forum.polymake.org

Posted: **30 May 2012, 19:17**

Hello,

I have a question regarding triangulations of point configurations and polytopes in polymake. Assume we have a point configuration $pc and $Q is its convex hull (represented by a polytope) in polymake. For example:

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$pc=new PointConfiguration<Rational> (POINTS=>[[1,0,0],[1,1,2],[1,4,2],[1,2,1]],  WEIGHTS=>[0,0,0,-1], WEIGHTS_GENERIC=>1);
$Q=new Polytope<Rational>(POINTS=>$pc->POINTS);

Given a face $f of the boundary of the triangulation of $pc I would like to find the unique face $F of $Q, of the same dimension, that $f lies. Then I would like to find all the points of $pc that lie on $F. Is there a simple way to do this in polymake?

Thanks a lot,
Vissarion

Posted: **31 May 2012, 00:26**

It seems you are using an older version of polymake, as WEIGHTS has been declared as a subproperty of TRIANGULATION in the newest. The following should also work with your version though.

You can access the convex hull of a point configuration $pc from within $pc by asking for its CONVEX_HULL. This property contains the convex hull of $pc is of type polytope, so it has all properties and methods you expect from a usual polytope (e.g. $pc->CONVEX_HULL->FACETS gives the facets). It has one additional property: $pc->CONVEX_HULL->VERTEX_POINT_MAP gives a map that assigns each vertex of the convex hull the index of the corresponding point in $pc->POINTS.

There is, however, no direct solution to your question. You will need some lines of perl within the shell. To find the facet of the convex hull a given set of points is contained in you would have to loop over all facets of $pc->CONVEX_HULL until you find one such that the scalar product of the facet with all points in the face evaluates to 0. The second question is similar: for a given facet loop over all points and store those that are on the facet.

I could suggest some lines of perl for this if wanted.

Posted: **31 May 2012, 09:56**

The following is a slight variation/addition to the solution suggested above.

Suppose you have a cube and some triangulation. This induces a triangulation of the boundary.

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$c=cube(3);
$t=$c->TRIANGULATION->BOUNDARY;

The following (naive) code checks all pairs of facets from the triangulation and the polytope. Extensions and speed up easily possible.

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foreach my $bnd_triangle (@{$t->FACETS}) {
  foreach my $poly_facet (@{$c->VERTICES_IN_FACETS}) {
    print $bnd_triangle, " in ", $poly_facet, "\n" if incl($bnd_triangle,$poly_facet)<0;
  }
}

The output of the latter is:

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{0 1 2} in {0 1 2 3}
{0 1 4} in {0 1 4 5}
{0 2 4} in {0 2 4 6}
{1 2 3} in {0 1 2 3}
{1 3 5} in {1 3 5 7}
{1 4 5} in {0 1 4 5}
{2 3 6} in {2 3 6 7}
{2 4 6} in {0 2 4 6}
{4 5 6} in {4 5 6 7}
{3 5 7} in {1 3 5 7}
{3 6 7} in {2 3 6 7}
{5 6 7} in {4 5 6 7}

Of course, if you work with a PointConfiguration instead of a Polytope you additionally need to employ CONVEX_HULL and VERTEX_POINT_MAP as indicated above.

Posted: **31 May 2012, 17:46**

Thanks a lot for your quick reply and your suggestions!

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