Related Problems in Spherical and Solid Geometry

TL;DR

This paper explores a simple yet unexamined problem involving four great circles on a sphere, relating it to solid geometry and practical applications like the Perspective 3-Point problem.

Contribution

It introduces a novel problem in spherical geometry, connects it to solid geometry, and demonstrates its relevance to practical pose estimation applications.

Findings

Established a relationship between spherical and solid geometry problems.

Proved a significant claim relating four great circles to tetrahedral angles.

Linked the geometric problem to the Perspective 3-Point (Pose) Problem.

Abstract

A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real projective plane. The problem on the sphere involves four great circles and their intersections. A substantial claim is made concerning this problem, and subsequently proved by relating the spherical problem to a compelling problem in solid geometry. This latter problem essentially concerns relationships between the angles of a tetrahedron, and has practical applications, particularly in connection with the Perspective 3-Point (Pose) Problem.