This package is a small, fast implementation of an algorithm for finding extremal rays of a polyhedral cone, with filtering. It is intended for finding normal surfaces in triangulated 3-manifolds, and therefore does not implement various features that might be useful for general extremal ray problems.

The setup is this. Define the support of a vector v in R^n to be the set of indices i such that v_i is non-zero. We are given an integer matrix M, typically with many more columns than rows, and a list of "illegal supports". The support of a vector is illegal if its support contains one of the illegal supports on the list.

We want to find all the extremal rays of the cone
(Null space of M) intersect (positive orthant),

which are generated by vectors with legal support. (The restriction to vector with legal support is what is meant by "filtering".) In the 3-manifold application the matrix describes the normal surface equations in quad form, and the illegal supports include all pairs of distinct quads in the same 3-simplex. (They could also include various types of "obvious compressions".)

The algorithm is due to Dave Letscher, and incorporates ideas of Komei Fukuda's. One constructs a sequence P_n of polyhedra as follows.

P_0 is the standard n-simplex.

Given P_n-1, construct P_n as follows. Take the hyperplane H_n defined by the nth row of the matrix M. Divide the vertices of P_n-1 into three classes: positive, negative and neutral, according to whether they lie on the positive side, the negative side, or are contained in, H_n. For each pair consisting of a positive vertex v and a negative vertex w we intersect H with the segment joining v to w. The intersection point u will give rise to a vertex of P_n if it satisfies two conditions:

  1. the support of u is legal; and
  2. the submatrix of M_u of M has nullity 1, where M_u consists of elements of which are contained in the first n rows and in the columns which are indexed by the support of u.

If u satisfies these conditions then its smallest integral multiple is a vertex of P_n. The neutral vertices of P_n-1 are also vertices of P_n.

If M has k rows, then the vertices of P_k generate the set of extremal rays of the cone which have legal supports.

Installation Instructions

This package can be built either as a standalone C module, which you can link with your own C code, or as a python module. The python module is needed by t3m.

To build the C module type make in the distribution directory. The code will run much faster. The object file FXrays.o and a number of test programs will be built.

To build the python module, if you have superuser privileges on your UNIX system, type:

python install

This will install FXrays in the site-packages directory of your python installation.

If you are not a superuser and wish to install the FXrays python module into your user site-packages (as described in do

python build --user

NOTE: On Windows, FXrays does not build correctly with mingw64. There are segfaults caused by linking with msvcrt instead of msvcr90. However, building a 64 bit FXrays with the default msvc compiler (in the Microsoft Visual C++ for Python 2.7 package) works.

Bugs and Comments

Please email bugs, comments, or offers of assistance, to either or both of:


Copyright (C) 2000-present by Marc Culler and others.

This program is released under the GNU General Public License version 2 or (at your option) any later version as published by the Free Software Foundation. See

for the complete license text.