Geometry of planar graphs with angles
SCG '86 Proceedings of the second annual symposium on Computational geometry
Theory of linear and integer programming
Theory of linear and integer programming
Angles of Planar Triangular Graphs
SIAM Journal on Discrete Mathematics
On the complexity of optimization problems for 3-dimensional convex polyhedra and decision trees
Computational Geometry: Theory and Applications
New results on drawing angle graphs
Computational Geometry: Theory and Applications - Special issue on geometric representations of graphs
Checking the convexity of polytopes and the planarity of subdivision
WADS '97 Selected papers presented at the international workshop on Algorithms and data structure
Checking geometric programs or verification of geometric structures
Selected papers from the 12th annual symposium on Computational Geometry
Robust spherical parameterization of triangular meshes
Computing - Geometric modelling dagstuhl 2002
Geometric Folding Algorithms: Linkages, Origami, Polyhedra
Geometric Folding Algorithms: Linkages, Origami, Polyhedra
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In this paper we explore, from an algorithmic point of view, the extent to which the facial angles and combinatorial structure of a convex polyhedron determine the polyhedron--in particular the edge lengths and dihedral angles of the polyhedron. Cauchy's rigidity theorem of 1813 states that the dihedral angles are uniquely determined. Finding them is a significant algorithmic problem which we express as a spherical graph drawing problem. Our main result is that the edge lengths, although not uniquely determined, can be found via linear programming. We make use of significant mathematics on convex polyhedra by Stoker, Van Heijenoort, Gale, and Shepherd.