---
title: Dense point sets have sparse Delaunay triangulations
url: https://www.emergentmind.com/papers/0110030
type: paper
arxiv_id: '0110030'
arxiv_url: https://arxiv.org/abs/0110030
published: '2001-10-15'
categories:
- cs.CG
- cs.DM
---

# Dense point sets have sparse Delaunay triangulations

## Abstract

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3). This bound is tight in the worst case for all D = O(sqrt{n}). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any n and D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D.