Polyhedron visualisation

Question

Is there a software that can create a 3D visualization of a polyhedron, based on inequations that define its delimiting planes, of the form

A_1 x < b_1
A_2 x < b_2
A_3 x < b_3

Here an example of the 2D equivalent using Grapher on OS X:

Answer 1

I would like to recommend free MATLAB based software that implements a new visualization technique for polyhedra. It is called "boundary interval method", and its primary purpose is to depict polyhedra determined by inequality systems, i.e. exactly your problem. The link to the relevant web-page is http://www.nsc.ru/interval/sharaya/index.html#codes and what you need is the package lineq2.

An example of polyhedron visualization by the package lineq2 is given below. Preliminaries: download lineq2.zip, unpack it, set MATLAB path to the directory with the package. In MATLAB command window, input the matrix A, right-hand side vector b, and type 3D visualization command SolSetIneq3D(A,b):

>> A = [1 2 3; 3 -2 1; 1 0 -1; 0 -1 0; 2 0 -1] 
   A = 
        1   2   3 
        3  -2   1 
        1   0  -1 
        0  -1   0 
        2   0  -1 
>> b = [ -2; -1; 0; -1; -2] 
   b = 
       -2 
       -1 
        0 
       -1 
       -2 
>> SolSetIneq3D(A,b) 
 ans = 
        1 
>> 

The result should look as follows:

These pictures are obtained by rotating the resulting image of the solution set to the linear inequality system Ax>=b with the above 5x3-matrix A and 5-vector b in the right-hand side. I have chosen the data randomly, and one can see that the solution set is unbounded (green faces depict pieces of bounding planes, while red faces are cuts of the solution set through which it continues to infinity). This is the strong point of the package and the visualization method in general: lineq2 can visualize unbounded and meager solution sets. I do not know of any other software that can do that.

If you want to get acquainted with an exposition of the new technique and its theoretical basics, these are given in the paper

Irene A. Sharaya, Boundary intervals method for visualization of polyhedral solution sets, Reliable Computing, vol. 19, issue 4, pp. 435-467.

Reliable Computing is an open electronic journal, and the direct link to the paper is http://interval.louisiana.edu/reliable-computing-journal/volume-19/reliable-computing-19-pp-435-467.pdf

             Thanks to Journeyman Geek for his friendly moderation of the post.