Editing Delaunay triangulation (section)

== Applications ==
{{see also|Voronoi diagram#Applications}}

The [[Euclidean minimum spanning tree]] of a set of points is a subset of the Delaunay triangulation of the same points,{{r|AK2013}} and this can be exploited to compute it efficiently.

For modelling [[terrain]] or other objects given a [[point cloud]], the Delaunay triangulation gives a nice set of triangles to use as polygons in the model.  In particular, the Delaunay triangulation avoids narrow triangles (as they have large circumcircles compared to their area). See [[triangulated irregular network]].

Delaunay triangulations can be used to determine the density or intensity of points samplings by means of the [[Delaunay tessellation field estimator|Delaunay tessellation field estimator (DTFE)]].

[[File:Delaunay Triangulation (100 Points).svg|right|thumb|250px|A Delaunay triangulation of a random set of 100 points in a plane.]]
Delaunay triangulations are often used to [[mesh generation|generate meshes]] for space-discretised solvers such as the [[finite element method]] and the [[finite volume method]] of physics simulation, because of the angle guarantee and because fast triangulation algorithms have been developed.  Typically, the domain to be meshed is specified as a coarse [[simplicial complex]]; for the mesh to be numerically stable, it must be refined, for instance by using [[Ruppert's algorithm]].

The increasing popularity of [[finite element method]] and [[boundary element method]] techniques increases the incentive to improve automatic meshing algorithms. However, all of these algorithms can create distorted and even unusable grid elements. Fortunately, several techniques exist which can take an existing mesh and improve its quality. For example, smoothing (also referred to as mesh refinement) is one such method, which repositions nodes to minimize element distortion. The [[stretched grid method]] allows the generation of pseudo-regular meshes that meet the Delaunay criteria easily and quickly in a one-step solution. 

[[Constrained Delaunay triangulation]] has found applications in [[path planning]] in automated driving and topographic surveying. {{r|AKI2012}}