# COVERING A SQUARE WITH UP TO 30 EQUAL CIRCLES

@inproceedings{Nurmela2000COVERINGAS, title={COVERING A SQUARE WITH UP TO 30 EQUAL CIRCLES}, author={Kari J. Nurmela and Patric R. J. Osterga}, year={2000} }

A computational method for finding good coverings of a square with equal circles is presented. The algorithm uses a quasi-Newton method with BFGS secant update to minimize the uncovered area by moving the circles. The radius of the circles is further adapted to find locally optimal coverings, and the algorithm is applied repeatedly to random initial configurations. The structures of the coverings are determined and the coordinates of each circle are calculated with high precision using a…

No Paper Link Available

## Figures and Tables from this paper

## 69 Citations

Conjecturally Optimal Coverings of an Equilateral Triangle with Up to 36 Equal Circles

- Computer Science, MathematicsExp. Math.
- 2000

This paper presents a computational method to find good, conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles, 19 of which are either new or improve on earlier published coverings.

Optimal Covering Points and Related Problems

- 2014

We consider the optimal covering of the unit square by N circles. By optimal, we mean the covering that can be done with N circles of minimum radius. Equivalently, we study the problem of the optimal…

A Shape-Newton approach to the problem of covering with identical balls

- Mathematics
- 2021

The problem of covering a region of the plane with a fixed number of minimum-radius identical balls is studied in the present work. An explicit construction of bi-Lipschitz mappings is provided to…

Covering a compact polygonal set by identical circles

- Mathematics, Computer ScienceComput. Optim. Appl.
- 2010

A modification of the Zoutendijk feasible directions method is developed to search local minima of compact polygonal set by identical circles of minimal radius based on Voronoi polygons.

Global Optimization with the use optimal covering

- Computer Science
- 2007

An abridged description of optimal stratified sampling and optimal covering algorithms containing only the essential of the methods is presented and a set of computational results is presented.

Congruent Circles Packing and Covering Problems for Multi-Connected Domains with Non-Euclidean Metric , and Their Applications to Logistics

The article is devoted to optimal covering and packing problems for a bounded set in a two-dimensional metric space with a given amount of congruous circles. Such problems are of both theoretical…

A novel circle covering algorithm based on Coulomb Force Model for the problem of locating repeaters

- Computer Science2011 6th International Conference on Computer Science & Education (ICCSE)
- 2011

A novel circle covering algorithm for the problem of locating repeaters when users in a certain area need to be covered, and a physical model — the Coulomb Force Model is proposed — to solve this multi-layer problem.

On reserve and double covering problems for the sets with non-Euclidean metrics

- Computer ScienceYugoslav Journal of Operations Research
- 2019

The article is devoted to Circle covering problem for a bounded set in a two-dimensional metric space with a given amount of circles, and considers the case where covering set is a multiply-connected domain.

Efficient algorithm for placing a given number of base stations to cover a convex region

- Computer Science, MathematicsJ. Parallel Distributed Comput.
- 2006

An efficient algorithm for the base-station placement problem is developed using Voronoi diagram which works for covering a convex region of arbitrary shape and the execution time is a fraction of a second.

On covering the square flat torus by congruent discs

- Computer ScienceAustralas. J Comb.
- 2019

These are periodic discs coverings of the Euclidean plane by congruent discs of minimal radius such that the square flat torus can be covered by these discs.

## References

SHOWING 1-10 OF 33 REFERENCES

Conjecturally Optimal Coverings of an Equilateral Triangle with Up to 36 Equal Circles

- Computer Science, MathematicsExp. Math.
- 2000

This paper presents a computational method to find good, conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles, 19 of which are either new or improve on earlier published coverings.

Packing up to 50 Equal Circles in a Square

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 1997

Computational methods to find good packings of more than 20 circles are discussed and a new packing of 49 circles settles the proof that when n is a square number, the best packing is the square lattice exactly when n≤ 36.

Improved Coverings of a Square with Six and Eight Equal Circles

- Mathematics, Computer ScienceElectron. J. Comb.
- 1996

Improved coverings with six and eight circles and a new, thin covering with eleven circles are given by the use of simulated annealing and a combinatorial method for constructing lower bounds for the optimal covering radius is presented.

Covering a Rectangle With Equal Circles

- Mathematics
- 1997

Recently, Tarnai and Gáspár [22] used mechanically inspired computer simulations to construct thin coverings of a square with up to ten equal circles. We generalise the problem to rectangles and…

More Optimal Packings of Equal Circles in a Square

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 1999

The computer-aided approach to densest packings of n equal circles in a square is further developed here and the range is extended to n ≤ 27 .

Packing circles in a square: A review and new results

- 1992

There are many interesting optimization problems associated with the packing and covering of objects in a closed volume or bounded surface Typical examples arise in classical physics or chemistry…

Covering a Rectangle with Six and Seven Circles

- Computer Science, MathematicsDiscret. Appl. Math.
- 2000

For six and seven circles, the thinnest possible covering is determined and thin coverings are given for the remaining range of values, thereby extending the previous conjecture for the square.

Dense packings of congruent circles in a circle

- Computer Science, MathematicsDiscret. Math.
- 1998

Algorithm 611: Subroutines for Unconstrained Minimization Using a Model/Trust-Region Approach

- Computer ScienceTOMS
- 1983

This paper describes a set of subroutines designed to solve general unconstrained optimization problems for which it is feasible to work explicitly with either the Hessian of the objective function…

Unsolved Problems In Geometry

- Computer Science
- 1991

A monograph on geometry, each section in the book describes a problem or a group of related problems, capable of generalization of variation in many directions.