Barycentric subdivision

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. 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. The barycentric subdivision (henceforth BCS) of an n {displaystyle n} -dimensional simplex S {displaystyle S} consists of (n + 1)! n {displaystyle n} -dimensional simplices. Each piece, with vertices v 0 , v 1 , … , v n {displaystyle v_{0},v_{1},dots ,v_{n}} , can be associated with a permutation p 0 , p 1 , … , p n {displaystyle p_{0},p_{1},dots ,p_{n}} of the vertices of S {displaystyle S} , in such a way that each vertex v i {displaystyle v_{i}} is the barycenter of the points p 0 , p 1 , … , p i {displaystyle p_{0},p_{1},dots ,p_{i}} . 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) S {displaystyle S} consists of two smaller segments, each connecting one endpoint (0-dimensional face) of S {displaystyle S} to the midpoint of S {displaystyle S} itself (1-dimensional face). The BCS of a triangle S {displaystyle S} divides it into six triangles; each part has one vertex v 2 {displaystyle v_{2}} at the barycenter of S {displaystyle S} , another one v 1 {displaystyle v_{1}} at the midpoint of some side, and the last one v 0 {displaystyle v_{0}} at one of the original vertices. The BCS of a tetrahedron S {displaystyle S} divides it into 24 tetrahedra; each part has one vertex at the center of S {displaystyle S} , one on some face, one along some edge, and the last one at some vertex of S {displaystyle S} . An important feature of BCS is the fact that the maximal diameter of an n {displaystyle n} -dimensional simplex shrinks at least by the factor n n + 1 {displaystyle {frac {n}{n+1}}} . Another way of defining the BCS of a simplex S {displaystyle S} is to associate each part to a sequence F 0 , F 1 , … , F n {displaystyle F_{0},F_{1},dots ,F_{n}} of faces of S {displaystyle S} , with increasing dimensions, such that F i {displaystyle F_{i}} is a facet of F i + 1 {displaystyle F_{i+1}} , for i {displaystyle i} from 0 to n − 1 {displaystyle n-1} . Then each vertex v i {displaystyle v_{i}} of the corresponding piece is the barycenter of face F i {displaystyle F_{i}} .