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15
Conformal mapping in linear time
, 2006
"... Abstract. Given any ɛ> 0 and any planar region Ω bounded by a simple ngon P we construct a (1 + ɛ)quasiconformal map between Ω and the unit disk in time C(ɛ)n. One can take C(ɛ) = C + C log 1 ɛ log log ..."
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Cited by 13 (11 self)
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Abstract. Given any ɛ> 0 and any planar region Ω bounded by a simple ngon P we construct a (1 + ɛ)quasiconformal map between Ω and the unit disk in time C(ɛ)n. One can take C(ɛ) = C + C log 1 ɛ log log
Convergence of the zipper algorithm for conformal mapping
, 2006
"... In the early 1980’s an elementary algorithm for computing conformal maps was discovered by R. Kühnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points z0,...,zn in the plane, the algorithm computes an explicit conformal map of the unit disk onto a ..."
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Cited by 11 (1 self)
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In the early 1980’s an elementary algorithm for computing conformal maps was discovered by R. Kühnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points z0,...,zn in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve γ with z0,...,zn ∈ γ. We prove convergence for Jordan regions in the sense of uniformly close boundaries, and give corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a C1 curve or a K−quasicircle. The algorithm was discovered as an approximate method for conformal welding, however it can also be viewed as a discretization of the Löwner differential equation. Conformal maps have useful applications to problems in physics, engineering and mathematics, but how do you find a conformal map say of the upper half plane H to a complicated region? Rather few maps can be given explicitly by hand, so that a computer must be used to find the map approximately. One reasonable way to describe a region numerically is to give a large number of
FaddeevVolkov solution of the YangBaxter Equation and Discrete Conformal Symmetry. Nucl. Phys. B784
, 2007
"... The FaddeevVolkov solution of the startriangle relation is connected with the modular double of the quantum group Uq(sl2). It defines an Isingtype lattice model with positive Boltzmann weights where the spin variables take continuous values on the real line. The free energy of the model is exactl ..."
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Cited by 7 (2 self)
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The FaddeevVolkov solution of the startriangle relation is connected with the modular double of the quantum group Uq(sl2). It defines an Isingtype lattice model with positive Boltzmann weights where the spin variables take continuous values on the real line. The free energy of the model is exactly calculated in the thermodynamic limit. The model describes quantum fluctuations of circle patterns and the associated discrete conformal transformations connected with the Thurston’s discrete analogue of the Riemann mappings theorem. In particular, in the quasiclassical limit the model precisely describe the geometry of integrable circle patterns with prescribed intersection angles. 1
Convergence of a variant of the Zipper algorithm for conformal mapping
"... In the early 1980’s an elementary algorithm for computing conformal maps was discovered by R. Kühnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points z0,...,zn in the plane, the algorithm computes an explicit conformal map of the unit disk onto a ..."
Abstract

Cited by 7 (1 self)
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In the early 1980’s an elementary algorithm for computing conformal maps was discovered by R. Kühnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points z0,...,zn in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve γ with z0,...,zn ∈ γ. We prove convergence for Jordan regions in the sense of uniformly close boundaries, and give corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a C 1 curve or a K−quasicircle. The algorithm was discovered as an approximate method for conformal welding, however it can also be viewed as a discretization of the Loewner differential equation. Conformal maps have applications to problems in physics, engineering and mathematics, but how do you find a conformal map say of the upper half plane H to a complicated region? Rather few maps can be given explicitly by hand, so that a computer must be used to find the map approximately. One reasonable way to describe a region numerically is to give a large number of
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.
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)
Irreducible Apollonian configurations and packings
"... An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One well studied method of forming an Apollonian configuration is to start with three mutually tangent ci ..."
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Cited by 2 (2 self)
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An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One well studied method of forming an Apollonian configuration is to start with three mutually tangent circles and fill a curvilinear triangle with a new circle, then repeat with each newly created curvilinear triangle. More generally, we can start with three mutually tangent circles and a rule (or rules) for how to fill a curvilinear triangle with circles. In this paper we consider the basic building blocks of these rules, irreducible Apollonian configurations. Our main result is to show how to find a small field that can realize such a configuration and also give a method to relate the bends of the new circles to the bends of the circles forming the curvilinear triangle. 1
Applications
"... Abstract. We provide an overview of the connections between circle packings and quasiconformal mappings, with particular attention to applications to string theory and image recognition. ..."
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Cited by 2 (0 self)
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Abstract. We provide an overview of the connections between circle packings and quasiconformal mappings, with particular attention to applications to string theory and image recognition.
Reflections on the Arbelos
"... 1. INTRODUCTION. Written in block capitals on lined paper, the letter bore a postmark from a Northeastern seaport. “I have a problem that I would like to solve, ” the letter began, “but unfortunately I cannot. I dropped out from 2nd year high school, and this problem is too tough for me. ” There fol ..."
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1. INTRODUCTION. Written in block capitals on lined paper, the letter bore a postmark from a Northeastern seaport. “I have a problem that I would like to solve, ” the letter began, “but unfortunately I cannot. I dropped out from 2nd year high school, and this problem is too tough for me. ” There followed the diagram shown in Figure 1 along with the