# regularSubdivision -- Computes the regular cell decomposition

## Synopsis

• Usage:
L = regularSubdivision(P,w)
L = regularSubdivision(M,w)
• Inputs:
• P, ,
• w, , Row matrix containing the weights.
• M, , Matrix containing the points.
• Outputs:
• L, a list, List of polyhedra or List of lists of indices indicated which points form a cell.

## Description

This function computes the regular subdivision of P given by the weight vector w. This is computed by placing the i-th lattice point of P on height w_i in n+1 space, taking the convexHull of these with the ray (0,...,0,1), and projecting the compact faces into n space. Note that the polyhedron must be compact, i.e. a polytope and the length of the weight vector must be the number of lattice points.

This function can also be used to compute the regular subdivision given a matrix M of points and a weight vector w. The points are lifted to the weights given by the matrix w, and the lower envelope is computed.

 i1 : P = crossPolytope 3 o1 = P o1 : Polyhedron i2 : w = matrix {{1,2,2,2,2,2,1}} o2 = | 1 2 2 2 2 2 1 | 1 7 o2 : Matrix ZZ <--- ZZ i3 : L = regularSubdivision(P,w) o3 = {Polyhedron{...1...}, Polyhedron{...1...}, Polyhedron{...1...}, ------------------------------------------------------------------------ Polyhedron{...1...}, Polyhedron{...1...}, Polyhedron{...1...}, ------------------------------------------------------------------------ Polyhedron{...1...}, Polyhedron{...1...}} o3 : List i4 : apply(L,vertices) o4 = {| -1 1 0 |, | -1 1 0 |, | 1 0 0 |, | 0 1 0 0 |, | 0 -1 0 0 |, | -1 | 0 0 1 | | 0 0 0 | | 0 -1 1 | | 0 0 -1 0 | | 0 0 1 0 | | 0 | 0 0 0 | | 0 0 1 | | 0 0 0 | | 0 0 0 1 | | 0 0 0 -1 | | 0 ------------------------------------------------------------------------ 0 0 |, | 0 0 0 |, | 0 0 0 |} 0 0 | | -1 1 0 | | -1 0 0 | -1 1 | | 0 0 -1 | | 0 -1 1 | o4 : List
 i5 : M = matrix {{1,0,1,0},{1,1,0,0}}; 2 4 o5 : Matrix ZZ <--- ZZ i6 : w = matrix {{1,0,0,1}}; 1 4 o6 : Matrix ZZ <--- ZZ i7 : S = regularSubdivision (M,w) o7 = {{0, 1, 2}, {0, 1, 3}} o7 : List

## Ways to use regularSubdivision :

• "regularSubdivision(Matrix,Matrix)"
• "regularSubdivision(Polyhedron,Matrix)"

## For the programmer

The object regularSubdivision is .