Billiards in Conics Revisited - Optical Properties and Geometric Consequences

Explore the fascinating world of optical properties in conics and billiard reflections in this comprehensive lecture. Delve into the classical understanding of conics dating back to antiquity and discover how the billiard inside an ellipse demonstrates complete integrability. Learn about the interior of an ellipse foliated by confocal ellipses acting as caustics, where light rays tangent to a caustic remain tangent after reflection. Examine classic results and their geometric consequences, including the Ivory lemma, which proves the equality of diagonals in curvilinear quadrilaterals formed by arcs of confocal ellipses and hyperbolas. Investigate applications such as the bicentennial Poncelet Porism, a renowned theorem in projective geometry, along with its lesser-known offshoots like the Poncelet Grid theorem and related circle patterns and configuration theorems. Gain insights into the mathematical principles underlying these concepts and their significance in geometric analysis.