ArticleOriginal scientific text
Title
Rotation indices related to Poncelet’s closure theorem
Authors , ,
Abstract
Let CRCr denote an annulus formed by two non-concentric circles CR, Cr in the Euclidean plane. We prove that if Poncelet’s closure theorem holds for k-gons circuminscribed to CRCr, then there exist circles inside this annulus which satisfy Poncelet’s closure theorem together with Cr, with n- gons for any n > k.
Bibliography
- Berger, M., Geometry, I and II, Springer, Berlin, 1987.
- Black, W. L., Howland, H. C., Howland, B., A theorem about zigzags between two circles, Amer. Math. Monthly 81 (1974), 754–757.
- Bos, H. J. M., Kers, C., Dort, F., Raven, D. W., Poncelet’s closure theorem, Expo. Math. 5 (1987), 289–364.
- Cima, A., Gasull, A., Manosa, V., On Poncelet’s maps, Comput. Math. Appl. 60 (2010), 1457–1464.
- Cieslak, W., The Poncelet annuli, Beitr. Algebra Geom. 55 (2014), 301–309.
- Cieslak, W., Martini, H., Mozgawa, W., On the rotation index of bar billiards and Poncelet’s porism, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), 287–300.
- Lion, G., Variational aspects of Poncelet’s theorem, Geom. Dedicata 52 (1994), 105– 118.
- Martini, H., Recent results in elementary geometry, Part II, Symposia Gaussiana, Proc. 2nd Gauss Symposium (Munich, 1993), de Gruyter, Berlin and New York, 1995, 419–443.
- Schwartz, R., The Poncelet grid, Adv. Geom. 7 (2007), 157-175.
- Weisstein, E. W., Poncelet’s Porism, http:/mathworld. wolfram. com/Ponceletsporism.html