  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
nontrivial behavior 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.
