An inequality for double tangents
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- by Benjamin Halpern PDF
- Proc. Amer. Math. Soc. 76 (1979), 133-139 Request permission
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.References
- Thomas F. Banchoff, Global geometry of polygons. I: The theorem of Fabricius-Bjerre, Proc. Amer. Math. Soc. 45 (1974), 237–241. MR 370599, DOI 10.1090/S0002-9939-1974-0370599-7
- Fr. Fabricius-Bjerre, On the double tangents of plane closed curves, Math. Scand. 11 (1962), 113–116. MR 161231, DOI 10.7146/math.scand.a-10656
- Benjamin Halpern, Global theorems for closed plane curves, Bull. Amer. Math. Soc. 76 (1970), 96–100. MR 262936, DOI 10.1090/S0002-9904-1970-12380-1
- Benjamin Halpern, Double normals and tangent normals for polygons, Proc. Amer. Math. Soc. 51 (1975), 434–437. MR 372797, DOI 10.1090/S0002-9939-1975-0372797-6
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 76 (1979), 133-139
- MSC: Primary 53A04; Secondary 70B15
- DOI: https://doi.org/10.1090/S0002-9939-1979-0534404-2
- MathSciNet review: 534404