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63
Distributed computation of virtual coordinates
 In Proc. 23rd Symp. Computational Geometry (SoCG ’97
, 2007
"... Sensor networks are emerging as a paradigm for future computing, but pose a number of challenges in the fields of networking and distributed computation. One challenge is to devise a greedy routing protocol – one that routes messages through the network using only information available at a node o ..."
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Cited by 12 (1 self)
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Sensor networks are emerging as a paradigm for future computing, but pose a number of challenges in the fields of networking and distributed computation. One challenge is to devise a greedy routing protocol – one that routes messages through the network using only information available at a node or its neighbors. Modeling the connectivity graph of a sensor network as a 3connected planar graph, we describe how to compute on the network in a distributed and local manner a special geometric embedding of the graph. This embedding supports a geometric routing protocol based on the ”virtual ” coordinates of the nodes derived from the embedding.
Face Numbers of 4Polytopes and 3Spheres
 Proceedings of the international congress of mathematicians, ICM 2002
, 2002
"... Steinitz (1906) gave a remarkably simple and explicit description of the set of all fvectors f(P ) = (f0 , f1 , f2) of all 3dimensional convex polytopes. His result also identifies the simple and the simplicial 3dimensional polytopes as the only extreme cases. Moreover, it can be extended to s ..."
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Cited by 10 (2 self)
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Steinitz (1906) gave a remarkably simple and explicit description of the set of all fvectors f(P ) = (f0 , f1 , f2) of all 3dimensional convex polytopes. His result also identifies the simple and the simplicial 3dimensional polytopes as the only extreme cases. Moreover, it can be extended to strongly regular CW 2spheres (topological objects), and further to Eulerian lattices of length 4 (combinatorial objects).
On Teichmüller spaces of surfaces with boundary
 MR 2350850 Zbl pre05196149
"... We characterize hyperbolic metrics on compact triangulated surfaces with boundary using a variational principle. As a consequence, a new parameterization of the Teichmüller space of compact surface with boundary is produced. In the new parameterization, the Teichmüller space becomes an open convex p ..."
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Cited by 9 (4 self)
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We characterize hyperbolic metrics on compact triangulated surfaces with boundary using a variational principle. As a consequence, a new parameterization of the Teichmüller space of compact surface with boundary is produced. In the new parameterization, the Teichmüller space becomes an open convex polytope. It is conjectured that the WeilPetersson symplectic form can be expressed explicitly in terms of the new coordinate. 1.1. The purpose of this paper is to produce a new parameterization of the Teichmüller space of compact surface with nonempty boundary so that the lengths of the boundary components are fixed. In this new parameterization, the Teichmüller space becomes an explicit open convex polytope. Our result can be considered as the counterpart of
Optimal surface parameterization using inverse curvature map
 Transactions on Visualization and Computer Graphics
"... Abstract—Mesh parameterization is a fundamental technique in computer graphics. The major goals during mesh parameterization are to minimize both the angle distortion and the area distortion. Angle distortion can be eliminated by the use of conformal mapping, in principle. Our paper focuses on solvi ..."
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Cited by 9 (2 self)
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Abstract—Mesh parameterization is a fundamental technique in computer graphics. The major goals during mesh parameterization are to minimize both the angle distortion and the area distortion. Angle distortion can be eliminated by the use of conformal mapping, in principle. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. First, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. On a mesh, the vertex curvature is related to edge lengths by the curvature map. Our result shows the map is invertible, i.e., the edge lengths can be computed from the curvature (by integration). Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Second, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Third, we give an energy form to measure the area distortions, and show that it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum; it has a natural physical interpretation. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced. The theoretical framework of the inverse curvature map can be applied to further study discrete conformal mappings.
A variational proof of Alexandrov’s convex cap theorem; arXiv: math/0703169. [333
"... We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are ..."
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Cited by 7 (3 self)
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We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are globally rigid, that is uniquely determined by their curvatures. 1
Packing circles and spheres on surfaces
 TO APPEAR IN THE ACM SIGGRAPH CONFERENCE PROCEEDINGS
"... Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose facesâ incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate circl ..."
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Cited by 7 (4 self)
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Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose facesâ incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate circle packings, hexagonal meshes which carry a torsionfree support structure, hybrid trihex meshes, and others. We show how triangle meshes can be optimized so as to have the incircle packing property. We explain their relation to conformal geometry and implications on solvability of optimization. The examples we give confirm that this kind of meshes is a rich source of geometric structures relevant to architectural geometry.
Wellcentered Planar Triangulation  An Iterative Approach
, 2007
"... We present an iterative algorithm to transform a given planar triangle mesh into a wellcentered one by moving the interior vertices while keeping the connectivity fixed. A wellcentered planar triangulation is one in which all angles are acute. Our approach is based on minimizing a certain energy ..."
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Cited by 7 (4 self)
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We present an iterative algorithm to transform a given planar triangle mesh into a wellcentered one by moving the interior vertices while keeping the connectivity fixed. A wellcentered planar triangulation is one in which all angles are acute. Our approach is based on minimizing a certain energy that we propose. Wellcentered meshes have the advantage of having nice orthogonal dual meshes (the dual Voronoi diagram). This may be useful in scientific computing, for example, in discrete exterior calculus, in covolume method, and in spacetime meshing. For some connectivities with no wellcentered configurations, we present preprocessing steps that increase the possibility of finding a wellcentered configuration. We show the results of applying our energy minimization approach to small and large meshes, with and without holes and gradations. Results are generally good, but in certain cases the method might result in inverted elements.
Quantum geometry of 3dimensional lattices
, 2008
"... We study geometric consistency relations between angles on 3dimensional (3D) circular quadrilateral lattices — lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable “ultralocal” Poisson bracket alge ..."
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Cited by 6 (3 self)
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We study geometric consistency relations between angles on 3dimensional (3D) circular quadrilateral lattices — lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable “ultralocal” Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the YangBaxter equation). These solutions generate an infinite number of nontrivial solutions of the YangBaxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasiclassical limit.
A VARIATIONAL PRINCIPLE FOR WEIGHTED DELAUNAY TRIANGULATIONS AND HYPERIDEAL
"... We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with nonintersecting sitecircles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperb ..."
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Cited by 6 (0 self)
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We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with nonintersecting sitecircles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus, the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, we consider weighted Delaunay triangulations in piecewise flat surfaces, allowing cone singularities with prescribed cone angles in the vertices. The material presented here extends work by Rivin on Delaunay triangulations and ideal polyhedra. 1.
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.