Editing Delaunay triangulation (section)

==Properties==
[[File: Example steps in Delauney triangularization.png|thumb|Example steps]]
[[File:Delaunay triangulation does not minimize edge length.gif|thumb|Each frame of the animation shows a Delaunay triangulation of the four points. Halfway through, the triangulating edge flips showing that the Delaunay triangulation maximizes the minimum angle, not the edge-length of the triangles.]]

Let {{mvar|n}} be the number of points and {{mvar|d}} the number of dimensions.

* The union of all simplices in the triangulation is the convex hull of the points.
* The Delaunay triangulation contains {{tmath|\textstyle O\bigl(n^{\lceil d/2 \rceil}\bigr) }} simplices.{{r|Seidel1995}}
* In the plane ({{math|1=''d'' = 2}}), if there are {{mvar|b}} vertices on the convex hull, then any triangulation of the points has at most {{math|2''n'' – 2 – ''b''}} triangles, plus one exterior face (see [[Euler characteristic]]).
* If points are distributed according to a [[Poisson process]] in the plane with constant intensity, then each vertex has on average six surrounding triangles. More generally for the same process in {{mvar|d}} dimensions the average number of neighbors is a constant depending only on {{mvar|d}}.{{r|Meijering}}
* In the plane, the Delaunay triangulation maximizes the minimum angle.  Compared to any other triangulation of the points, the smallest angle in the Delaunay triangulation is at least as large as the smallest angle in any other.  However, the Delaunay triangulation does not necessarily minimize the maximum angle.{{r|ETW1992}} The Delaunay triangulation also does not necessarily minimize the length of the edges. 
* A circle circumscribing any Delaunay triangle does not contain any other input points in its interior.
* If a circle passing through two of the input points doesn't contain any other input points in its interior, then the segment connecting the two points is an edge of a Delaunay triangulation of the given points.
* Each triangle of the Delaunay triangulation of a set of points in {{mvar|d}}-dimensional spaces corresponds to a facet of [[convex hull]] of the projection of the points onto a ({{math|''d'' + 1}})-dimensional [[paraboloid]], and vice versa.
* The closest neighbor {{mvar|b}} to any point {{mvar|p}} is on an edge {{mvar|bp}} in the Delaunay triangulation since the [[nearest neighbor graph]] is a subgraph of the Delaunay triangulation.
* The Delaunay triangulation is a [[geometric spanner]]: In the plane ({{math|1=''d'' = 2}}), the shortest path between two vertices, along Delaunay edges, is known to be no longer than 1.998 times the Euclidean distance between them.{{r|Xia}}