Points, lines, and surfaces of light

Charlista: Pierre Coullet - INPHYNI Université de la Côte d’Azur

Resumen: The envelope of light rays coming from a point source, reflected on curved mirrors, and refracting in diopters was one of the central topics of differential geometry in the 19th century. The mathematical nature of "caustic" surfaces was elucidated by Gaspard Monge and his students Dupin and Malus. The history of optics from Euclid to Newton is essentially geometric. We will show in this seminar, as Gaspard Monge did in his famous article "Les Déblais et les Remblais", that these phenomena are in no way specific to optics. It is simply the study of families of lines in the third dimension depending on two parameters, the "congruence of lines". We will illustrate these issues with several elementary optical experiments and interactive simulations. We will also show the impact of teaching geometry in middle school via optics, where only the compass and the non-graduated ruler are used by the students.