---
tag: 259Y10BYJC
kind: concept
name: Delaunay triangulation
msc: [52C99, 68U05]
specializes: [V0B6C6TMWD]
sources:
  - url: https://en.wikipedia.org/wiki/Delaunay_triangulation
same_as:
  wikidata: Q192445
  wikipedia: Delaunay triangulation
---

## Summary

A Delaunay triangulation of a finite point set in the plane is a special [[triangulation|V0B6C6TMWD]] of it.
The circumcircle of each of its triangles has no point of the set inside.
Among all triangulations of the set, it maximizes the smallest angle, so it avoids thin triangles.
It is the geometric dual of the [[Voronoi diagram|QDNEBBVZ84]], and it is widely used for meshes.

## Explanations

### L1 human

Take a triangulation of a set of points and look at the circle through the three corners of each triangle.
The triangulation is Delaunay when no other point of the set sits strictly inside any of these circles.
For four points at the corners of a square, all four lie on one circle, and both diagonals give a Delaunay triangulation.
If you push one corner slightly outward, only one diagonal remains Delaunay.