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Computing the Intrinsic Delaunay Triangulation of a Closed Polyhedral Surface

Loïc Dubois

Abstract

Every surface that is intrinsically polyhedral can be represented by a portalgon: a collection of polygons in the Euclidean plane with some pairs of equally long edges abstractly identified. While this representation is arguably simpler than meshes (flat polygons in R3 forming a surface), it has unbounded happiness: a shortest path in the surface may visit the same polygon arbitrarily many times. This pathological behavior is an obstacle towards efficient algorithms. On the other hand, Löffler, Ophelders, Staals, and Silveira (SoCG 2023) recently proved that the (intrinsic) Delaunay triangulations have bounded happiness. In this paper, given a closed polyhedral surface S, represented by a triangular portalgon T, we provide an algorithm to compute the Delaunay triangulation of S whose vertices are the singularities of S (the points whose surrounding angle is distinct from 2pi). The time complexity of our algorithm is polynomial in the number of triangles and in the logarithm of the aspect ratio r of T. Within our model of computation, we show that the dependency in log(r) is unavoidable. Our algorithm can be used to pre-process a triangular portalgon before computing shortest paths on its surface, and to determine whether the surfaces of two triangular portalgons are isometric.

Computing the Intrinsic Delaunay Triangulation of a Closed Polyhedral Surface

Abstract

Every surface that is intrinsically polyhedral can be represented by a portalgon: a collection of polygons in the Euclidean plane with some pairs of equally long edges abstractly identified. While this representation is arguably simpler than meshes (flat polygons in R3 forming a surface), it has unbounded happiness: a shortest path in the surface may visit the same polygon arbitrarily many times. This pathological behavior is an obstacle towards efficient algorithms. On the other hand, Löffler, Ophelders, Staals, and Silveira (SoCG 2023) recently proved that the (intrinsic) Delaunay triangulations have bounded happiness. In this paper, given a closed polyhedral surface S, represented by a triangular portalgon T, we provide an algorithm to compute the Delaunay triangulation of S whose vertices are the singularities of S (the points whose surrounding angle is distinct from 2pi). The time complexity of our algorithm is polynomial in the number of triangles and in the logarithm of the aspect ratio r of T. Within our model of computation, we show that the dependency in log(r) is unavoidable. Our algorithm can be used to pre-process a triangular portalgon before computing shortest paths on its surface, and to determine whether the surfaces of two triangular portalgons are isometric.
Paper Structure (51 sections, 71 theorems, 7 figures)

This paper contains 51 sections, 71 theorems, 7 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Our results
  4. Overview of the proof of Theorem \ref{['thm:main result']}
  5. Preliminaries
  6. Portalgons, tessellations, and polyhedral surfaces
  7. Aspect ratio, systole, happiness, and segment-happiness
  8. Tubes and bifaces
  9. Description of the algorithm
  10. Inserting vertices and edges
  11. Deleting vertices
  12. Simplifying tubes
  13. Data structure for marking bifaces as inactive
  14. Algorithm
  15. Analysis of the algorithm
  16. ...and 36 more sections

Key Result

Theorem 1.1

Let $T$ be a portalgon of $n$ triangles, of aspect ratio $r$, whose surface $\mathcal{S}(T)$ is closed. One can compute the portalgon of the Delaunay tessellation of $\mathcal{S}(T)$ in $O(n^3 \log^2(n) \cdot \log^4(r))$ time.

Figures (7)

  • Figure 2.1: (Left) A triangular portalgon $T$: two triangles in the Euclidean plane, with two sides matched in red. (Right) The surface $\mathcal{S}(T)$, and the 1-skeleton $T^1$.
  • Figure 3.1: (From left to right) A good biface, a biface not good, a thin biface, a thick biface.
  • Figure 4.1: Data structure for Algorithm: a portalgon whose polygons are partitioned, here by color, inducing sub-portalgons called regions, and a region singularized as active, here in red.
  • Figure 5.1: The red loop encloses the blue segment in the surface.
  • Figure D.1: The setting of Lemma \ref{['inapolygon']}.
  • ...and 2 more figures

Theorems & Definitions (130)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.2
  • Proposition 1.2
  • Proposition 1.2
  • proof : Proof of Theorem \ref{['thm:main result']}
  • Lemma 3.1
  • Proposition 3.1
  • Proposition 5.0
  • proof : Sketch of proof
  • ...and 120 more