presents a new algorithm for Voronoi diagram computation of surface patches using 4D hyperplane cutting, addressing inaccuracies in existing medial axis extraction methods.

Findings

Produces higher fidelity medial axes with fewer artifacts.

Can compute offset surfaces alongside Voronoi diagrams.

Outperforms existing methods in accuracy and quality.

Abstract

Extraction of a high-fidelity 3D medial axis is a crucial operation in CAD. When dealing with a polygonal model as input, ensuring accuracy and tidiness becomes challenging due to discretization errors inherent in the mesh surface. Commonly, existing approaches yield medial-axis surfaces with various artifacts, including zigzag boundaries, bumpy surfaces, unwanted spikes, and non-smooth stitching curves. Considering that the surface of a CAD model can be easily decomposed into a collection of surface patches, its 3D medial axis can be extracted by computing the Voronoi diagram of these surface patches, where each surface patch serves as a generator. However, no solver currently exists for accurately computing such an extended Voronoi diagram. Under the assumption that each generator defines a linear distance field over a sufficiently small range, our approach operates by tetrahedralizing the region of interest and computing the medial axis within each tetrahedral element. Just as SurfaceVoronoi computes surface-based Voronoi diagrams by cutting a 3D prism with 3D planes (each plane encodes a linear field in a triangle), the key operation in this paper is to conduct the hyperplane cutting process in 4D, where each hyperplane encodes a linear field in a tetrahedron. In comparison with the state-of-the-art, our algorithm produces better outcomes. Furthermore, it can also be used to compute the offset surface.