Combinatorial Yamabe Flow on Surfaces

Luo, Feng

doi:10.1142/s0219199704001501

Cited by 202 publications

(271 citation statements)

References 11 publications

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“…Iterative? Shape-preserving [6] Fixed Yes No MIPS [16] Free Yes Yes ABF/ABF++ [33,34] Free Local (no flips) Yes LSCM/DNCP [4,23] Free No No Holomorphic 1-form [12] Fixed No No Mean-value [7] Fixed Yes No Yamabe Riemann map [26] Fixed Yes Yes Circle patterns [22] Free Local (no flips) Yes Genus-0 surface conformal map [19] Free No Yes Discrete Ricci flow [20] Fixed Yes Yes Spectral conformal [27] Free No No Generalized Ricci flow [36] Fixed Yes Yes Two-step iteration [3] Fixed Yes Yes [13] and an iterative scheme for genus-0 surface conformal mapping in [12] to obtain a planar conformal parameterization. In [27], Mullen et al reported a spectral approach to discrete conformal parameterizations, which involves solving a sparse symmetric generalized eigenvalue problem.…”

Section: Methodsmentioning

confidence: 99%

“…By integrating the holomorphic 1-forms on a mesh, a globally conformal parameterization can be obtained. In [26], Luo proposed the combinatorial Yamabe flow on the space of all piecewise flat metrics associated with a triangulated surface for the parameterization. In [20], Jin et al suggested the discrete Ricci flow method for conformal parameterizations, based on a variational framework and circle packing.…”

Section: Methodsmentioning

confidence: 99%

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A linear formulation for disk conformal parameterization of simply-connected open surfaces

Choi

Lui

2017

Adv Comput Math

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Surface parameterization is widely used in computer graphics and geometry processing. It simplifies challenging tasks such as surface registrations, morphing, remeshing and texture mapping. In this paper, we present an efficient algorithm for computing the disk conformal parameterization of simply-connected open surfaces. A double covering technique is used to turn a simply-connected open surface into a genus-0 closed surface, and then a fast algorithm for parameterization of genus-0 closed surfaces can be applied. The symmetry of the double covered surface preserves the efficiency of the computation. A planar parameterization can then be obtained with the aid of a Möbius transformation and the stereographic projection. After that, a normalization step is applied to guarantee the circular boundary. Finally, we achieve a bijective disk conformal parameterization by a composition of quasi-conformal mappings. Experimental results demonstrate a significant improvement in the computational time by over 60%. At the same time, our proposed method retains comparable accuracy, bijectivity and robustness when compared with the state-of-the-art approaches. Applications to texture mapping are presented for illustrating the effectiveness of our proposed algorithm.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

A linear formulation for disk conformal parameterization of simply-connected open surfaces

Choi

Lui

2017

Adv Comput Math

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“…discrete) variant of the Ricci flow by Chow and Luo [7] (see also [16]). In their paper the surface is triangulated, and the metric is given as a function which assigns to each vertex v i (i = 1, .…”

Section: Remarks Concerning Related Workmentioning

confidence: 99%

Stochastic Particle Approximations for the Ricci Flow on Surfaces and the Yamabe Flow

Philipowski

2011

Potential Anal

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We present stochastic particle approximations for the normalized Ricci flow on surfaces and for the non-normalized Yamabe flow on manifolds of arbitrary dimension.Keywords: Ricci flow on surfaces, Yamabe flow, stochastic interacting particle system. AMS subject classification: 53C44, 82C22. Ricci and Yamabe flowThanks to Perelman's seminal work on the geometrization and hence the Poincaré conjecture [19,20,21] the Ricci flow has attracted worldwide attention and led to many new developments (see e.g. [17]). The importance of this subject has been underlined by the award of the Fields medal to Perelman. There is now a strong interest in understanding the microscopic structure of the Ricci flow.The evolution of a Riemannian metric g = g t on a connected d-dimensional closed manifold M under the (normalized) Ricci flow is described by the partial differential equationHere Ric is the Ricci curvature andR the average scalar curvature of M , i.e.R := 1 vol(M ) M R, where R is the scalar curvature (all quantities taken with respect to g t ).In dimension d = 2 we have Ric = 1 2 Rg, so that in this case (1) 2. As t → ∞, g t converges in any C k -norm to a smooth metric g ∞ of constant curvature.

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“…It is motivated by Thurston's circle packings as a discrete analog of holomorphic functions [34]. Another version of discrete conformality is based on conformal equivalence of triangle meshes [29,39], where the conformal structure is defined by the length cross ratios of neighboring triangles. Luo introduced this notion when studying a discrete Yamabe flow.…”

Section: Introductionmentioning

confidence: 99%

“…Note that Theorem 1.4 concerns the theory of conformal equivalence of triangle meshes [29,39] while Theorem 1.5 deals with the notion of circle patterns [36].…”

Section: Introductionmentioning

confidence: 99%

Isothermic triangulated surfaces

Lam

Pinkall

2016

Math. Ann.

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Abstract. We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes.A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.

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Combinatorial Yamabe Flow on Surfaces

Cited by 202 publications

References 11 publications

A linear formulation for disk conformal parameterization of simply-connected open surfaces

A linear formulation for disk conformal parameterization of simply-connected open surfaces

Stochastic Particle Approximations for the Ricci Flow on Surfaces and the Yamabe Flow

Isothermic triangulated surfaces

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