# Singular

#### D.13.2.5 secondaryPolytope

Procedure from library `polymake.lib` (see polymake_lib).

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

Assume:
- 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

Purpose:
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

Return:
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

Note:
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
http://www.rambau.wm.uni-bayreuth.de/TOPCOM/);

Example:
 ```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 print(secpoly[1]); ==> 1 2 2 1 ==> 2 1 1 2 // the corresponding triangulations secpoly[2]; ==> [1]: ==> [1]: ==> 1,2,3 ==> [2]: ==> 2,3,4 ==> [2]: ==> [1]: ==> 1,3,4 ==> [2]: ==> 1,2,4 ```