Rotation indices related to Poncelet’s closure theorem
Waldemar Cieślak ; Horst Martini ; Witold Mozgawa
Annales UMCS, Mathematica, Tome 68 (2015), / Harvested from The Polish Digital Mathematics Library
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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 ngons for any n > k.

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:269955 
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     journal = {Annales UMCS, Mathematica},
     volume = {68},
     year = {2015},
     zbl = {1312.51005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_umcsmath-2015-0003}
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Waldemar Cieślak; Horst Martini; Witold Mozgawa. Rotation indices related to Poncelet’s closure theorem. Annales UMCS, Mathematica, Tome 68 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_umcsmath-2015-0003/

[1] Berger, M., Geometry, I and II, Springer, Berlin, 1987.

[2] Black, W. L., Howland, H. C., Howland, B., A theorem about zigzags between two circles, Amer. Math. Monthly 81 (1974), 754-757. | Zbl 0291.50008

[3] Bos, H. J. M., Kers, C., Dort, F., Raven, D. W., Poncelet’s closure theorem, Expo. Math. 5 (1987), 289-364.[WoS] | Zbl 0633.51014

[4] Cima, A., Gasull, A., Manosa, V., On Poncelet’s maps, Comput. Math. Appl. 60 (2010), 1457-1464.[WoS] | Zbl 1201.51024

[5] Cieślak, W., The Poncelet annuli, Beitr. Algebra Geom. 55 (2014), 301-309. | Zbl 1298.53004

[6] 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), 287-300. | Zbl 1278.53006

[7] Lion, G., Variational aspects of Poncelet’s theorem, Geom. Dedicata 52 (1994), 105-118. | Zbl 0808.51025

[8] 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. | Zbl 0852.51013

[9] Schwartz, R., The Poncelet grid, Adv. Geom. 7 (2007), 157-175.[WoS] | Zbl 1123.51027

[10] Weisstein, E. W., Poncelet’s Porism, http:/mathworld.wolfram.com/Ponceletsporism.html