Combinatorial Ricci Flows on Surfaces

Chow, Bennett; Luo, Feng

doi:10.4310/jdg/1080835659

Cited by 307 publications

(415 citation statements)

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“…The uniform flat metric computed using discrete Ricci flow is guaranteed to induce a diffeomorphism between P i to D P i (and M i to D M i ) [14]. Note that a harmonic map f : A ⊂ R 2 → B ⊂ R 2 is a diffeomorphism if ∂ B is convex and the boundary condition f (∂ A) = ∂ B is a homeomorphism.…”

Section: Appendixmentioning

confidence: 99%

“…However, the harmonic map is not suitable for this step since it is very hard to specify the position of boundary points for surface with multiple boundaries. Therefore, we use discrete Ricci flow [14,9,15] for the parameterization step since we only need to specify the target curvatures (rather than their positions) of the boundary points.…”

Section: Uniform Flat Metric and Multi-hole Diskmentioning

confidence: 99%

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A divide-and-conquer approach for automatic polycube map construction

Wang

et al. 2009

Computers & Graphics

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Polycube map is a global cross-surface parameterization technique, where the polycube shape can roughly approximate the geometry of modeled objects while retaining the same topology. The large variation of shape geometry and its complex topological type in real world applications make it difficult to effectively construct a high-quality polycube that can serve as a good global parametric domain for a given object. In practice, existing polycube-map construction algorithms typically require a large amount of user interaction for either pre-constructing the polycubes with great care or interactively specifying the geometric constraints to arrive at the user-satisfied maps. Hence, it is tedious and labor intensive to construct polycube maps for surfaces of complicated geometry and topology. This paper aims to develop an effective method to construct polycube maps for surfaces with complicated topology and geometry. Using our method, users can simply specify how close the target polycube mimics a given shape in a quantitative way. Our algorithm can both construct a similar polycube of high geometric fidelity and compute a high-quality polycube map in an automatic fashion. In addition, our method is theoretically guaranteed to output a one-to-one map. To demonstrate the efficacy of our method, we apply the automatically-constructed polycube maps in a number of computer graphics applications, such as seamless texture tiling, T-spline construction, and quadrilateral mesh generation.

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Section: Appendixmentioning

confidence: 99%

Section: Uniform Flat Metric and Multi-hole Diskmentioning

confidence: 99%

A divide-and-conquer approach for automatic polycube map construction

Wang

et al. 2009

Computers & Graphics

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“…The main theorem in [7] is as follows: Under an approriate condition on the triangulation the solution of (7) exists for all time and converges exponentially fast to a metric of constant curvature.…”

Section: Remarks Concerning Related Workmentioning

confidence: 99%

“…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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“…The discrete algorithms for heat flow can be found in [29], the discrete Ricci flow method can be found in [30]. Figure 4 and 7 demonstrate global conformal parameterizations for high genus surfaces.…”

Section: Global Conformal Surface Parameterizationsmentioning

confidence: 99%