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arXiv · 1401.2660

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

Raul Rojas
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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.

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Raul Rojas. 2014-01-12. The straight line, the catenary, the brachistochrone, the circle, and Fermat. https://arxiv.org/abs/1401.2660

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