A linearized circle packing algorithm (original) (raw)

A circle packing algorithm

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.

A note on circle packing

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.

Discretization-Based Solution Approaches for the Circle Packing Problem

arXiv (Cornell University), 2023

The problem of packing a set of circles into the smallest surrounding container is considered. This problem arises in different application areas such as automobile, textile, food, and chemical industries. The so-called circle packing problem can be cast as a nonconvex quadratically constrained program, and is difficult to solve in general. An iterative solution approach based on a bisection-type algorithm on the radius of the larger circle is provided. The present algorithm discretizes the container into small cells and solves two different integer linear programming formulations proposed for a restricted and a relaxed version of the original problem. The present algorithm is enhanced with solution space reduction, bound tightening and variable elimination techniques. Then, a computational study is performed to evaluate the performance of the algorithm. The present algorithm is compared with BARON and Gurobi that solve the original nonlinear formulation and heuristic methods from literature, and obtain promising results.

A Literature Review on Circle and Sphere Packing Problems: Models and Methodologies

Advances in Operations Research, 2009

This paper reviews the most relevant literature on efficient models and methods for packing circular objects/items into Euclidean plane regions where the objects/items and regions are either two-or three-dimensional. These packing problems are NP hard optimization problems with a wide variety of applications. They have been tackled using various approaches-based algorithms ranging from computer-aided optimality proofs, to branch-and-bound procedures, to constructive approaches, to multi-start nonconvex minimization, to billiard simulation, to multiphase heuristics, and metaheuristics.

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.

Polynomial-Time Approximation Schemes for Circle Packing Problems

We give an asymptotic approximation scheme (APTAS) for the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height 1 + γ, for some arbitrarily small number γ > 0. Our algorithm is polynomial on log 1/γ, and thus γ is part of the problem input. For the special case that γ is constant, we give a (one dimensional) resource augmentation scheme, that is, we obtain a packing into bins of unit width and height 1 + γ using no more than the number of bins in an optimal packing. Additionally, we obtain an APTAS for the circle strip packing problem, whose goal is to pack a set of circles into a strip of unit width and minimum height. These are the first approximation and resource augmentation schemes for these problems. Our algorithm is based on novel ideas of iteratively separating small and large items, and may be extended to a wide range of packing problems that satisfy certain conditions. These extensions comprise problems with different kinds of items, such as regular polygons, or with bins of different shapes, such as circles and spheres. As an example, we obtain APTAS's for the problems of packing d-dimensional spheres into hypercubes under the L p-norm.

A Reliable Area Reduction Technique for Solving Circle Packing Problems

Computing, 2006

We are dealing with the optimal, i.e. densest packings of congruent circles into the unit square. In the recent years we have built a numerically reliable, verified method using interval arithmetic computations, which can be regarded as a 'computer-assisted proof'. An efficient algorithm has been published earlier for eliminating large sets of suboptimal points of the equivalent point packing problem. The present paper discusses an interval arithmetic based version of this tool, implemented as an accelerating device of an interval branch-and-bound optimization algorithm. In order to satisfy the rigorous requirements of a computational proof, a detailed algorithmic description and a proof of correctness are provided. This elimination method played a key role in solving the previously open problem instances of packing 28, 29, and 30 circles.

Using Different Norms in Packing Circular Objects

Lecture Notes in Computer Science, 2015

A problem of packing unequal circles in a fixed size rectangular container is considered. The circle is considered in a general sense, as a set of points that are all the same distance (not necessary Euclidean) from a given point. An integer formulation is proposed using a grid approximating the container and considering the nodes of the grid as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. Valid inequalities are proposed to strengthening the original formulation. Nesting circles inside one another is considered tacking into account the thickness of the circles. Numerical results on packing circles, ellipses, rhombuses and octagons are presented to demonstrate the efficiency of the proposed approach.

Packing circular-like objects in a rectangular container

Известия Российской академии наук. Теория и системы управления, 2015

Packing problems generally consist of packing a set of items of known dimensions into one or more large objects in order to minimize a certain objective (e.g. the unused part of the objects or waste).

Mathematical analysis of 2D packing of circles on bounded and unbounded planes: Analytic Formulation and Simulation

Cornell University, New York, U.S. (arXiv:2208.08222), Aug, 2022

This paper encompasses the mathematical derivations of the analytic and generalized formula and recurrence relations to find out the radii of n number of circles inscribed or packed in the plane region bounded by circular arcs (including sectors, semi and quarter circles) & the straight lines. The values of radii obtained using analytic formula and recurrence relations have been verified by comparing with those obtained using MATLAB codes. The methods used in this paper for packing circles are deterministic unlike heuristic strategies and optimization techniques. The analytic formulae derived for plane packing of tangent circles can be generalized and used for packing of spheres in 3D space and packing of circles on the spherical surface which is analogous to distribution of non-point charges. The packing density of identical circles, externally tangent to each other, the most densely packed on the regular hexagonal and the infinite planes have been formulated and analysed. This study paves the way for mathematically solving the problems of dense packing of circles in 2D containers, the packing of spheres in the voids (tetrahedral and octahedral) and finding the planar density on crystallographic plane.