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In mathematics and computational geometry, a Delaunay triangulation for a set P of points in a 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 is named after Boris Delaunay for his work on this topic from 1934.〔Delaunay, Boris: ''(Sur la sphère vide. A la mémoire de Georges Voronoï ), Bulletin de l'Académie des Sciences de l'URSS, Classe des sciences mathématiques et naturelles, No. 6: 793–800, 1934''〕 For a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case). 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. ==Relationship with the Voronoi diagram== 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. File:Delaunay_circumcircles_centers.svg|The Delaunay triangulation with all the circumcircles and their centers (in red). Image:Delaunay_Voronoi.svg|Connecting the centers of the circumcircles produces the Voronoi diagram (in red). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Delaunay triangulation」の詳細全文を読む スポンサード リンク
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