We start with a polyhedron in 2-space which is the
convexHull of a given set of points.
V = matrix {{0,2,-2,0},{-1,1,1,1}} |
P = convexHull V |
This gives an overview of the characteristics of the polyhedron. If we want to know more details, we can ask for them.
Here we see that the point (0,1) is not a vertex and
P is actually a triangle.
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. To get the hyperplanes we use:
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.
Furthermore, we can construct the convex hull of a set of points and a set of rays.
R = matrix {{1},{0},{0}} |
V1 = V || matrix {{1,1,1,1}} |
P1 = convexHull(V1,R) |
vertices P1 |
This polyhedron is not compact anymore and also not of full dimension.
On the other hand we can construct a polyhedron as the
intersection of affine half-spaces and affine hyperplanes.
HS = transpose (V || matrix {{-1,2,0,1}}) |
v = matrix {{1},{1},{1},{1}} |
hyperplanesTmp = matrix {{1,1,1}} |
w = matrix {{3}} |
P2 = intersection(HS,v,hyperplanesTmp,w) |
This is a triangle in 3-space with the following vertices.
If we don't intersect with the hyperplane we get a full dimensional polyhedron.
P3 = intersection(HS,v) |
vertices P3 |
linealitySpace P3 |
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.
P4 = hypercube(3,2) |
vertices P4 |
Another on is the
crossPolytope, in this case with diameter six.
P5 = crossPolytope(3,3) |
vertices P5 |
Furthermore the standard simplex (
stdSimplex).
P6 = stdSimplex 2 |
vertices P6 |
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:
P7 = convexHull(P4,P5) |
vertices P7 |
Or we can intersect them by using
intersection:
P8 = intersection(P4,P5) |
vertices P8 |
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:
P9 = convexHull {(V1,R),P2,P6} |
vertices P9 |
Further functions are for example the Minkowski sum (
minkowskiSum) of two polyhedra.
Q = convexHull (-V) |
P10 = P + Q |
vertices P10 |
In the other direction, we can also determine all Minkowski summands (see
minkSummandCone) of a polyhedron.
(C,L,M) = minkSummandCone P10 |
apply(values L, vertices) |
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.
The result is in QQ^4.
To find out more about this polyhedron use for example.
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:
L = faces(1,P11) |
vertP11 = vertices P11 |
apply(L, l -> vertP11_(l#0)) |
We can compute all lattice points of the polyhedron with
latticePoints.
Evenmore the tail/recession cone of a polyhedron with
tailCone.
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:
P12 = polar P11 |
vertices P12 |