On the Motion of Billiards in Ellipses

Abstract

For billiards in an ellipse with an ellipse as caustic, there exist canonical coordinates such that the billiard transformation from vertex to vertex is equivalent to a shift of coordinates. A kinematic analysis of billiard motions paves the way to an explicit canonical parametrization of the billiard and even of the associated Poncelet grid. This parametrization uses Jacobian elliptic functions to the numerical eccentricity of the caustic as modulus.

The Geometry of Billiards in Ellipses and their Poncelet Grids

“…Finally we recall that, based on the Arnold-Liouville theorem from the theory of completely integrable systems, it is proved in [18] that there exist canonical coordinates u on the ellipses e and c such that for any billiard the transitions from P i → P i+1 and Q i → Q i+1 correspond to shifts of the respective canonical coordinates u i and u i+1 by 2∆u. Explicit formulas for the parameter transformation t → u are provided in [24]. (1) , this time with origin Q 1 and unit point Q 3 Figure 9.…”

“…11.2.3.9]. Equivalent conditions in terms of elliptic functions can be deduced from [24,Corollary 3].…”

Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids

“…According to (14), the normal vector of the ellipsoid along the curve e = E ∩ H 1 with the parametrization e(t) from ( 17) is…”

“…According to [14, Theorem 2], a canonical parametrization can be expressed in terms of Jacobian elliptic functions to the modulus d/a c . After replacing u by u := a ′ c u , the three Jacobian elliptic base functions (see, e.g., [8]) are…”

“…Now we transfer two results on elliptic billiards (note [14,Theorems 8 and 9]) to the isometric focal billiards . The second result is a refinement of Corollary 8.…”