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 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 then *I* is even and . Furthermore, examples are given which show that if the four tuplet (*E, I, X, F*) of nonnegative integers satisfies (a) *F* even, (b) , and (c) if then *I* is even and , then there is a regular closed plane curve which realizes these values.

**[B]**Thomas F. Banchoff,*Global geometry of polygons. I: The theorem of Fabricius-Bjerre*, Proc. Amer. Math. Soc.**45**(1974), 237–241. MR**0370599**, 10.1090/S0002-9939-1974-0370599-7**[F]**Fr. Fabricius-Bjerre,*On the double tangents of plane closed curves*, Math. Scand**11**(1962), 113–116. MR**0161231****[H1]**Benjamin Halpern,*Global theorems for closed plane curves*, Bull. Amer. Math. Soc.**76**(1970), 96–100. MR**0262936**, 10.1090/S0002-9904-1970-12380-1**[H2]**Benjamin Halpern,*Double normals and tangent normals for polygons*, Proc. Amer. Math. Soc.**51**(1975), 434–437. MR**0372797**, 10.1090/S0002-9939-1975-0372797-6

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1979-0534404-2

Keywords:
Double tangent,
self-intersection,
inflexion point,
tangent winding number

Article copyright:
© Copyright 1979
American Mathematical Society