# Delaunay triangulation of manifolds

arXiv: https://arxiv.org/abs/1311.0117 · HTML: https://mlsanity.com/paper/1311.0117/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Nov 2013.

## Built on by (most cited first, 2 of 2)

- [A Discussion On the Validity of Manifold Learning](https://mlsanity.com/paper/2106.01608/) (2021)
- [An adaptive sampling algorithm for data-generation to build a data-manifold for physical problem surrogate modeling](https://mlsanity.com/paper/2505.08487/) (2025)

## Abstract (arXiv)

We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a manifold Delaunay complex is produced provided the transition functions are bi-Lipschitz with a constant close to 1, and the sample points meet a local density requirement; no smoothness assumptions are required. If the transition functions are smooth, the output is a triangulation of the manifold. The output complex is naturally endowed with a piecewise flat metric which, when the original manifold is Riemannian, is a close approximation of the original Riemannian metric. In this case the ouput complex is also a Delaunay triangulation of its vertices with respect to this piecewise flat metric.