The problem of finding the Delaunay triangulation of a set of points in d-dimensional Euclidean space can be converted to the problem of finding the convex hull of a set of points in (d 1)-dimensional space, by giving each point p an extra coordinate equal to |p|, taking the bottom side of the convex hull, and mapping back to d-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 simplices.
We can store the history of the splits and flips performed: each triangle stores a pointer to the two or three triangles that replaced it.
To find the triangle that contains v, we start at a root triangle, and follow the pointer that points to a triangle that contains v, until we find a triangle that has not yet been replaced. Over all vertices, then, this takes O(n log n) time.), the runtime can be exponential in the dimension even if the final Delaunay triangulation is small.
The Delaunay triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi diagram for P.
Special cases include the existence of three points on a line and four points on circle.
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For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors.
By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions.
Generalizations are possible to metrics other than Euclidean.
However, in these cases a Delaunay triangulation is not guaranteed to exist or be unique.
Nonsimplicial facets only occur when d 2 of the original points lie on the same d-hypersphere, i.e., the points are not in general position.