Measure of overlap between two arbitrary ellipses on a sphere.

Area of Science:

• Physics
• Computer Science
• Geometry

Background:

• Packing problems and particle simulations often involve calculating intersections and interactions between ellipsoids.
• The behavior of particles on a spherical surface is complex due to curvature and compactness.
• While disk packing (Tammes problem) is studied, ellipse packing on a sphere remains less explored.

Purpose of the Study:

• To develop an algorithm for detecting collisions between ellipses on a sphere.
• To investigate the distinct ways ellipses can interact on a spherical surface.
• To define a contact function for pair potentials in simulations.

Main Methods:

• Developed an algorithm based on solving an eigenvalue problem to detect ellipse collisions.
• Analyzed topologically distinct intersection and contact configurations of ellipses on a sphere.

Main Results:

• Successfully created an algorithm for detecting ellipse-sphere collisions.
• Identified and categorized various topological arrangements of touching or intersecting ellipses.
• Defined a contact function applicable to short- and long-range pair potentials.

Conclusions:

• The developed algorithm provides a method for simulating anisotropic particle interactions on spherical surfaces.
• This work opens new avenues for exploring complex packing problems involving elliptical shapes.
• The defined contact function can be utilized in advanced physics and computer science models.