I'm interested in the following problem: I am given the combinatorial structure (vertices, edges, faces) and edge lengths of a polyhedron. From this I'd like to infer the vertex positions.
Now, I know that the above information does not uniquely determine the polyhedron. For instance, position and rotation are deliberate. But if I also pre-specify the positions of two connected vertices, then position and rotation of the polyhedron are determined.
My question is: Is the polyhedron completely determined then? If not, which additional information is needed? And is there a known algorithm for constructing at least one of the possible polyhedra from this information?
The application is to construct a 3D model of a house given the side lengths. If there is no algorithm for general polyhedra, maybe there is one for a subset of all possible house shapes? I assume simple house shapes here, i.e. all walls and roof sides are just single faces. The faces can, however, have 5 or more vertices and the house shapes do not have to be convex.
Thanks a lot in advance!