Measure of overlap between two arbitrary ellipses on a sphere

Abstract

Various packing problems and simulations of hard and soft interacting particles, such as microscopic models of nematic liquid crystals, reduce to calculations of intersections and pair interactions between ellipsoids. When constrained to a spherical surface, curvature and compactness lead to non-trivial behaviour that finds uses in physics, computer science and geometry. A well-known idealized isotropic example is the Tammes problem of finding optimal non-intersecting packings of equal hard disks. The anisotropic case of elliptic particles remains, on the other hand, comparatively unexplored. We develop an algorithm to detect collisions between ellipses constrained to the two-dimensional surface of a sphere based on a solution of an eigenvalue problem. We investigate and discuss topologically distinct ways two ellipses may touch or intersect on a sphere, and define a contact function that can be used for construction of short- and long-range pair potentials.

Dense packings of geodesic hard ellipses on a sphere

“…The details of this description are given in ref. 50. As α → π/2, the semi-major axis of this ellipsoid approaches R , resulting in a sharper and sharper vertex of the spherical ellipse (see Fig.…”

“…29,34,52,56 The procedure requires a measure of overlap to be defined as the system energy; however, conventional overlap parameters used for measuring distances between ellipsoids in Euclidean geometries 57–59 do not work for purely 2D ellipses on a sphere. We therefore use a recently developed algorithm by Gnidovec et al 50 to determine the overlap function λ between two spherical ellipses. This algorithm is based on calculating the eigenvalues of a linear interpolation between quadratic forms that define spherical ellipses, and returns λ < 1 if two ellipses overlap.…”

“…The ellipses are defined as having a constant sum of geodesic distances to two foci, which also corresponds to having elliptical orthogonal projections onto a plane. 50 We first introduce the model and explain the packing simulation procedure used to generate jammed configurations of spherical ellipses. We then present packing results, demonstrating that the packing fraction depends non-monotonously on the ellipse aspect ratio and can change with system size.…”