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

2014/01/12 by Raul Rojas, Raúl Rojas, Rojas, Raul · 1 voice
Engineering · Mathematics · #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Mathematics and Applications #math.HO

paper · pdf · doi:10.48550/arxiv.1401.2660

openalex publication_date 2014/01/12 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

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

Discussions

• In my opinion, a fundamental insight into the nature of the Brachistochrone problem was put forward by Jacob Bernoulli. Jacob Bernoulli was the older brother of Johann Bernoulli. Johann Bernoulli was [stackexchange, 5 points]

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