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

An inequality for double tangents


Author: Benjamin Halpern
Journal: Proc. Amer. Math. Soc. 76 (1979), 133-139
MSC: Primary 53A04; Secondary 70B15
MathSciNet review: 534404
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Abstract: For a regular closed curve on the plane it is known that $ E = I + X + \tfrac{1}{2}F$ where E, I, X and F are the numbers of external double tangents, internal double tangents, self-intersections, and inflexion points respectively. It is proven here that if $ F = 0$ then I is even and $ I \leqslant (2X + 1)(X - 1)$ . Furthermore, examples are given which show that if the four tuplet (E, I, X, F) of nonnegative integers satisfies (a) F even, (b) $ E = I + X + \tfrac{1}{2}F$, and (c) if $ F = 0$ then I is even and $ I \leqslant X(X - 1)$, then there is a regular closed plane curve which realizes these values.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0534404-2
PII: S 0002-9939(1979)0534404-2
Keywords: Double tangent, self-intersection, inflexion point, tangent winding number
Article copyright: © Copyright 1979 American Mathematical Society