Mobility spaces and geodesics for the n-sphere

TL;DR

This paper introduces a new algebraic framework called mobility spaces to model geodesic paths on the n-sphere, connecting algebraic structures with geometric properties of spherical geodesics.

Contribution

It develops the concept of mobility spaces based on mobility algebras, linking modules over rings with affine and spherical geodesic structures.

Findings

Mobility spaces model geodesics on the n-sphere.

The formula for spherical linear interpolation is an example of a mobility space.

Strong connection established between modules over rings and affine mobility spaces.

Abstract

We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers. We show that there is a strong connection between modules over a ring and affine mobility spaces over a mobility algebra. However, geodesics in general fail to be affine thus giving rise to the new algebraic structure of mobility space. We show that the so called formula for spherical linear interpolation, which gives geodesics on the n-sphere, is an example of a mobility space over the unit interval mobility algebra.