Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices

Bücking, Ulrike

doi:10.1007/978-3-662-50447-5_3

Cited by 10 publications

(9 citation statements)

References 24 publications

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“…In this section, we show that every simply connected umbilic-free CMC-1 surface can be approximated by our discrete CMC-1 surfaces. Based on the convergence result on circle patterns by He-Schramm [16] and Bücking [11], we deduce that discrete osculating Möbius transformations converge to their smooth counterparts and so do discrete CMC-1 surfaces as a result of the Weierstrass-type representation.…”

Section: A 1-parameter Family Of Cross Ratio Systemsmentioning

confidence: 83%

“…By restricting the combinatorics to triangle lattices, we prove that every smooth CMC-1 surfaces without umbilic points can be approximated by our discrete CMC-1 surfaces (Theorem 5.7). It is achieved by proving the convergence of the osculating Möbius transformations to their smooth counterparts, which relies on the results by He-Schramm [16] and Bücking [11].…”

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confidence: 99%

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CMC-1 surfaces via osculating Möbius transformations between circle patterns

Lam¹

2020

Preprint

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Given two circle patterns of the same combinatorics in the plane, the Möbius transformations mapping circumdisks of one to the other induces a P SL(2, C)-valued function on the dual graph. Such a function plays the role of an osculating Möbius transformation and induces a realization of the dual graph in hyperbolic space. We characterize the realizations and obtain a one-to-one correspondence in the cases that the two circle patterns share the same shear coordinates or the same intersection angles. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature H ≡ 1 in hyperbolic space. We further establish convergence on triangular lattices.

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Section: A 1-parameter Family Of Cross Ratio Systemsmentioning

confidence: 83%

mentioning

confidence: 99%

CMC-1 surfaces via osculating Möbius transformations between circle patterns

Lam¹

2020

Preprint

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“…This innocentlooking denition leads to a rich discrete theory which is just as exible as the smooth one [Bobenko et al 2015]. Bücking [2016Bücking [ , 2018 and Gu et al [2019] consider convergence under renement.…”

Section: Related Work 21 Discrete Conformal Equivalencementioning

confidence: 99%

Discrete conformal equivalence of polyhedral surfaces

2021

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This paper describes a numerical method for surface parameterization, yielding maps that are locally injective and discretely conformal in an exact sense. Unlike previous methods for discrete conformal parameterization, the method is guaranteed to work for any manifold triangle mesh, with no restrictions on triangulatiothat each task can be formulated as a convex problem where the triangulation is allowed to change---we complete the picture by introducing the machinery needed to actually construct a discrete conformal map. In particular, we introduce a new scheme for tracking correspondence between triangulations based on normal coordinates , and a new interpolation procedure based on layout in the light cone. Stress tests involving difficult cone configurations and near-degenerate triangulations indicate that the method is extremely robust in practice, and provides high-quality interpolation even on meshes with poor elements.

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“…For results in convergence, Gu-Luo-Wu proved the convergence of discrete conformal maps to the uniformization map if the triangulation is a "δ triangulation" and no edge flip is required [GLW19]. Bücking showed the convergence of discrete conformal mapping to the smooth Riemann mapping through subdivision [Bü16], [Bü17].…”

Section: Computation Of Discrete Conformal Mapsmentioning

confidence: 99%

Conformal Parametrization of Surfaces of Genus Zero and 3D Reconstruction of Nuclear Fusion Hotspots

Wong

2020

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To my beloved mother and sister, your love is the biggest reason for me to wake up every day and to fight hard for a better future ii ContentsAbstract iv Acknowledgments v Funding sources and disclaimer vii Chapter 0. Introduction Chapter 1. Surface Parametrizations 1.1. Background and overview on conformal maps in the continuous setting 1.2. Computation of discrete conformal maps 1.3. Conformalized Mean Curvature Flow Chapter 2. Improvement on conformalized mean curvature flow 2.1. Flow behaviors and limitations 2.2. Initialization procedure of cMCF using Tutte embedding 2.3. Sphericalized cMCF -using cMCF to construct a homotopy of degree one maps Chapter 3. 3D reconstruction of nuclear fusion hotspot x-ray emission distributions from very few two-dimensional projections 3.1. Introduction to Inertial Confinement Fusion at the National Ignition Facility 3.2. Algebraic Reconstruction Technique (ART) 3.3. 3D x-ray reconstruction of nuclear fusion hotspot using two or three lines of sight with synthetic and experimental data 3.4. 3D electron temperature measurement of nuclear fusion hotspot using 3D X-ray reconstructions Appendix A. Mathematical derivation for formulas in chapter 1 Appendix B. Synthetic models and their 3D electron temperature measurement Bibliography iii

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Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices

Cited by 10 publications

References 24 publications

CMC-1 surfaces via osculating Möbius transformations between circle patterns

CMC-1 surfaces via osculating Möbius transformations between circle patterns

Discrete conformal equivalence of polyhedral surfaces

Conformal Parametrization of Surfaces of Genus Zero and 3D Reconstruction of Nuclear Fusion Hotspots

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