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

We present some new results on the relations between the rotation index of bar billiards of two nested circles [C.sub.R] and [C.sub.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 transvers...

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Bibliographic Details
Published in:Bulletin of the Belgian Mathematical Society, Simon Stevin Simon Stevin, 2013-04, Vol.20 (2), p.287-300
Main Authors: Cieślak, W., Martini, H., Mozgawa, W.
Format: Article
Language:English
Subjects:
51M04
51N20
53A04
bar billiard
Billiards
Homeomorphisms
invariant measure
Poncelet porism
rotation index
self-intersections
Steiner formula
Theorems (Mathematics)
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Summary:We present some new results on the relations between the rotation index of bar billiards of two nested circles [C.sub.R] and [C.sub.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 1/3, 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 2/5. Finally we show that, close to a pair of circles having Poncelet's porism property for index 1/3, there exist always circle pairs having indices 1/4 they and 1/6; in the case 1/4 they are even unique. Key words and phrases: bar billiard, invariant measure, Poncelet porism, rotation index, self-intersections, Steiner formula.
ISSN:1370-1444
2034-1970
DOI:10.36045/bbms/1369316545