The Delaunay triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi diagram for P. Relationship with the Voronoi diagram Ĭonnecting the centers of the circumcircles produces the Voronoi diagram (in red). However, in these cases a Delaunay triangulation is not guaranteed to exist or be unique. Generalizations are possible to metrics other than Euclidean distance. 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.īy considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions. įor a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case). The triangulation is named after Boris Delaunay for his work on this topic from 1934. Delaunay triangulations maximize the minimum of all the angles of the triangles in the triangulation they tend to avoid sliver triangles. In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT( P) such that no point in P is inside the circumcircle of any triangle in DT( P). Triangulation method A Delaunay triangulation in the plane with circumcircles shown
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