Bulletin of the Belgian Mathematical Society - Simon Stevin

On the rotation index of bar billiards and Poncelet's porism

Abstract

We present some new results on the relations between the rotation index of bar billiards of two nested circles $C_R$ and $C_r$, of radii $R$ and $r$ and with distance $d$ between their centers, satisfying Poncelet's porism property. The rational indices correspond to closed Poncelet transverses, without or with self-intersections. We derive an interesting series arising from the theory of special functions. This relates the rotation number $\frac 13$, of a triangle of Poncelet transverses, to a double series involving $R, r$, and $d$. We also provide a Steiner-type formula which gives a necessary condition for a bar billiard to be a pentagon with self-intersections and rotation index $\frac 25$. Finally we show that, close to a pair of circles having Poncelet's porism property for index $\frac{1}{3}$, there exist always circle pairs having indices $\frac{1}{4}$ they and $\frac{1}{6}$; in the case $\frac{1}{4}$ they are even unique.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2 (2013), 287-300.

Dates
First available in Project Euclid: 23 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1369316545

Digital Object Identifier
doi:10.36045/bbms/1369316545

Mathematical Reviews number (MathSciNet)
MR3082765

Zentralblatt MATH identifier
1278.53006

Citation

Cieślak, W.; Martini, H.; Mozgawa, W. On the rotation index of bar billiards and Poncelet's porism. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 2, 287--300. doi:10.36045/bbms/1369316545. https://projecteuclid.org/euclid.bbms/1369316545