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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Closed planar curves without inflections


Authors: Shuntaro Ohno, Tetsuya Ozawa and Masaaki Umehara
Journal: Proc. Amer. Math. Soc. 141 (2013), 651-665
MSC (2010): Primary 53A04; Secondary 53A15, 53C42
Published electronically: June 1, 2012
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Abstract: We define a computable topological invariant $ \mu (\gamma )$ for generic closed planar regular curves $ \gamma $, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves without inflections) whose numbers of crossings are less than or equal to five. Moreover, we discuss the relationship between the number of double tangents and the invariant $ \mu (\gamma )$ of a given $ \gamma $.


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Additional Information

Shuntaro Ohno
Affiliation: Department of Mathematics, Kyoto Koka Senior High School, Nishi-kyogoku Nodacho, Kyoto 615-0861, Japan
Email: tkmskr0329@yahoo.co.jp

Tetsuya Ozawa
Affiliation: Department of Mathematics, Meijo University, Tempaku, Nagoya, 468-8502 Japan
Email: ozawa@meijo-u.ac.jp

Masaaki Umehara
Affiliation: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-34, O-okayama Meguro-ku, Tokyo 152-8552, Japan
Email: umehara@is.titech.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11319-X
PII: S 0002-9939(2012)11319-X
Keywords: Plane curve, inflection point, double tangent
Received by editor(s): December 21, 2010
Received by editor(s) in revised form: June 21, 2011, and June 26, 2011
Published electronically: June 1, 2012
Additional Notes: The third author was partially supported by the Grant-in-Aid for Scientific Research (A) No. 22244006, Japan Society for the Promotion of Science.
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.