Apollonian packing of circles within ellipses

The purpose of a circle packing procedure is to fill up a predefined, geometrical, closed contour with a maximum finite number of circles. The subject has received considerable attention in pure and applied sciences and has proved to be highly effective in connection with many a problem in logistics and technology. The well-known Apollonian circle packing achieves the packing of an infinite number of mutually tangent smaller circles of decreasing radii, internal or tangent to the outer boundary. Algorithms are available in the literature for the packing of equal-radius circles within an ellipse for global optimization purposes. In this paper, we propose a new algorithm for the Apollonian packing of circles within an ellipse, based on fundamental numerical methods, granting suitable speed, accuracy and stability. The novelty of the proposed approach consists in its applicability to the Apollonian packing of circles within a generic, closed, convex contour, if the parametrization of its outer boundary is given.