Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, and computational geometry, a Delaunay triangulation for a set P of points in the plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid skinny triangles. The triangulation was invented by Boris Delaunay in 1934. Based on Delaunay's definition, the circumcircle of a triangle formed by three points from the original point set is empty if it does not contain vertices other than the three that define it. The Delaunay condition states that a triangle net is a Delaunay triangulation if all the circumcircles of all the triangles in the net are empty. This is the original definition for two-dimensional spaces. It is possible to use it in three-dimensional spaces by using a circumscribed sphere in place of the circumcircle. For a set of points on the same line there is no Delaunay triangulation.

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