## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

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

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

#### 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

**Subjects**

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

**Keywords**

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

#### 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