another questions re: vert(hom(p, q))
Posted: 27 Dec 2012, 00:03
Can polymake express a vertex map of hom(p, q) explicitly as a map, i.e., as a linear-map-matrix and a shift vector?
Thank you!
Thank you!
That matrix encodes a projective transformation (yielding affine coordinates, however), such that it can be applied to points in the polytope corresponding to the first argument directly.
Here is a use case (which will become part of the interactive help in the next polymake release):
Let us look at the mapping polytope of the unit interval and the standard unimodular triangle.
Code: Select all
> $I=simplex(1); $T=simplex(2); $Hom_IT=mapping_polytope($I,$T);
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> print $Hom_IT->DIM
4
> print rows_labeled($Hom_IT->FACETS,$Hom_IT->FACET_LABELS);
v0*F0:0 1 0 0 0
v0*F1:0 0 0 1 0
v0*F2:1 -1 0 -1 0
v1*F0:0 1 1 0 0
v1*F1:0 0 0 1 1
v1*F2:1 -1 -1 -1 -1
> print $Hom_IT->N_VERTICES;
9
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> $v=$Hom_IT->VERTICES->[0];
> $M=new Matrix([$v->slice([1..2]),$v->slice([3..4])]);
> print $I->VERTICES * transpose($M);
0 0
1 0