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Rigidity of polyhedral surfaces
, 2006
"... We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational a ..."
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Cited by 13 (5 self)
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We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law and the Lengendre transformation of them. These include energies used by Colin de Verdiere, Braegger, Rivin, CohenKenyonPropp, Leibon and BobenkoSpringborn for variational principles on triangulated surfaces. Our study is based on a set of identities satisfied by the derivative of the cosine law. These identities which exhibit similarity in all spaces of constant curvature are probably a discrete analogous of the Bianchi identity.
TRIANGULATED RIEMANN SURFACES WITH BOUNDARY AND THE WEILPETERSSON POISSON STRUCTURE
, 2006
"... Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at ∂S perpendicularly are coordinates on the Teichmüller space T (S). We compute the WeilPetersson Poisson structure on T (S) in this system of coordinates and we prov ..."
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Cited by 9 (2 self)
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Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at ∂S perpendicularly are coordinates on the Teichmüller space T (S). We compute the WeilPetersson Poisson structure on T (S) in this system of coordinates and we prove that it limits pointwise to the piecewiselinear Poisson structure defined by Kontsevich on the arc complex of S. As a byproduct of the proof, we obtain a formula for the firstorder variation of the distance between two closed geodesic under FenchelNielsen deformation.
Computing Teichmüller Shape Space
 SUBMITTED TO IEEE TVCG
"... Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmüller theory. Two surfaces are conformal equivalent, if there exists a bijective anglepreserving map between them. ..."
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Cited by 6 (3 self)
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Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmüller theory. Two surfaces are conformal equivalent, if there exists a bijective anglepreserving map between them. The Teichmüller space for surfaces with the same topology is a finite dimensional manifold, where each point represents a conformal equivalence class, and the conformal map is homotopic to Identity. A curve in the Teichmüller space represents a deformation process from one class to the other. In this work, we apply Teichmüller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic, invariant under the rigid motions and scalings, insensitive to resolutions. Furthermore, the method has solid theoretic foundation, and the computation of Teichmüller coordinates is practical, stable and efficient. The algorithms for the Teichmüller coordinates of surfaces with positive or zero Euler numbers have been studied before. This work focuses on the surfaces with negative Euler numbers, which have a unique conformal Riemannian metric with −1 Gaussian curvature. The coordinates which we will compute are the lengths of a special set of geodesics under this special metric. The metric can be obtained by the curvature flow algorithm, the geodesics can be calculated using algebraic topological method. We tested our method extensively for indexing and comparison of about one hundred of surfaces with various topologies, geometries and resolutions. The experimental results show the efficacy and efficiency of the length coordinate of the Teichmüller space.
Riemann surfaces with boundary and natural triangulations of the Teichmüller space
, 2008
"... We compare some natural triangulations of the Teichmüller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt ScannellWolf’s proof to show that grafting semiinfinite cylinders at the ends of hyperbolic surfaces with fixed boundary lengths is a homeomorphism. T ..."
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Cited by 2 (2 self)
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We compare some natural triangulations of the Teichmüller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt ScannellWolf’s proof to show that grafting semiinfinite cylinders at the ends of hyperbolic surfaces with fixed boundary lengths is a homeomorphism. This way, we construct a family of triangulations of the Teichmüller space of punctures surfaces that interpolates between PennerBowditchEpstein’s (using the spine construction) and HarerMumfordThurston’s (using Strebel’s differentials). Finally, we show (adapting arguments of Dumas) that on a fixed punctured surface, when the triangulation approaches HMT’s, the associated Strebel differential is wellapproximated by the Schwarzian of the associated projective structure and by the Hopf differential of the collapsing map.
A NOTE ON CIRCLE PATTERNS ON SURFACES
, 2007
"... Abstract. In this paper we give two different proofs of Bobenko and Springborn’s theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface up to isometry (or similarity). 1. ..."
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Cited by 1 (1 self)
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Abstract. In this paper we give two different proofs of Bobenko and Springborn’s theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface up to isometry (or similarity). 1.
RIGIDITY OF POLYHEDRAL SURFACES, II
, 2007
"... We study the rigidity of polyhedral surfaces using variational principle. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first dis ..."
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We study the rigidity of polyhedral surfaces using variational principle. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover a uniform approach on several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, BobenkoSpringborn and Thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context.
Variational Principles on Triangulated Surfaces
, 2008
"... We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles. ..."
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We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.