Bulletin of the Belgian Mathematical Society - Simon Stevin

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

W. Cieślak, H. Martini, and W. Mozgawa

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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

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

First available in Project Euclid: 23 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A04: Curves in Euclidean space 51M04: Elementary problems in Euclidean geometries 51N20: Euclidean analytic geometry

bar billiard invariant measure Poncelet porism rotation index self-intersections Steiner formula


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

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