Rotation indices related to Poncelet’s closure theorem

Waldemar Cieślak, Horst Martini, Witold Mozgawa

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.

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References


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




DOI: http://dx.doi.org/10.17951/a.2014.68.2.19
Date of publication: 2015-05-23 16:29:44
Date of submission: 2015-05-09 13:18:40


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