Variational Principles for Discrete Surfaces
This new volume introduces readers to some of the current topics of research in the geometry of polyhedral surfaces, with applications to computer graphics. The main feature of the volume is a systematic introduction to the geometry of polyhedral surfaces based on the variational principle. The authors focus on using analytic methods in the study of some of the fundamental results and problems of polyhedral geometry: for instance, the Cauchy rigidity theorem, Thurston's circle packing theorem, rigidity of circle packing theorems, and Colin de Verdiere's variational principle. The present book is the first complete treatment of the vast, and expansively developed, field of polyhedral geometry.
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Spherical Geometry and Cauchy Rigidity Theorem
A Brief Introduction to Hyperbolic Geometry
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2-dimensional adjacent algorithm angle structure angles facing B-arcs circle packing metric closed surface closed triangulated surface Compute conformal mappings converges convex polytope convex set Corollary cosh cosx defined Delaunay denoted dihedral angles discrete curvature discrete Ricci flow edge lengths energy function Euclidean polyhedral metric Euclidean polyhedral surface Euclidean triangles Euler characteristic facing the edge follows forall Furthermore Gauss-Bonnet theorem Gaussian curvature genus geodesic boundary geometry half-edge Hessian matrix Hyperbolic embedding hyperbolic metric hyperbolic triangle inequality inner angles intersection isometry Legendre transformation Leibon Lemma limm metric g metrics on 5,7 Mobius transformation open convex parameterization plane Poincare disk positive definite projective structure Proof Ricci flow Riemannian metric rigidity theorem Rivin shows sinh sinx sphere spherical triangle strictly concave strictly convex Suppose surface Ricci flow tangent target curvature Teichmuller space topological totally geodesic triangular mesh universal covering space variational principle vertex vertices