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. It provides a systematic introduction to the geometry of polyhedral surfaces based on the variational principle. 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 this 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. This present book is the first complete treatment of the vast, and expansively developed, field of polyhedral geometry.
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The Cosine Law and Polyhedral Surfaces
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adjacent algorithm angles assume B-arcs boundary called centered circle packing metric classes closed compact Compute concave conformal connected consider convex polytope coordinates corner cosh cosine law cosx covering curvature defined definition denoted determined discrete discrete curvature domain edge lengths embedding energy equivalent Euclidean Euclidean polyhedral exists facing follows formula function fundamental Furthermore Gaussian curvature geodesic geometry given half-edge Hessian hexagon hyperbolic metric ideal triangulation induced inner angles intersection isometry Lemma matrix mesh method metrics on S,T Möbius transformation normal Note obtain oriented plane polyhedral metrics polyhedral surface positive preserving projective Proof proved result Ricci flow Riemannian metric shows sinh smooth space sphere spherical strictly structure Suppose Take tangent tangent law Theorem triangle triangulated surface unique unit universal vertex vertices