i1 : convexHull(matrix {{0,0,-1,-1},{2,-2,1,-1},{0,0,0,0}},matrix {{1},{0},{0}}) o1 = {ambient dimension => 3 } dimension of lineality space => 0 dimension of polyhedron => 2 number of facets => 5 number of rays => 1 number of vertices => 4 o1 : Polyhedron |
This table displays a short summary of the properties of the Polyhedron. Note that the number of rays and vertices are modulo the lineality space. So for example a line in QQ^2 has one vertex and no rays. However, one can not access the above information directly, because this is just a virtual hash table generated for the output. The data defining a Polyhedron is extracted by the functions included in this package. A Polyhedron can be constructed as the convex hull (convexHull) of a set of points and a set of rays or as the intersection (intersection) of a set of affine half-spaces and affine hyperplanes.
For example, consider the square and the square with an emerging ray for the convex hull:
i2 : V = matrix {{1,1,-1,-1},{1,-1,1,-1}} o2 = | 1 1 -1 -1 | | 1 -1 1 -1 | 2 4 o2 : Matrix ZZ <--- ZZ |
i3 : convexHull V o3 = {ambient dimension => 2 } dimension of lineality space => 0 dimension of polyhedron => 2 number of facets => 4 number of rays => 0 number of vertices => 4 o3 : Polyhedron |
i4 : R = matrix {{1},{1}} o4 = | 1 | | 1 | 2 1 o4 : Matrix ZZ <--- ZZ |
i5 : convexHull(V,R) o5 = {ambient dimension => 2 } dimension of lineality space => 0 dimension of polyhedron => 2 number of facets => 4 number of rays => 1 number of vertices => 3 o5 : Polyhedron |
If we take the intersection of the half-spaces defined by the directions of the vertices and 1 we get the crosspolytope:
i6 : HS = transpose V o6 = | 1 1 | | 1 -1 | | -1 1 | | -1 -1 | 4 2 o6 : Matrix ZZ <--- ZZ |
i7 : v = R || R o7 = | 1 | | 1 | | 1 | | 1 | 4 1 o7 : Matrix ZZ <--- ZZ |
i8 : P = intersection(HS,v) o8 = {ambient dimension => 2 } dimension of lineality space => 0 dimension of polyhedron => 2 number of facets => 4 number of rays => 0 number of vertices => 4 o8 : Polyhedron |
i9 : vertices P o9 = | -1 1 0 0 | | 0 0 -1 1 | 2 4 o9 : Matrix QQ <--- QQ |
This can for example be embedded in 3-space on height 1:
i10 : HS = HS | matrix {{0},{0},{0},{0}} o10 = | 1 1 0 | | 1 -1 0 | | -1 1 0 | | -1 -1 0 | 4 3 o10 : Matrix ZZ <--- ZZ |
i11 : HP = matrix {{0,0,1}} o11 = | 0 0 1 | 1 3 o11 : Matrix ZZ <--- ZZ |
i12 : w = matrix {{1}} o12 = | 1 | 1 1 o12 : Matrix ZZ <--- ZZ |
i13 : P = intersection(HS,v,HP,w) o13 = {ambient dimension => 3 } dimension of lineality space => 0 dimension of polyhedron => 2 number of facets => 4 number of rays => 0 number of vertices => 4 o13 : Polyhedron |
i14 : vertices P o14 = | -1 1 0 0 | | 0 0 -1 1 | | 1 1 1 1 | 3 4 o14 : Matrix QQ <--- QQ |
See also Working with polyhedra.
The object Polyhedron is a type, with ancestor classes PolyhedralObject < HashTable < Thing.