The shortest way to the geodesics of spheres

TL;DR

This paper provides an elementary geometric proof that geodesics on spheres are the minor arcs of great circles, applicable to any sphere in any inner product space.

Contribution

It offers a simple, elementary proof of the geodesic characterization on spheres, extending to all spheres in inner product spaces.

Findings

Geodesics on spheres are the minor arcs of great circles.

The proof uses only elementary geometric arguments.

Results apply universally to spheres in any inner product space.

Abstract

In this paper, we prove, using only elementary geometric arguments and only assuming that the curves are continuous, that the geodesics on a sphere are the minor arcs of the great circles. Our result are valid for any sphere in any inner product space.