A Pseudopolynomial Algorithm for Alexandrov's Theorem

Details A Pseudopolynomial Algorithm for Alexandrov's Theorem A Pseudopolynomial Algorithm for Alexandrov's Theorem

2008-12-30

arxiv.org/abs/0812.5030v3

Computer Science

Computational Geometry

25 pages; new Delaunay triangulation algorithm, minor other changes; an abbreviated v2 was at WADS 2009

Scientific paper

Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution leads to the polyhedron corresponding to a given metric. We describe an algorithm based on this differential equation to compute the polyhedron to arbitrary precision given the metric, and prove a pseudopolynomial bound on its running time. Along the way, we develop pseudopolynomial algorithms for computing shortest paths and weighted Delaunay triangulations on a polyhedral surface, even when the surface edges are not shortest paths.

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