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
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i3 : L = regularSubdivision(P,w)
o3 = {Polyhedron{...1...}, Polyhedron{...1...}, Polyhedron{...1...},
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Polyhedron{...1...}, Polyhedron{...1...}, Polyhedron{...1...},
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Polyhedron{...1...}, Polyhedron{...1...}}
o3 : List
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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
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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
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i5 : M = matrix {{1,0,1,0},{1,1,0,0}};
2 4
o5 : Matrix ZZ <--- ZZ
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i6 : w = matrix {{1,0,0,1}};
1 4
o6 : Matrix ZZ <--- ZZ
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i7 : S = regularSubdivision (M,w)
o7 = {{0, 1, 2}, {0, 1, 3}}
o7 : List
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The object regularSubdivision is a method function.