Con el objetivo de empezar el semestre de la mejor manera posible invitamos muy cordialmente a toda la comunidad DIM a un nuevo ciclo del de Coloquio DIM.
El próximo coloquio es el martes 31 de marzo a las 16:15, en la sala von Neumann ubicada en el séptimo piso del Centro de Modelamiento Matemático (CMM). En esta ocasión, Pierre Coullet nos presentará una pregunta física que se conecta con ecuaciones geométricas interesantes, ilustrada con simulaciones numéricos y un experimento simple en vivo.
Título de la charla: «The Geometry of Shadows: From Alhazen to Euler and Monge».
Abstract: We study the geometry of shadows produced by a curvilinear light source partially occluded by a curvilinear obstacle. The resulting shadow patterns exhibit a surprising geometric richness, including structures reminiscent of optical caustics and displaying stable singularities such as folds, cusps and swallow tails.
The starting point of our analysis can be traced back to Alhazen. In his treatise « On the Quality of Shadows », he introduced a photometric viewpoint closely related to eclipse geometry: from a given point on the screen, one determines which portions of the source are visible or hidden by the obstacle.
From a geometric point of view, the problem leads naturally to the study of a congruence of rays joining the source and the obstacle. The effective rays responsible for the shadow envelope correspond to singular directions of this congruence. Their union forms ruled developable surfaces.
This perspective connects the optical problem to the theory of developable surfaces developed by Euler and later formalized by Monge. The envelope of effective rays can be interpreted as a developable surface whose edge of regression governs the singular structure of the shadow boundary.
The photometric structure of the shadow reveals nontrivial scaling laws near these singular curves. The corresponding exponents are directly related to the type of underlying singularity, showing that the geometry of the congruence controls not only the shape of the shadow but also the distribution of light intensity.
A simple tabletop experiment and numerical simulations will be used to visualize the shadow patterns
