## Variational Principles for Circle Patterns and Koebe's Theorem (0)

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Venue: | Trans. Amer. Math. Soc |

Citations: | 56 - 11 self |

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@ARTICLE{Bobenko_variationalprinciples,

author = {Alexander I. Bobenko and Boris A. Springborn},

title = {Variational Principles for Circle Patterns and Koebe's Theorem},

journal = {Trans. Amer. Math. Soc},

year = {},

volume = {356},

pages = {659--689}

}

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### Abstract

The subject of this paper is a special class of configurations, or patterns, of intersecting circles in constant curvature surfaces. The combinatorial aspect of such a pattern is described by a cellular decomposition of the surface. The faces of the cellular decomposition correspond to circles and the vertices correspond to points where circles intersect. (See figures 1 and 2.) In the most general case that we consider, the surface may have cone-like singularities in the centers of the circles and in the points of intersection. In oarticular, we treat...

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Citation Context ...od to construct some kinds of circle patterns and thus prove existence and uniqueness theorems. These methods were adapted by Brightwell and Scheinerman [BS93], who state and prove Theorem 1. Schramm =-=[Sch92]-=- proves a theorem that is much more general than Theorem 2. In it, the sphere is replaced by an arbitrary convex body. The proof is not constructive. 2 The original Steinitz theorem only asserts the e... |

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