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References
V. Arnold. Remarks on the extactic points of plane curves, The Gelfand Mathematical Seminars, Birkhäuser, 1996, 11–22.
E. Ghys. Cercles osculateurs et géométrie lorentzienne. Talk at the journée inaugurale du CMI, Marseille, February 1995.
E. Ghys. Osculating curves, talk at the “Geometry and Imagination” Conference, Princeton 2007, http://www.umpa.ens-lyon.fr/~ghys/articles/.
A. Kneser. Bemerkungen über die Anzahl der Extrema der Krümmung auf geschlossenen Kurven und über verwandte Fragen in einer nichteuklidischen Geometrie, Festschrift H. Weber, 1912, 170–180.
S. Mukhopadhyaya. New methods in the geometry of a plane arc, Bull. Calcutta Math. Soc. 1 (1909), 32–47.
V. Ovsienko, S. Tabachnikov. Projective differential geometry, old and new: from Schwarzian derivative to cohomology of diffeomorphism groups, Cambridge University Press, 2005.
V. Ovsienko, S. Tabachnikov. What is ... the Schwazian derivative, Notices of AMS, 56 (2009), 34–36.
S. Tabachnikov, V. Timorin. Variations on the Tait-Kneser theorem. arXiv math.DG/0602317.
P. G. Tait. Note on the circles of curvature of a plane curve, Proc. Edinburgh Math. Soc. 14 (1896), 403.
Acknowledgments
We are grateful to Jos Leys for producing images used in this article. S. T. was partially supported by the Simons Foundation grant No. 209361 and by the NSF grant DMS-1105442. V. T. was partially supported by the Deligne fellowship, the Simons-IUM fellowship, RFBR grants 10-01-00739-a, 11-01-00654-a, MESRF grant MK-2790.2011.1, and AG Laboratory NRU-HSE, MESRF grant ag. 11 11.G34.31.0023.
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This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on.
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Ghys, É., Tabachnikov, S. & Timorin, V. Osculating Curves: Around the Tait-Kneser Theorem. Math Intelligencer 35, 61–66 (2013). https://doi.org/10.1007/s00283-012-9336-6
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DOI: https://doi.org/10.1007/s00283-012-9336-6