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I have some origami polyhedra which I know the type of faces it has and how they are connected (such as this torus) and I want to calculate the co-ordinates of the vertices to use as an input to script.

My question is how should I go about translating knowledge of the faces into the locations of the vertices? Are there geometric tricks I can apply? or are there any software libraries/tools I could use to help out here? To make thinks a bit more complicated I don't think all the faces are regular polygons, all the edges should be the same length and all the angles should be the same but being made of paper there is a little wiggle room.

Gorloth
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  • CGAL is great among all computational geometry libs, but it still can't do what you described: given a 2D collection of faces, wrapping them to a 3d polyhderon... – Shuhao Cao Jun 11 '13 at 05:27

1 Answers1

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A standard tool to do this sort of thing is the polymake program, which google will find in seconds.

But from the description in your question (which is not very explicit) it seems that polymake would not work for you. For example, tori are not convex, and most things work on covex solds, and so on.

Mariano Suárez-Álvarez
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  • I don't think polymake could wrap a 2d skeleton into a polyhedron... – Shuhao Cao Jun 11 '13 at 05:16
  • Well, one of its make abilities is to compute vertices of polyhedra given by faces, when these are given by equations. I doubt anything can compute vertices from a combinatorial description of the 2-skeleton. – Mariano Suárez-Álvarez Jun 11 '13 at 05:18
  • Yeah, it is an NP-hard problem, well. – Shuhao Cao Jun 11 '13 at 05:23
  • @MarianoSuárez-Alvarez: A graph's adjacency matrix provides vertices of "harmonious" (automorphisms = isometries) geometric realizations, one for each eigenvalue. (See this answer.) These eigen-realizations may or may not correspond to convex polyhedra (as points, edges, faces may coincide, intersect, or span egregiously-high dimensions), but they give rise to all possible realizations under a proper notion of figural addition (the one corresponding to simple matrix addition). I don't (yet) know a strategy to ensure a convex polyhedral combination. – Blue Jun 11 '13 at 05:44