Directed packings of circles in the plane (original) (raw)
Related papers
2012
The problem of packing circles into a domain of prescribed topology is considered. The circles need not have equal radii. The Collins-Stephenson algorithm computes such a circle packing. This algorithm is parallelized in two different ways and its performance is reported for a triangular, planar domain test case. The implementation uses the highly parallel graphics processing unit (GPU) on commodity hardware. The speedups so achieved are discussed based on a number of experiments.
International Journal of Mathematics and Mathematical Sciences, 2005
Given a bounded sequence of integers {d 0 ,d 1 ,d 2 ,...}, 6 ≤ d n ≤ M, there is an associated abstract triangulation created by building up layers of vertices so that vertices on the nth layer have degree d n . This triangulation can be realized via a circle packing which fills either the Euclidean or the hyperbolic plane. We give necessary and sufficient conditions to determine the type of the packing given the defining sequence {d n }.
Computational Geometry, 2003
A circle packing is a configuration P of circles realizing a specified pattern of tangencies. Radii of packings in the euclidean and hyperbolic planes may be computed using an iterative process suggested by William Thurston. We describe an efficient implementation, discuss its performance, and illustrate recent applications. A central role is played by new and subtle monotonicity results for "flowers" of circles.
Circle packings in the approximation of conformal mappings
Bulletin of the American Mathematical Society, 1990
Connections between circle packings and analytic functions were first suggested by William Thurston [T2], who conjectured that the conformai mapping of a simply connected plane domain Q to the unit disc A could be approximated by manipulating hexagonal circle configurations lying in Q. The conjecture was confirmed by Rodin and Sullivan [RS]. Their proof relies heavily on the hexagonal combinatorics of the circle configurations, a restriction not suggested by the underlying intuition. The purpose of this note is to announce that Thurston's conjecture is true with much weaker combinatoric hypotheses and to outline the proof. The main lines of argument are those developed by Rodin and Sullivan, but the proof is independent. The deepest part of their work-a uniqueness result of Sullivan's which depends on Mostow rigidity-is replaced here by probabilistic arguments. We work in the setting of hyperbolic geometry and make use of the discrete Schwarz-Pick lemma proven in [BS] to understand the behavior of circle configurations. We analyze how curvature distributes itself around a packing as successive differential changes are made to boundary circles, ultimately modelling this process as a random walk. The proof that a certain limiting random walk is recurrent replaces the uniqueness result of Sullivan. Details will appear elsewhere. The author gratefully acknowledges support of the National Science Foundation and the Tennessee Science Alliance. STATEMENT OF THE MAIN RESULT Let P denote a finite collection of circles in the plane having mutually disjoint interiors. Connect centers of tangent circles with euclidean line segments. If a triangulation of a simply connected closed region of the plane results, we say that P is a circle packing.
Spiral hexagonal circle packings in the plane
Geometriae Dedicata, 1994
We discuss an intriguing geometric algorithm which generates infinite spiral patterns of packed circles in the plane. Using Kleinian group and covering theory, we construct a complex parametrization of all such patterns and characterize those whose circles have mutually disjoint interiors. We prove that these 'coherent' spirals, along with the regular hexagonal packing, give all possible hexagonal circle packings in the plane. Several examples are illustrated.
Circle packings of maps —The Euclidean case
Rendiconti del Seminario Matematico e Fisico di Milano, 1997
In an earlier work, the author extended the Andreev-Koebe-Thurston circle packing theorem. Additionally, a polynomial time algorithm for constructing primal-dual circle packings of arbitrary (essentially) 3-connected maps was found. In this note, additional details concerning surfaces of constant curvature 0 (with special emphasis on planar graphs where a slightly different treatment is necessary) are presented.
A polynomial time circle packing algorithm
Discrete Mathematics, 1993
Mohar, B., A polynomial time circle packing algorithm, Discrete Mathematics 117 (1993) 2577263. The Andreev-Koebe-Thurston circle packing theorem is generalized and improved in two ways. Simultaneous circle packing representations of the map and its dual map are obtained such that any two edges dual to each other cross at the right angle. The necessary and sufficient condition for a map to have such a primal-dual circle packing representation is that its universal cover graph is 3-connected. A polynomial time algorithm is obtained that given such a map M and a rational number E > 0 finds an a-approximation for the primal-dual circle packing representation of M. In particular, there is a polynomial time algorithm that produces simultaneous geodesic line convex drawings of a given map and its dual in a surface with constant curvature, so that only edges dual to each other cross.