The unified discrete surface Ricci flow

Zhang, Min; Guo, Ren; Zeng, Wei; Luo, Feng; Yau, Shing–Tung; Gu, Xianfeng

doi:10.1016/j.gmod.2014.04.008

Cited by 51 publications

(63 citation statements)

References 41 publications

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“…In [YGL*09], Yang et al incorporated inversive distance circle packing metric to make discrete Ricci flow more flexible and more robust for meshes with low qualities. Later in [ZGZ*14], Zhang et al introduced a unified theoretic framework for discrete Ricci flow including all common schemes, like tangential circle packing, Thurston's circle packing, inversive distance circle packing and virtual radius circle packing. Rich applications of discrete Ricci flow have also been explored, such as medical analysis [WGC*07, WYZ*08, QFY*08], shape analysis [ZSG10], shape registration [ZYZ*08,ZG11] and network routing [SYG*09,GGL14].…”

Section: Related Workmentioning

confidence: 99%

“…On the other hand, flow‐based methods like discrete Ricci flow [CL*03, JKLG08, YGL*09, ZGZ*14], Yamabe flow [Gli05] and Calabi flow [Ge18, GX18, GH18, ZX18, ZLG*18] are inspired by corresponding smooth geometric flows that deform the metric of a Riemannian manifold in differential geometry. They do not operate the coordinates in the parameter domain directly and are intrinsically independent of topology.…”

Section: Introductionmentioning

confidence: 99%

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Discrete Calabi Flow: A Unified Conformal Parameterization Method

Zhou

et al. 2019

Computer Graphics Forum

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Conformal parameterization for surfaces into various parameter domains is a fundamental task in computer graphics. Prior research on discrete Ricci flow provided us with promising inspirations from methods derived via Riemannian geometry, which is rigorous in theory and effective inpractice. In this paper, we propose a unified conformal parameterization approachfor turning triangle meshes into planar and spherical domains using discrete Calabi flow onpiecewise linear metric. We incorporate edge‐flipping surgery to guarantee convergence as well as other significant improvements including approximate Newton's method, optimal step‐lengths, priority embedding and boundary customizing, which achieve better performance and functionality with robustness and accuracy.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Discrete Calabi Flow: A Unified Conformal Parameterization Method

Zhou

et al. 2019

Computer Graphics Forum

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“…A large class of these was studied in [27], and it was shown in [32] (see also [66]) that all of these conformal deformations can be described by certain functions. Previously, different discrete conformal deformations were considered in [3,41,47,57] and many other places.…”

Section: Application Of Curvature Flowmentioning

confidence: 99%

“…in the Euclidean background, and there is an extra linear reaction term in the other constant curvature backgrounds (see [31] for Euclidean background and [32] and [66] for other constant curvature backgrounds). Thus the stability of the zero curvature problem is related to the definiteness of the Laplacian.…”

Section: Application Of Curvature Flowmentioning

confidence: 99%

Book Review: Ricci flow for shape analysis and surface registration: theories, algorithms and applications

Chow¹,

Glickenstein²,

Luo³

2016

Bull. Amer. Math. Soc.

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One of the goals in the field of modern geometric analysis is to study canonical structures on manifolds and vector bundles-existence, uniqueness, and moduli. One of the pioneers of analytical approaches in this field is Yau, who, with Schoen and other coauthors solved a number of outstanding geometric problems, many of which are related to elliptic PDEs. A personal account of these developments is in the two-volume set [65]. Around the same time, there were remarkable developments in other areas of geometry. A more geometric theory, including the fundamental theories of compactness and collapse, was developed by Cheeger, Gromov, and others. Thurston revolutionized low-dimensional topology by developing a geometric theory aimed at understanding 3-manifolds via the geometrization conjecture and related ideas. Soon after, inspired by the earlier work of Eells and Sampson on harmonic maps, Hamilton invented the Ricci flow, a parabolic PDE, and through a number of very original works nearly single-handedly developed it into a compelling program to approach the Poincaré and geometrization conjectures. Two decades later, Perelman solved the Poincaré and geometrization conjectures by introducing a plethora of deep and powerful ideas and methods into Ricci flow to complete Hamilton's program. At the present time, more than another decade later, geometric analysis is a thriving field with many active subfields, including Kähler geometry, influenced by the work of Tian, Donaldson, Chen, and others, the study of Ricci curvature by Cheeger, Colding, Minicozzi, and others, geometric flows such as the mean curvature flow by Huisken, White, and others, and the work of Brendle to solve long-standing conjectures, to just name a few of the many directions this field has taken.The idea that a discrete set of values determines geometry is the main tenet of discrete differential geometry. These ideas come from a number of disparate directions. Whitney used numerical analysis to connect de Rham theory and simplicial homology [64], and these ideas form the central tenets of discrete exterior calculus [20]. In a separate intellectual stream, Regge introduced a notion of discrete geometry that he considered a way of developing computational and quantum gravity from first principles that were motivated by, but not dependent on, the theory of smooth manifolds (see [34,51]). Thurston introduced the idea of circle packings for approximating Riemann mappings, and this developed into a whole field of circle packing and related topics (see Stephenson's book [58], reviewed in [6]). A discrete form of integrable systems was developed by Bobenko and Suris [5] into a geometric theory, while other groups [1,23] have developed similar or competing theories arising from numerical analysis and mathematical physics. All of these directions may be characterized as structure-preserving discretizations.

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“…It was also proved that circle patterns converge to minimal surfaces (see [10,11]). Current development of research related to circle patterns can be found in [12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Approximation of quasiconformal mappings by SG circle patterns

Lan

2014

Applicable Analysis

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Given a measurable function λ with λ ∞ < 1, the approximating solutions of Beltrami equation for λ are directly constructed by using the techniques of SG circle pattern and conformal welding. It is proved that the sequence of approximating solutions converges uniformly on to compact subsets to a quasiconformal mapping with complex dilation λ. This provides a new approach of constructing discrete quasiconformal mappings.

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The unified discrete surface Ricci flow

Cited by 51 publications

References 41 publications

Discrete Calabi Flow: A Unified Conformal Parameterization Method

Discrete Calabi Flow: A Unified Conformal Parameterization Method

Book Review: Ricci flow for shape analysis and surface registration: theories, algorithms and applications

Approximation of quasiconformal mappings by SG circle patterns

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