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49
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.
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
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.
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 5 (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.
Globally optimal surface mapping for surfaces with arbitrary topology
 IEEE Trans. on Visualization and Computer Graphics
, 2008
"... Abstract — Computing smooth and optimal onetoone maps between surfaces of same topology is a fundamental problem in graphics and such a method provides us a ubiquitous tool for geometric modeling and data visualization. Its vast variety of applications includes shape registration/matching, shape b ..."
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Cited by 4 (1 self)
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Abstract — Computing smooth and optimal onetoone maps between surfaces of same topology is a fundamental problem in graphics and such a method provides us a ubiquitous tool for geometric modeling and data visualization. Its vast variety of applications includes shape registration/matching, shape blending, material/data transfer, data fusion, information reuse, etc. The mapping quality is typically measured in terms of angular distortions among different shapes. This paper proposes and develops a novel quasiconformal surface mapping framework to globally minimize the stretching energy inevitably introduced between two different shapes. The existing stateoftheart intersurface mapping techniques only afford local optimization either on surface patches via boundary cutting or on the simplified base domain, lacking rigorous mathematical foundation and analysis. We design and articulate an automatic variational algorithm that can reach the global distortion minimum for surface mapping between shapes of arbitrary topology, and our algorithm is solely founded upon the intrinsic geometry structure of surfaces. To our best knowledge, this is the first attempt towards rigorously and numerically computing globally optimal maps. Consequently, we demonstrate our mapping framework offers a powerful computational tool for graphics and visualization tasks such as data and texture transfer, shape morphing, and shape matching. Index Terms — Quasiconformal surface mapping, harmonic map, uniformization metric, surface parameterization.
Local Rigidity of Inversive Distance Circle Packing
 Tech. Rep. arXiv.org, Mar
"... Abstract. A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle. ..."
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Cited by 4 (1 self)
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Abstract. A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.
A Conformal Energy for Simplicial Surfaces
 COMBINATORIAL AND COMPUTATIONAL GEOMETRY
, 2005
"... A new functional for simplicial surfaces is suggested. It is invariant with respect to Möbius transformations and is a discrete analogue of the Willmore functional. Minima of this functional are investigated. As an application a bending energy for discrete thinshells is derived. ..."
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Cited by 4 (1 self)
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A new functional for simplicial surfaces is suggested. It is invariant with respect to Möbius transformations and is a discrete analogue of the Willmore functional. Minima of this functional are investigated. As an application a bending energy for discrete thinshells is derived.
Circle patterns on singular surfaces
 In preparation
, 2005
"... We consider “hyperideal ” circle patterns, i.e. patterns of disks which do not cover the whole surface, which are associated to hyperideal hyperbolic polyhedra. The main result is that, on a Euclidean or hyperbolic surface with conical singularities, those hyperideal circle patterns are uniquely det ..."
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Cited by 4 (0 self)
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We consider “hyperideal ” circle patterns, i.e. patterns of disks which do not cover the whole surface, which are associated to hyperideal hyperbolic polyhedra. The main result is that, on a Euclidean or hyperbolic surface with conical singularities, those hyperideal circle patterns are uniquely determined by the intersection angles of the circles and the singular curvatures. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra. The results presented here extend those in [Sch05a], however the proof is completely different (and more intricate) since [Sch05a] used a shortcut which is not available here. Résumé On considère des motifs de cercles “hyperidéaux”, c’estàdire des motifs de disques qui ne couvrent pas toute la surface sousjacente, et qui sont associés aux polyèdres hyperboliques hyperidéaux. Le résultat principal est que ces motifs de cercles, sur les surfaces euclidiennes ou hyperboliques à singularités coniques, sont uniquement déterminés par les angles d’intersection des cercles et par les courbures singulières. C’est lié à des résultats sur les angles dièdres des polyèdres hyperboliques idéaux ou hyperidéaux. Les résultats présentés ici étendent ceux de [Sch05a], mais les preuves sont complètement différentes (et plus élaborés)
Conformal surface parameterization using euclidean ricci flow
"... Surface parameterization is a fundamental problem in graphics. Conformal surface parameterization is equivalent to finding a Riemannian metric on the surface, such that the metric is conformal to the original metric and induces zero Gaussian curvature for all interior points. Ricci flow is a theoret ..."
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Cited by 3 (1 self)
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Surface parameterization is a fundamental problem in graphics. Conformal surface parameterization is equivalent to finding a Riemannian metric on the surface, such that the metric is conformal to the original metric and induces zero Gaussian curvature for all interior points. Ricci flow is a theoretic tool to compute such a conformal flat metric. This paper introduces an efficient and versatile parameterization algorithm based on Euclidean Ricci flow. The algorithm can parameterize surfaces with different topological structures in an unified way. In addition, we can obtain a novel class of parameterization, which provides a conformal invariant of a surface that can be used as a surface signature. 1.
Spherical Representation and Polyhedron Routing for Load Balancing in Wireless Sensor Networks
"... Abstract—In this paper we address the problem of scalable and load balanced routing for wireless sensor networks. Motivated by the analog of the continuous setting that geodesic routing on a sphere gives perfect load balancing, we embed sensor nodes on a convex polyhedron in 3D and use greedy routin ..."
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Cited by 2 (2 self)
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Abstract—In this paper we address the problem of scalable and load balanced routing for wireless sensor networks. Motivated by the analog of the continuous setting that geodesic routing on a sphere gives perfect load balancing, we embed sensor nodes on a convex polyhedron in 3D and use greedy routing to deliver messages between any pair of nodes with guaranteed success. This embedding is known to exist by the KoebeAndreevThurston Theorem for any 3connected planar graphs. In our paper we use discrete Ricci flow to develop a distributed algorithm to compute this embedding. Further, such an embedding is not unique and differs from one another by a Möbius transformation. We employ an optimization routine to look for the Möbius transformation such that the nodes are spread on the polyhedron as uniformly as possible. We evaluated the load balancing property of this greedy routing scheme and showed favorable comparison with previous schemes. I.