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11
A Pseudopolynomial Algorithm for Alexandrov’s Theorem
, 2009
"... 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 some convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution is the polyhedron corresponding t ..."
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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 some convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution is 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.
On Flat Polyhedra deriving from Alexandrov’s Theorem
, 2010
"... We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov’s gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n 3) time for polygons of n vertices. ..."
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Cited by 2 (1 self)
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We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov’s gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n 3) time for polygons of n vertices.
INFINITESIMAL RIGIDITY OF CONVEX SURFACES THROUGH THE SECOND DERIVATIVE OF THE HILBERTEINSTEIN FUNCTIONAL I Polyhedral Case
, 2011
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On weakly convex starshaped polyhedra
, 2007
"... Abstract. Weakly convex polyhedra which are starshaped with respect to one of their vertices are infinitesimally rigid. This is partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of ..."
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Abstract. Weakly convex polyhedra which are starshaped with respect to one of their vertices are infinitesimally rigid. This is partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.
Variational properties of the discrete . . .
, 2013
"... This is a survey on rigidity and geometrization results obtained with the help of the discrete HilbertEinstein functional, written for the proceedings of the “Discrete Curvature” colloquium in Luminy. ..."
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This is a survey on rigidity and geometrization results obtained with the help of the discrete HilbertEinstein functional, written for the proceedings of the “Discrete Curvature” colloquium in Luminy.
Computational Geometry Column 49
"... 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 ..."
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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 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 gluingtogether of a collection
Research Statement
, 2009
"... My main interest is analytic number theory and random matrix theory (especially the distribution of zeros of ..."
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My main interest is analytic number theory and random matrix theory (especially the distribution of zeros of
Cauchy’s Theorem and Edge Lengths of Convex Polyhedra
, 2007
"... 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 dihe ..."
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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.