On the rotation index of bar billiards and Poncelet's porism
Artykuł w czasopiśmie
MNiSW
15
Lista A
| Status: | |
| Autorzy: | Cieślak Waldemar, Martini Horst, Mozgawa Witold |
| Rok wydania: | 2013 |
| Wersja dokumentu: | Drukowana | Elektroniczna |
| Język: | angielski |
| Numer czasopisma: | 2 |
| Wolumen/Tom: | 20 |
| Strony: | 287 - 300 |
| Web of Science® Times Cited: | 5 |
| Scopus® Cytowania: | 5 |
| Bazy: | Web of Science | Scopus |
| Efekt badań statutowych | NIE |
| Materiał konferencyjny: | NIE |
| Publikacja OA: | NIE |
| Abstrakty: | angielski |
| 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. |
