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Towards Voronoi Diagrams of Surface Patches

Pengfei Wang, Jiantao Song, Lei Wang, Shiqing Xin, Dongming Yan, Shuangmin Chen, Changhe Tu, Wenping Wang

TL;DR

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, where each hyperplane encodes a linear field in a tetrahedron.

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.

Towards Voronoi Diagrams of Surface Patches

TL;DR

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, where each hyperplane encodes a linear field in a tetrahedron.

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.

Paper Structure

This paper contains 38 sections, 25 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. RELATED WORKS
  3. Conventional Voronoi Diagrams
  4. Extended Voronoi Diagrams
  5. Methodology
  6. Review on SurfaceVoronoi
  7. Distance over-propagation.
  8. Incremental half-plane cutting.
  9. Linear Representation of Scalar Field in a Tetrahedron
  10. Geometric View on Tetrahedron-range Linear Scalar Field
  11. ALGORITHM
  12. Tetrahedralization
  13. Incremental Hyperplane Cutting
  14. Handling numerical issues.
  15. Robust implementation.
  16. ...and 23 more sections

Figures (18)

  • Figure 1: This paper suggests computing the medial axis of CAD models via Voronoi diagrams of surface patches, treating each patch as an individual computational unit. Interestingly, this computational technique can also be used to calculate offsets. Top: Input polygonal models (surface patches are visualized in a color-coded style). Middle: Medial-axis surfaces. Bottom: Inward offset surfaces.
  • Figure 2: SurfaceVoronoi involves a step of incremental plane cutting, as illustrated in (a) the original triangular prism, (b) after one cutting operation, and (c) after two cutting operations. The red-colored line segment represents the triangle-restricted Voronoi diagram.
  • Figure 3: Just as a rectangle (a) and a box (b) can be generated by sweeping a line segment and a rectangle along an additional dimension, respectively, we envision the creation of an initial 4D triangular prism by sweeping a 3D tetrahedron along the fourth dimension (c). In (b) and (c), a 2D side face and a 3D side face are visualized in brown.
  • Figure 4: Illustration of how an initial 4D triangular prism is intersected by a 4D hyperplane: (a) Base face. (b-e) Four 3D side faces generated by sweeping the tetrahedral sides along the fourth dimension. (f) Top face.
  • Figure 5: Variant versions of Voronoi diagram. Here we take four identical Koala models as generators.
  • ...and 13 more figures