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Computational Geometry Column 19

JOSEPH O'ROURKE

1993
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International journal of computational geometry and applications
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The new algorithm of Bobenko and Izmestiev for reconstructing the unique polyhedron determined by given gluings of polygons is described. One form of Cauchy's rigidity theorem states that the combinatorial structure of a triangulated convex polyhedron together with all its edge lengths determines a unique convex polyhedron P : the 3D vertex coordinates are uniquely determined (up to rigid motions) by this information. However, it has long been an unsolved problem to algorithmically reconstruct
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... ically reconstruct the geometric shape. Sabitov found an exponential algorithm to solve this problem based on the "volume polynomial" [Sab96] . A recent extension of Sabitov's work [FP05] establishes that the unknown internal diagonal lengths between each pair of vertices are roots of a polynomial of degree at most 4 m , where m is the number of edges of the polyhedron P . Knowing these internal diagonals permits reconstruction. Exponential lower bounds on the polynomial degree left practical reconstruction unresolved. One can view the information input to Cauchy's result as a gluing-together of a collection of polygons to form a topological sphere. In Cauchy's theorem, the polygons are in fact the faces of the polyhedron. Alexandrov [Ale05] proved a significant strengthening: any gluingtogether of a collection of polygons to form a topological sphere leads to a unique convex polyhedron, 1 as long as no more than 2π of angle is glued at any one point. The polygons now have no relation to the faces. In fact, his theorem applies even to just one polygon whose perimeter is glued to, or identified with itself. Alexandrov's proof is, alas, nonconstructive, and did not suggest an algorithm. Now Alexander Bobenko and Ivan Izmestiev have found a constructive proof of Alexandrov's theorem, which leads to an effective numerical algorithm to reconstruct the 3D structure of the unique polyhedron guaranteed by the theorem, and therefore of Cauchy's theorem as well [BI06] . They and Stefan Sechelmann have implemented the algorithm and have made it available through a

doi:10.1142/s0218195993000142
fatcat:wftpyycl5vfuxp6rwgefqe64cy