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D.13.2.5 secondaryPolytope

Procedure from library polymake.lib (see polymake_lib).

secondaryPolytope(polygon[,#]); list polygon, list #

- polygon is a list of integer vectors of the same size representing the affine coordinates of lattice points
- if the triangulations of the corresponding polygon have already been computed with the procedure triangulations then these can be given as a second (optional) argument in order to avoid doing this computation again

the procedure considers the marked polytope given as the convex hull of the lattice points and with these lattice points as markings; it then computes the lattice points of the secondary polytope given by this marked polytope which correspond to the triangulations computed by the procedure triangulations

list, say L, such that:
L[1] = intmat, each row gives the affine coordinates of a lattice point in the secondary polytope given by the marked
polytope corresponding to polygon
L[2] = the list of corresponding triangulations

if the triangluations are not handed over as optional argument the procedure calls for its computation of these triangulations the program points2triangs from the program topcom by Joerg Rambau, Universitaet Bayreuth; it therefore is necessary that this program is installed in order to use this procedure; see

LIB "polymake.lib";
==> Welcome to polymake version
==> Copyright (c) 1997-2015
==> Ewgenij Gawrilow, Michael Joswig (TU Darmstadt)
==> http://www.polymake.org
// the lattice points of the unit square in the plane
list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1);
// the secondary polytope of this lattice point configuration is computed
list secpoly=secondaryPolytope(polygon);
==> Evaluating Commandline Options ...
==> ... done.
==> 0
==> 0
// the points in the secondary polytope
==>      1     2     2     1
==>      2     1     1     2
// the corresponding triangulations
==> [1]:
==>    [1]:
==>       1,2,3
==>    [2]:
==>       2,3,4
==> [2]:
==>    [1]:
==>       1,3,4
==>    [2]:
==>       1,2,4