Local rigidity of inversive distance circle packing

Abstract

Abstract. A Euclidean (or hyperbolic) circle packing on a triangulated closed surface with prescribed inversive distance is locally determined by its cone angles. We prove this by establishing a variational principle.

Virtual coordinates in hyperbolic space based on Ricci flow for WLANs

“…; b; cÞ ¼ ðcosh l i ; cosh l j ; cosh l k Þ; ðx; y; zÞ ¼ ðcosh r i ; cosh r j ; cosh r k Þ Lemma 2 [22]. The discrete hyperbolic Ricci energy is convex in the admissible metric space.…”

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.…”

“…Recent results [Guo 2009;Luo 2010] have shown the existence of an energy (with no known explicit form) that recovers, at its critical point, an augmented metric (l, w) from vertex angles {θ i } and inversive distances {η ij }. They also showed that this energy relies on the change of variables u i = 1 2 log w i , and Eq.…”

“…We can then use the augmented metric set L W in order to guarantee that the operator ∆ w is positive semi-definite and thence that the circle packing energy is convex. Lastly, Guo [2009] proved that the space of vertex scalings is convex once η ≥ 0, and the minimization of the circle packing energy is thus reduced to a convex optimization. In the following theorem, we summarize the metric characterization of T w based on the inversive distance circle packing.…”

Ricci flow-based spherical parameterization and surface registration

“…In order to solve this problem, inversive distance metric [31] was introduced to Ricci flow with euclidean and hyperbolic geometry in [32]. Guo proved the convexity of discrete Ricci energy with the inversive distance circle packing for Euclidean and Hyperbolic case [33]. Thus, Guo's method is more flexible, more robust and conformal for meshes with low quality triangulations.…”