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Geodesics of a Torus Solved with a Method of Lagrange

A geodesic is the equivalent of a straight line on a surface; locally a geodesic is the shortest path between two points. Lagrange's method can be used to find the differential equations describing the geodesic for a torus, which are then solved with Mathematica's built-in function NDSolve. You can place the frame anywhere on the torus and rotate it to set the initial position and directions of geodesics of a given length.
French
Une géodésique est l'équivalent d'une ligne droite sur une surface; une géodésique est localement le plus court chemin entre deux points. On peut analyser par la méthode de Lagrange pour trouver les équations différentielles des géodésiques, ensuite résolues par méthode numérique avec NDSolve. On peut placer le repère en n'importe quel endroit du tore et lui faire subir une rotation pour afficher les positions et directions initiales des géodésiques selon une longueur reglable.

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