Singular

D.13.2.8 secondaryFan

Usage:

secondaryFan(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, the ith entry of L[1] contains information about the ith cone in the secondary fan of the polygon, i.e. the cone dual to the ith vertex of the secondary polytope
L[1][i][1] = integer matrix representing the inequalities which describe the cone dual to the ith vertex
L[1][i][2] = a list which contains the inequalities represented by L[1][i][1] as a list of strings, where we use the
variables x(1),...,x(n)
L[1][i][3] = only present if 'er' is set to 1; in that case it is an interger matrix whose rows are the extreme rays
of the cone
L[2] = is an integer matrix whose rows span the linearity space of the fan, i.e. the linear space which is contained in each cone
L[3] = the secondary polytope in the format of the procedure polymakePolytope
L[4] = the list of triangulations corresponding to the vertices of the secondary polytope

Note:

- the procedure calls for its computation polymake by Ewgenij Gawrilow, TU Berlin and Michael Joswig, so it only works if polymake is installed; see http://www.polymake.org/
- in the optional argument # it is possible to hand over other names for the variables to be used -- be careful, the format must be correct and that is not tested, e.g. if you want the variable names to be u00,u10,u01,u11 then you must hand over the string 'u11,u10,u01,u11'
- 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/);

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 secfan=secondaryFan(polygon);
==> Evaluating Commandline Options ...
==> ... done.
==> 0
==> 0
==> polymake: used package ppl
==>   The Parma Polyhedra Library (PPL): A C++ library for convex polyhedra
==>   and other numerical abstractions.
==>   http://www.cs.unipr.it/ppl/
==> 
// the number of cones in the secondary fan of the polygon
size(secfan[1]);
==> 2
// the inequalities of the first cone as matrix are:
print(secfan[1][1][1]);
==> 1,-1,-1,1
// the inequalities of the first cone as string are:
print(secfan[1][1][2]);
==> [1]:
==>    x(1)+x(4) > x(2)+x(3)
// the rows of the following matrix are the extreme rays of the first cone:
print(secfan[1][1][3]);
==> 1,-1,-1,1
// each cone contains the linearity space spanned by:
print(secfan[2]);
==> 0,0,-1,-1,
==> 1,0, 0,-1,
==> 0,1,-1, 0
// the points in the secondary polytope
print(secfan[3][1]);
==> 2,1,1,2,
==> 1,2,2,1
// the corresponding triangulations
secfan[4];
==> [1]:
==>    [1]:
==>       1,2,3
==>    [2]:
==>       2,3,4
==> [2]:
==>    [1]:
==>       1,3,4
==>    [2]:
==>       1,2,4