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Delaunay triangulation

In mathematics, and computational geometry, the Delaunay triangulation, for a set P of points in the plane, is the triangulation DT(P) of P such that no point in P is inside the circumcircle of any triangle in DT(P).

This is the Delaunay triangulation of a random set of points in the plane.

In the n-dimensional case it is stated as follows.

For a set P of points in the (n-dimensional) Euclidean space, the Delaunay triangulation is the triangulation DT(P) of P such that no point in P is inside the circum-hypersphere of any simplex in DT(P).

Another, equivalent, definition is:

The Delaunay triangulation of a discrete point set P is the dual of the Voronoi tesselation for P.

It is known that the Delaunay triangulation exists and is unique for P, if P is a set of points in general position, i.e., no three points are on the same line and no four are on the same circle, for a two dimensional set of points, or no n+1 points are on the same hyperplane and no n+2 points are on the same hypersphere, for a n-dimensional set of points. An elegant proof of this fact is outlined below. It is worth mentioning, because it reveals connections between the two constructs fundamental for computational and combinatorial geometry.

The problem of finding the Delaunay triangulation of a set of points in n-dimensional euclidean space can be converted to the problem of finding the convex hull of a set of points in n+1-dimensional space, by giving all points p an extra coordinate equal to p, taking the bottom side of the convex hull, and mapping back to n-dimensional space by deleting the last coordinate. As the convex hull is unique, so is the triangulation, assuming all facets of the convex hull are simplexes. A facet not being a simplex implies that n+2 of the original points lay on the same d-hypersphere, and the points were not in general position.

On the other hand, it is easily seen that for the set of three points on the same line there is no Delaunay trianguation (in fact, no triangulation at all). On the other hand, for 4 points on the same circle (e.g., the vertices of a rectangle) the Delaunay tringulation is not unique: clearly, the two possible triangulations that split the quadrangle into two triangles satisfy the Delaunay condition.

Generalizations are possible to metrics other than Euclidean. However in these cases the Delaunay triangulation is not guaranteed to exist or be unique.

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