[Paper Review] Geometric angle structures on triangulated surfaces

Ren Guo|arXiv (Cornell University)|Jan 20, 2006

Computational Geometry and Mesh Generation3 references4 citations

TL;DR

This paper characterizes the existence of spherical, hyperbolic, and Euclidean angle structures on triangulated surfaces via edge and Delaunay invariants. Using duality in linear programming and adaptations of Rivin and Leibon's methods, it establishes necessary and sufficient conditions for non-empty solution sets, resolving open linear programming problems for hyperbolic and spherical geometries.

ABSTRACT

In this paper we characterize a function defined on the set of edges of a triangulated surface such that there is a spherical angle structure having the function as the edge invariant (or Delaunay invariant). We also characterize a function such that there is a hyperbolic angle structure having the function as the edge invariant.

Motivation & Objective

• To resolve the open linear programming problem for the existence of hyperbolic angle structures with prescribed edge invariant.
• To solve the linear programming problem for the existence of spherical angle structures with prescribed Delaunay invariant.
• To extend known results on Euclidean cone metrics to spherical and hyperbolic geometries via variational and duality methods.
• To establish necessary and sufficient conditions for non-empty solution sets of angle structures using combinatorial and geometric constraints.

Proposed method

• Uses duality in linear programming to transform existence problems into feasibility conditions on subsets of triangles.
• Applies Leibon's characterization of hyperbolic angle structures via Delaunay invariants to derive new conditions for edge invariants.
• Employs variable transformation via $ y_i = \frac{\pi + x_i - x_j - x_k}{2} $ to relate hyperbolic and spherical angle structures.
• Derives necessary and sufficient conditions through combinatorial inequalities involving triangle counts and edge invariants over subsets of the triangulation.
• Leverages the identity $ 2D(e) + \mathcal{D}(e) = 2\pi $ to relate edge and Delaunay invariants across geometries.
• Uses contradiction arguments in linear programming to prove strict inequality conditions for non-emptiness of solution sets.

Experimental results

Research questions

• RQ1 What conditions on an edge invariant $ D:E \to (0,2\pi) $ ensure the existence of a hyperbolic angle structure?
• RQ2 What conditions on a Delaunay invariant $ \mathcal{D}:E \to (-2\pi,2\pi) $ ensure the existence of a spherical angle structure?
• RQ3 How can the existence of spherical angle structures be characterized in terms of edge invariants?
• RQ4 What is the relationship between edge invariants and Delaunay invariants in the context of hyperbolic and spherical geometries?
• RQ5 Can the linear programming feasibility conditions for hyperbolic and spherical angle structures be derived using duality and known results from Euclidean and hyperbolic cases?

Key findings

• The set $ AS(S,T;D) $ of spherical angle structures with edge invariant $ D $ is nonempty if and only if $ \pi|X| < \sum_{e \in E(X)} D(e) $ for all subsets $ X \subseteq F $.
• The set $ AH(S,T;D) $ of hyperbolic angle structures with edge invariant $ D $ is nonempty if and only if $ \pi(|F| - |X|) > \sum_{e \notin E(X)} D(e) $ for all subsets $ X \subset F $.
• The set $ AS(S,T;\mathcal{D}) $ of spherical angle structures with Delaunay invariant $ \mathcal{D} $ is nonempty if and only if $ \pi(|F| - |X|) > \sum_{e \notin E(X)} \left(\pi - \frac{1}{2}\mathcal{D}(e)\right) $ for all subsets $ X \subset F $.
• The existence of hyperbolic angle structures with edge invariant $ D $ is equivalent to the existence of spherical angle structures with Delaunay invariant $ \mathcal{D}(e) = 2\pi - 2D(e) $.
• The maximal value of the dual linear program is strictly negative under the derived conditions, ensuring non-emptiness of solution sets.
• The proof establishes that the solution set is nonempty only when strict inequalities hold, ruling out equality cases via contradiction.