Fronts and caustics in billiard tables

By: Gil Bor, CIMAT, gil@cimat.mx

(Under construction. Last modified: .)


Definitions. Pick an initial point inside a convex smooth curve and shine light from it (or shoot billiard balls) in all directions. When a light ray reaches the boundary it is reflected as usual. The envelope of the beam after $n$-reflections is the $n$-th caustic wrt the initial point. Theorem: for a generic initial point each caustic has $\geq 4$ cusps. Conjecture: for an ellipse there are exactly 4 cusps. The wave front at time $t$ is the locus of pts reached by all light rays after $t$ time units.


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