Visualizing Cones

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uramakrishna
Posts: 8
Joined: 13 May 2011, 18:13

Visualizing Cones

Postby uramakrishna » 17 May 2011, 13:45

Hello,

I am a new user of Polymake and am fascinated by it! Thanks to the authors.

I have a simple questions to begin with, as my current work involves (homogeneous) cones and their polars and I am having trouble using "polytope versions" of them. Could someone help me on some tricks on how to visualize 3d and 4d cones and their polars?

Also, is there some way to see the co-ordinates of the vertices of the polytopes, instead of numbers (0,1,2 ... etc.)?

Thanks
Ramakrishna

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joswig
Main Author
Posts: 280
Joined: 24 Dec 2010, 11:10

Re: Visualizing Cones

Postby joswig » 18 May 2011, 09:39

Listing the vertex coordinates (e.g., of an associahedron) is accomplished by

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print associahedron(3)->VERTICES;
Did you read our tutorials? Please look at http://www.polymake.org/doku.php/tutorial/coordinates.

Visualizing 3-dimensional unbounded polyhedra works like for ordinary polytopes. However, in four dimensions there is no automatic way (since it is not so obvious what one would like to see, in general). The best thing probably is to explicitly apply a projective transformation via

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$projectively_equivalent_polytope = bound($my_cone);
and then to visualize the polytope as a Schlegel diagram

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$projectively_equivalent_polytope->SCHLEGEL;
Notice that the latter command takes an optional parameter for choosing the facet to project on such as in:

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cube(4)->SCHLEGEL(FACET=>0);
The probably best thing for visualizing an unbounded polyhedron in four dimensions would then be to choose a facet for projection which contains the far face; it's the easiest if the far face is a facet.

Notice that the combinatorics of a cone is entirely described by the face at infinity. So, in this case, it is best to visualize just this (at most) three-dimensional face.


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