Delaunay Triangulation - d-dimensional Delaunay

d-dimensional Delaunay

For a set P of points in the (d-dimensional) Euclidean space, a Delaunay triangulation is a triangulation DT(P) such that no point in P is inside the circum-hypersphere of any simplex in DT(P). It is known that there exists a unique Delaunay triangulation for P, if P is a set of points in general position; that is, there exists no k-flat containing k + 2 points nor a k-sphere containing k + 3 points, for 1 ≤ k ≤ d − 1 (e.g., for a set of points in ℝ3; no three points are on a line, no four on a plane, no four are on a circle, and no five on a sphere).

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|2, 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. 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.