Packing circles within ellipses

Abstract

. AbstractThe problem of packing circles within ellipses is considered in the present paper. A new ellipse-based system of coordinates is introduced by means of which a closed formula to compute the distance of an arbitrary point to the boundary of an ellipse exists. Nonlinear programming models for some variants of 2D and 3D packing problems involving circular items and elliptical objects are given. The resulting models are medium-sized highly nonlinear challenging nonlinear programming problems for which a global solution is sought. For this purpose, multistart strategies are carefully and thoroughly explored. Numerical experiments are exhibited.

A study of ellipse packing in the high-dimensionality problems

“…Authors of study [8] consider the problem of circle packing into an ellipse. This problem is formulated as a nonlinear programming problem [9].…”

“…When using the second approach, affine space transformations result in a transformation of one of the ellipses (ellipsoids) into a circle (sphere), and the other one into a certain ellipse (ellipsoid). Followed the aforementioned transformations, the idea of the method developed in [8] for modeling geometrical relations of a circle and an ellipse was used to formalize conditions of nonintersection of the obtained objects. Generation of "good" starting points and application of the Algencan solver [17] in solution of nonlinear programming problems allowed the authors of paper [16] to improve the majority of results of study [10].…”

Packing ellipsoids by nonlinear optimization

“…Consider the ellipsoid E = {x ∈ R n | x M −1 x ≤ 1}, where M ∈ R n×n is a symmetric and positive definite matrix. After applyingwhere S ij is given by (7). In order to guarantee that E ii and E ij do not overlap, it is enough to require that the distance between the center c ii of the ball E ii and the ellipsoid E ij be greater than or equal to one.…”

“…In particular, many works have tackled the problem with optimization tools. See, for example, [7,9,17,18,19,23,37,50,51,52,53] and the references therein. On the other hand, the problem of packing ellipsoids has received more attention only in the past few years.…”

Augmented Lagrangians with possible infeasibility and finite termination for global nonlinear programming

“…By packing, we mean that the circles must be placed within the ellipse without overlapping. Considering continuous variables u, v, s ∈ IR N , this problem can be modeled [17,28] as a continuous and differentiable nonlinear programming problem as follows:…”