An introduction to geodesics: the shortest distance between two points

TL;DR

This paper provides an accessible overview of geodesics, explaining how they are computed as shortest paths on surfaces using variational methods, with applications in navigation, optics, and robotics.

Contribution

It introduces the methods for deriving geodesics through variational calculus, making the concept more understandable and highlighting its practical significance.

Findings

Geodesics are the shortest paths on surfaces.

Variational methods are used to compute geodesics.

Applications include aircraft navigation, light travel, and robot motion planning.

Abstract

We give an accessible introduction and elaboration on the methods used in obtaining a geodesic, which is the curve of shortest length connecting two points lying on the surface of a function. This is found through computing what's known as the variation of a functional, a "function of functions" of sorts. Geodesics are of great importance with wide applications, e.g. dictating the path followed by aircraft (great-circles), how light travels through space, assist in the process of mapping a 2D image to a 3D surface, and robot motion planning.