|
In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way. The name is also used in topology for a similar operation on cell complexes. The result is topologically equivalent to that of the geometric operation, but the parts have arbitrary shape and size. This is an example of a finite subdivision rule. Both operations have a number of applications in mathematics and in geometric modeling, especially whenever some function or shape needs to be approximated piecewise, e.g. by a spline. ==Barycentric subdivision of a simplex== The barycentric subdivision (henceforth ''BCS'') of an -dimensional simplex consists of (''n'' + 1)! simplices. Each piece, with vertices , can be associated with a permutation of the vertices of , in such a way that each vertex is the barycenter of the points . In particular, the BCS of a single point (a 0-dimensional simplex) consists of that point itself. The BCS of a line segment (1-simplex) consists of two smaller segments, each connecting one endpoint (0-dimensional face) of to the midpoint of itself (1-dimensional face). The BCS of a triangle divides it into six triangles; each part has one vertex at the barycenter of , another one at the midpoint of some side, and the last one at one of the original vertices. The BCS of a tetrahedron divides it into 24 tetrahedra; each part has one vertex at the center of , one on some face, one along some edge, and the last one at some vertex of . An important feature of BCS is the fact that the maximal diameter of an dimensional simplex shrinks at least by the factor .〔 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「barycentric subdivision」の詳細全文を読む スポンサード リンク
|