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 |
To find out more about this polyhedron use for example.
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
|