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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 22 (10 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.
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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Cited by 1 (1 self)
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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.
A DEFORMATION OF PENNER’S SIMPLICIAL COORDINATE
"... We find a oneparameter family of coordinates {Ψh}h∈R which is a deformation of Penner’s simplicial coordinate of the decorated Teichmüller space of an ideally triangulated punctured surface (S, T) of negative Euler characteristic. If h> 0, the decorated Teichmüller space in the Ψh coordinate ..."
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We find a oneparameter family of coordinates {Ψh}h∈R which is a deformation of Penner’s simplicial coordinate of the decorated Teichmüller space of an ideally triangulated punctured surface (S, T) of negative Euler characteristic. If h> 0, the decorated Teichmüller space in the Ψh coordinate becomes an explicit convex polytope P (T) independent of h, and if h < 0, the decorated Teichmüller space becomes an explicit bounded