# Working with polyhedra

Just like cones, polyhedra have two descriptions. One description as the convex hull of finitely many points (and optionally rays and lineality), the V-representation. Another description as the intersection of finitely many half-spaces, the H-representation. Using the method convexHull we can create a polyhedron in 2-space which is the convexHull of a given set of points.

 i1 : V = matrix {{0,2,-2,0},{-1,1,1,1}} o1 = | 0 2 -2 0 | | -1 1 1 1 | 2 4 o1 : Matrix ZZ <--- ZZ i2 : P = convexHull V o2 = P o2 : Polyhedron

Polyhedra uses the principle of lazy evaluation: Properties of the combinatorial objects are only computed on demand and then they are stored with the object. For example we can ask for the vertices of P using vertices:

 i3 : vertices P o3 = | 0 -2 2 | | -1 1 1 | 2 3 o3 : Matrix QQ <--- QQ

Here we see that the point (0,1) is not a vertex and P is actually a triangle.

 i4 : (HS,v) = facets P o4 = (| -1 -1 |, | 1 |) | 1 -1 | | 1 | | 0 1 | | 1 | o4 : Sequence

This gives the defining affine half-spaces, i.e. P is given by all p such that HS*p <= v and that lie in the defining affine hyperplanes. The rows of the matrix HS are the outer normals of the polyhedron P. To get the defining hyperplanes we use:

 i5 : hyperplanes P o5 = (0, 0) o5 : Sequence

There are none, so the polyhedron is of full dimension. It is also compact, since P has no rays and the lineality space is of dimension zero.

 i6 : isFullDimensional P o6 = true i7 : ambDim P o7 = 2 i8 : dim P o8 = 2 i9 : rays P o9 = 0 2 o9 : Matrix QQ <--- 0 i10 : linealitySpace P o10 = 0 2 o10 : Matrix QQ <--- 0

Internally, polyhedra are realized as cones, by embedding the polyhedron at height one and then taking the positive hull. To get at this cone, use cone. The height is the first coordinate of the rays of the cone, comparing the matrices of rays and vertices for the example one can see the correspondence:

 i11 : C = cone P o11 = C o11 : Cone i12 : rays C o12 = | 1 1 1 | | 0 -2 2 | | -1 1 1 | 3 3 o12 : Matrix ZZ <--- ZZ i13 : vertices P o13 = | 0 -2 2 | | -1 1 1 | 2 3 o13 : Matrix QQ <--- QQ

We can also construct the convex hull of a set of points and a set of rays.

 i14 : R = matrix {{1},{0},{0}} o14 = | 1 | | 0 | | 0 | 3 1 o14 : Matrix ZZ <--- ZZ i15 : V1 = V || matrix {{1,1,1,1}} o15 = | 0 2 -2 0 | | -1 1 1 1 | | 1 1 1 1 | 3 4 o15 : Matrix ZZ <--- ZZ i16 : P1 = convexHull(V1,R) o16 = P1 o16 : Polyhedron i17 : vertices P1 o17 = | 0 -2 | | -1 1 | | 1 1 | 3 2 o17 : Matrix QQ <--- QQ

This polyhedron is not compact anymore and also not of full dimension.

 i18 : isCompact P1 o18 = false i19 : isFullDimensional P1 o19 = false i20 : rays P1 o20 = | 1 | | 0 | | 0 | 3 1 o20 : Matrix QQ <--- QQ i21 : hyperplanes P1 o21 = (| 0 0 -1 |, | -1 |) o21 : Sequence

On the other hand we can construct a polyhedron as the intersection of affine half-spaces and affine hyperplanes, given via inequalities and equations:

 i22 : inequalities = transpose (V || matrix {{-1,2,0,1}}) o22 = | 0 -1 -1 | | 2 1 2 | | -2 1 0 | | 0 1 1 | 4 3 o22 : Matrix ZZ <--- ZZ i23 : v = matrix {{1},{1},{1},{1}} o23 = | 1 | | 1 | | 1 | | 1 | 4 1 o23 : Matrix ZZ <--- ZZ i24 : equations = matrix {{1,1,1}} o24 = | 1 1 1 | 1 3 o24 : Matrix ZZ <--- ZZ i25 : w = matrix {{3}} o25 = | 3 | 1 1 o25 : Matrix ZZ <--- ZZ i26 : P2 = polyhedronFromHData(inequalities,v,equations,w) o26 = P2 o26 : Polyhedron

This is a triangle in 3-space with the following vertices.

 i27 : isFullDimensional P2 o27 = false i28 : vertices P2 o28 = | 4 4 2 | | 9 5 5 | | -10 -6 -4 | 3 3 o28 : Matrix QQ <--- QQ

If we don't intersect with the hyperplane we get a full dimensional polyhedron.

 i29 : P3 = polyhedronFromHData(inequalities,v) o29 = P3 o29 : Polyhedron i30 : vertices P3 o30 = | 0 0 0 | | 1 1 -3 | | 0 -2 2 | 3 3 o30 : Matrix QQ <--- QQ i31 : linealitySpace P3 o31 = | 1 | | 2 | | -2 | 3 1 o31 : Matrix QQ <--- QQ i32 : isFullDimensional P3 o32 = true

Note that the vertices are given modulo the lineality space. Besides constructing polyhedra by hand, there are also some basic polyhedra implemented such as the hypercube, in this case with edge-length four.

 i33 : P4 = hypercube(3,2) o33 = P4 o33 : Polyhedron i34 : vertices P4 o34 = | -2 2 -2 2 -2 2 -2 2 | | -2 -2 2 2 -2 -2 2 2 | | -2 -2 -2 -2 2 2 2 2 | 3 8 o34 : Matrix QQ <--- QQ

Another on is the crossPolytope, in this case with diameter six.

 i35 : P5 = crossPolytope(3,3) o35 = P5 o35 : Polyhedron i36 : vertices P5 o36 = | -3 3 0 0 0 0 | | 0 0 -3 3 0 0 | | 0 0 0 0 -3 3 | 3 6 o36 : Matrix QQ <--- QQ

Furthermore the standard simplex (stdSimplex).

 i37 : P6 = stdSimplex 2 o37 = P6 o37 : Polyhedron i38 : vertices P6 o38 = | 1 0 0 | | 0 1 0 | | 0 0 1 | 3 3 o38 : Matrix QQ <--- QQ

Now that we can construct polyhedra, we can turn to the functions that can be applied to polyhedra. First of all, we can apply the convexHull function also to a pair of polyhedra:

 i39 : P7 = convexHull(P4,P5) o39 = P7 o39 : Polyhedron i40 : vertices P7 o40 = | -3 3 0 0 0 -2 2 -2 2 -2 2 -2 2 0 | | 0 0 -3 3 0 -2 -2 2 2 -2 -2 2 2 0 | | 0 0 0 0 -3 -2 -2 -2 -2 2 2 2 2 3 | 3 14 o40 : Matrix QQ <--- QQ

Or we can intersect them by using intersection:

 i41 : P8 = intersection(P4,P5) o41 = P8 o41 : Polyhedron i42 : vertices P8 o42 = | -1 1 -2 2 -2 2 -1 1 -1 1 0 0 -2 2 0 0 -2 2 0 0 -1 1 0 0 | | -2 -2 -1 -1 1 1 2 2 0 0 -1 1 0 0 -2 2 0 0 -2 2 0 0 -1 1 | | 0 0 0 0 0 0 0 0 -2 -2 -2 -2 -1 -1 -1 -1 1 1 1 1 2 2 2 2 | 3 24 o42 : Matrix QQ <--- QQ

Furthermore, both functions can be applied to a list containing any number of polyhedra and matrices defining vertices/rays or affine half-spaces/hyperplanes. All of these must be in the same ambient space. For example:

 i43 : P9 = convexHull {(V1,R),P2,P6} o43 = P9 o43 : Polyhedron i44 : vertices P9 o44 = | 4 4 2 0 -2 | | 9 5 5 -1 1 | | -10 -6 -4 1 1 | 3 5 o44 : Matrix QQ <--- QQ

Further functions are for example the Minkowski sum (minkowskiSum) of two polyhedra.

 i45 : Q = convexHull (-V) o45 = Q o45 : Polyhedron i46 : P10 = P + Q o46 = P10 o46 : Polyhedron i47 : vertices P10 o47 = | -4 4 -2 2 -2 2 | | 0 0 -2 -2 2 2 | 2 6 o47 : Matrix QQ <--- QQ

In the other direction, we can also determine all Minkowski summands (see minkSummandCone) of a polyhedron.

 i48 : (C,L,M) = minkSummandCone P10 o48 = (C, HashTable{0 => Polyhedron{...1...}}, | 1 0 |) 1 => Polyhedron{...1...} | 0 1 | 2 => Polyhedron{...1...} | 1 0 | 3 => Polyhedron{...1...} | 1 0 | 4 => Polyhedron{...1...} | 0 1 | o48 : Sequence i49 : apply(values L, vertices) o49 = {| 0 4 |, | 0 4 2 |, | 0 2 |, | 0 2 |, | 0 4 2 |} | 0 0 | | 0 0 -2 | | 0 2 | | 0 -2 | | 0 0 2 | o49 : List

Here the polyhedra in the hash table L are all possible Minkowski summands up to scalar multiplication and the columns of M give the minimal decompositions. So the hexagon P10 is not only the sum of two triangles but also the sum of three lines. Furthermore, we can take the direct product of two polyhedra.

 i50 : P11 = P * Q o50 = P11 o50 : Polyhedron i51 : vertices P11 o51 = | 0 -2 2 0 -2 2 0 -2 2 | | -1 1 1 -1 1 1 -1 1 1 | | -2 -2 -2 2 2 2 0 0 0 | | -1 -1 -1 -1 -1 -1 1 1 1 | 4 9 o51 : Matrix QQ <--- QQ

The result is in QQ^4.

 i52 : ambDim P11 o52 = 4

 i53 : fVector P11 o53 = {9, 18, 15, 6, 1} o53 : List

The function fVector gives the number of faces of each dimension, so it has 9 vertices, 18 edges and so on. We can access the faces of a certain codimension via:

 i54 : L = faces(1,P11) o54 = {({0, 1, 3, 4, 6, 7}, {}), ({0, 2, 3, 5, 6, 8}, {}), ({1, 2, 4, 5, 7, ----------------------------------------------------------------------- 8}, {}), ({0, 1, 2, 3, 4, 5}, {}), ({0, 1, 2, 6, 7, 8}, {}), ({3, 4, 5, ----------------------------------------------------------------------- 6, 7, 8}, {})} o54 : List i55 : vertP11 = vertices P11 o55 = | 0 -2 2 0 -2 2 0 -2 2 | | -1 1 1 -1 1 1 -1 1 1 | | -2 -2 -2 2 2 2 0 0 0 | | -1 -1 -1 -1 -1 -1 1 1 1 | 4 9 o55 : Matrix QQ <--- QQ i56 : apply(L, l -> vertP11_(l#0)) o56 = {| 0 -2 0 -2 0 -2 |, | 0 2 0 2 0 2 |, | -2 2 -2 2 -2 2 |, | | -1 1 -1 1 -1 1 | | -1 1 -1 1 -1 1 | | 1 1 1 1 1 1 | | | -2 -2 2 2 0 0 | | -2 -2 2 2 0 0 | | -2 -2 2 2 0 0 | | | -1 -1 -1 -1 1 1 | | -1 -1 -1 -1 1 1 | | -1 -1 -1 -1 1 1 | | ----------------------------------------------------------------------- 0 -2 2 0 -2 2 |, | 0 -2 2 0 -2 2 |, | 0 -2 2 0 -2 2 |} -1 1 1 -1 1 1 | | -1 1 1 -1 1 1 | | -1 1 1 -1 1 1 | -2 -2 -2 2 2 2 | | -2 -2 -2 0 0 0 | | 2 2 2 0 0 0 | -1 -1 -1 -1 -1 -1 | | -1 -1 -1 1 1 1 | | -1 -1 -1 1 1 1 | o56 : List

We can compute all lattice points of the polyhedron with latticePoints.

 i57 : L = latticePoints P11 o57 = {| 1 |, | -2 |, | 2 |, | 0 |, | 1 |, | -1 |, | 1 |, | -1 |, | 0 | 0 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 0 | | 0 | -2 | | -2 | | -2 | | 2 | | 2 | | -2 | | -2 | | -2 | | -2 | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 ----------------------------------------------------------------------- |, | 0 |, | 0 |, | 0 |, | 0 |, | 1 |, | -1 |, | 0 |, | 0 |, | 0 | | -1 | | 1 | | -1 | | -1 | | 0 | | 0 | | 0 | | -1 | | -1 | | -2 | | -2 | | -1 | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | -1 | | -1 | | -1 | | 0 | | -1 | | -1 | | -1 | | -1 | | 0 ----------------------------------------------------------------------- |, | 1 |, | -1 |, | 0 |, | 0 |, | -2 |, | 2 |, | -1 |, | 1 |, | 0 | | 0 | | 0 | | 0 | | -1 | | 1 | | 1 | | 1 | | 1 | | 1 | | -1 | | -1 | | -1 | | 0 | | -1 | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | 0 | | 1 | | -1 | | -1 | | -1 | | -1 | | -1 ----------------------------------------------------------------------- |, | 1 |, | -1 |, | 0 |, | 0 |, | 0 |, | 1 |, | -1 |, 0, | -2 |, | | | 0 | | 0 | | 0 | | -1 | | -1 | | 0 | | 0 | | 1 | | | | 0 | | 0 | | 0 | | 1 | | 1 | | 0 | | 0 | | -1 | | | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | 0 | | 0 | | ----------------------------------------------------------------------- 2 |, | -1 |, | 1 |, | 0 |, | 1 |, | -1 |, | 0 |, | -2 |, | 2 |, | 1 | | 1 | | 1 | | 1 | | 0 | | 0 | | 0 | | 1 | | 1 | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 1 | | 1 | | 1 | | -1 | | -1 | | ----------------------------------------------------------------------- -1 |, | 1 |, | 0 |, | 1 |, | -1 |, | 0 |, | 0 |, | 1 |, | -1 |, | 1 | | 1 | | 1 | | 0 | | 0 | | 0 | | -1 | | 0 | | 0 | | 0 | | 0 | | 0 | | 1 | | 1 | | 1 | | 2 | | 1 | | 1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | ----------------------------------------------------------------------- 0 |, | -2 |, | 2 |, | -1 |, | 1 |, | 0 |, | -2 |, | 2 |, | -1 |, | 1 |, 0 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | 1 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 1 | | 1 | | 1 | | 1 | ----------------------------------------------------------------------- | 0 |, | -2 |, | 2 |, | -1 |, | 1 |, | 0 |, | 1 |, | -1 |, | 0 |, | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 0 | | 0 | | 0 | | 0 | | 1 | | 1 | | 1 | | 1 | | 1 | | 2 | | 2 | | 2 | | 1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | ----------------------------------------------------------------------- | -2 |, | 2 |, | -1 |, | 1 |, | 0 |, | -2 |, | 2 |, | -1 |} | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 2 | | 2 | | 2 | | 0 | | 0 | | 0 | | 0 | | 0 | | -1 | | -1 | | -1 | o57 : List i58 : #L o58 = 81

Evenmore the tail/recession cone of a polyhedron with tailCone.

 i59 : C = tailCone P1 o59 = C o59 : Cone i60 : rays C o60 = | 1 | | 0 | | 0 | 3 1 o60 : Matrix ZZ <--- ZZ

Finally, there is also a function to compute the polar of a polyhedron, i.e. all points in the dual space that are greater than -1 on all points of the polyhedron:

 i61 : P12 = polar P11 o61 = P12 o61 : Polyhedron i62 : vertices P12 o62 = | 1 -1 0 0 0 0 | | 1 1 -1 0 0 0 | | 0 0 0 0 1 -1 | | 0 0 0 1 -1 -1 | 4 6 o62 : Matrix QQ <--- QQ