[Paper Review] Delaunay triangulation of manifolds

Jean‐Daniel Boissonnat, Ramsay Dyer+1 more|arXiv (Cornell University)|Nov 1, 2013

Topological and Geometric Data Analysis20 references4 citations

TL;DR

This paper introduces a novel algorithmic framework for constructing Delaunay triangulations of abstract manifolds without requiring them to be embedded in Euclidean space. By leveraging bi-Lipschitz coordinate patches with distortion close to 1, the method produces a manifold simplicial complex that is homeomorphic to the original manifold under sufficient sampling density, even without smoothness assumptions on transition maps.

ABSTRACT

We present an algorithmic framework for producing Delaunay triangulations of manifolds. The input to the algorithm is a set of sample points together with coordinate patches indexed by those points. The transition functions between nearby coordinate patches are required to be bi-Lipschitz with a constant close to 1. The primary novelty of the framework is that it can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. The output is a manifold simplicial complex that is the Delaunay complex of a perturbed set of points on the manifold. The guarantee of a manifold output complex demands no smoothness requirement on the transition functions, beyond the bi-Lipschitz constraint. In the smooth setting, when the transition functions are defined by common coordinate charts, such as the exponential map on a Riemannian manifold, the output manifold is homeomorphic to the original manifold, when the sampling is sufficiently dense.

Motivation & Objective

• To develop a general method for Delaunay triangulation of manifolds that are not embedded in Euclidean space.
• To handle abstract manifolds using only coordinate patches with bi-Lipschitz transition functions.
• To guarantee that the output is a valid manifold simplicial complex without requiring smoothness of transition maps.
• To ensure topological correctness—homeomorphism to the original manifold—under dense sampling.
• To extend Delaunay triangulation techniques to general Riemannian and non-smooth manifolds.

Proposed method

• The algorithm takes as input a set of sample points and associated coordinate patches with bi-Lipschitz transition functions.
• It perturbs the sample points on the manifold to ensure Delaunay properties in the local coordinate charts.
• It constructs a Delaunay complex in each local coordinate patch using standard Euclidean Delaunay triangulation.
• It glues the local Delaunay complexes together using the bi-Lipschitz structure to maintain global manifold consistency.
• It ensures that the resulting complex is a simplicial manifold by verifying local neighborhood conditions.
• It relies on the bi-Lipschitz condition to control distortion and preserve geometric fidelity across patches.

Experimental results

Research questions

• RQ1 Can Delaunay triangulation be generalized to abstract manifolds not embedded in Euclidean space?
• RQ2 What conditions on coordinate patches ensure that the resulting triangulation is a valid manifold simplicial complex?
• RQ3 How does the bi-Lipschitz condition on transition functions affect the topological correctness of the output?
• RQ4 What sampling density guarantees that the output complex is homeomorphic to the original manifold?
• RQ5 Can the framework handle non-smooth transition functions while still producing a topologically correct Delaunay complex?

Key findings

• The framework produces a Delaunay triangulation of an abstract manifold using only bi-Lipschitz coordinate patches, without requiring smoothness of transition functions.
• The output is a manifold simplicial complex that is guaranteed to be topologically correct when the sampling is sufficiently dense.
• The method ensures that the resulting complex is homeomorphic to the original manifold in the smooth setting, such as when using exponential maps on Riemannian manifolds.
• The bi-Lipschitz condition with constant close to 1 is sufficient to control geometric distortion and maintain Delaunay properties across patches.
• The framework extends Delaunay triangulation to general manifolds beyond submanifolds of Euclidean space.
• The algorithmic construction is robust under perturbations of the sample points, ensuring stability of the output complex.