Generalized Discrete Ricci Flow

Abstract

Surface Ricci flow is a powerful tool to design

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

“…In [20], Jin et al suggested the discrete Ricci flow method for conformal parameterizations, based on a variational framework and circle packing. Yang et al [36] generalized the discrete Ricci flow and improved the computation by allowing two circles to intersect or separate from each other, unlike the conventional circle packing-based method [20]. In [3], Choi and Lui presented a two-step iterative scheme to correct the conformality distortions at different regions of the unit disk.…”

“…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.…”

“…Here a T , b T , c T and d T are certain linear combinations of the x-coordinates and y-coordinates of the desired quasi-conformal map f . Hence, we can obtain the x-coordinate and ycoordinate function of f by solving the linear system in Equation (36). For more details, please refer to [25].…”

“…We then reconstruct another quasi-conformal map h : R → D with the given Beltrami coefficient µ, using LBS [25]. Note that boundary constraints are needed in solving Equation (36). In our case, we give the Dirichlet condition on the whole boundary of the normalized disk R. More explicitly, the boundary condition used in solving Equation (36) for the map h : R → D is…”

Weighted Triangulations for Geometry Processing

“…Another example, used in graphics to compute geometric flows [Jin et al 2008], is the Andreev-Thurston circle packing [Stephenson 2003;Chow and Luo 2003] which defines a family of vertex-centered circles such that the circles incident to any edge intersect. This idea was further extended to non-intersecting circles through inversive distance circle packing [Guo 2009;Yang et al 2009;Luo 2010], while tangency of neighboring circles corresponds to sphere packing [Colin de Verdière 1991]. Schiftner et al [2009] showed that sphere packing only exists for triangle meshes in which the incircles of neighboring triangles are also tangent.…”