The straight line, the catenary, the brachistochrone, the circle, and Fermat

TL;DR

This paper presents a unified formalism for classical curve optimization problems, deriving geodesics and metrics for various curves like the straight line, catenary, and circle, and finds numerical solutions using Fermat's principle.

Contribution

It introduces a unified differential equation framework for classical curves and demonstrates how to derive metrics and geodesics for new functions like the parabola.

Findings

Unified formalism for classical curves

Derivation of metrics and geodesics for new functions

Numerical solutions using Fermat's principle

Abstract

This paper shows that the well-known curve optimization problems which lead to the straight line, the catenary curve, the brachistochrone, and the circle, can all be handled using a unified formalism. Furthermore, from the general differential equation fulfilled by these geodesics, we can guess additional functions and the required metric. The parabola, for example, is a geodesic under a metric guessed in this way. Numerical solutions are found for the curves corresponding to geodesics in the various metrics using a ray-tracing approach based on Fermat's principle.