4D space : visualize polyhedra intersection

[BernardH]

Hey,

I would like to make a picture about Dantzig Wolfe decomposition.
I have two polyhedra, in R^4 (there is no simpler example).
Their intersection is the feasible solution region of a linear program.
Is there a way to make that look good with polymake.
I've juste tryed it with the live cd, I'm compiling it at the moment.
I made a Schlegel diagram of the intersection, but it is not what I need.
I would like to have the two polyhedra at the same time on the picture, in the same space.
I'm asking for help.
And I'm bad with Perl, totally newb.

[BernardH]

Nice software BTW !
So much work involved.
I grabbed the vertices from my polytopes and I'm gonna use a projection operator to plot them together in a 2d space.
Keep the good work up, I'll come back from time to time.

[joswig]

I grabbed the vertices from my polytopes and I'm gonna use a projection operator to plot them together in a 2d space.

In general, this is one of the very few sensible options. A special situation occurs if the vertices of the two polytopes in 4-space are in convex position. Then one could introduce an artificial extra facet to project a Schlegel diagram on.

Another option is to visualize the (primal or dual) graphs without any coordinate information; instead use the geometry to introduce extra forces for the spring embedder; color code the two polytopes and the intersection. This sometimes gives meaningful pictures, even in dimensions higher than four. In case you are interested such techniques are discussed in:

Drawing polytopal graphs with polymake, Computing and Visualization in Science 13 (2010), no. 2, 99–110 DOI:10.1007/s00791-009-0127-3

[gawrilow]

A special situation occurs if the vertices of the two polytopes in 4-space are in convex position. Then one could introduce an artificial extra facet to project a Schlegel diagram on.

Actually, one could take the union of both polytopes and draw its Schlegel diagram, adding the skeleton of the intersection to its interior. The tricky part is, however, to color the ridges appropriately, and to make the ridges of the union invisible. Without decent perl skills it would be really hard. And, still worse, the client schlegel_interactive would also need some slight refactoring, because now it can only add unconnected points like STEINER_POINTS. It is not difficult, but still needs to be done. On the other hand, if you are going to draw Schlegel diagrams of polytopal complexes, you would reuse this, so the effort spent won't be just for this particular request.