Curves of constant diameter and inscribed polygons

Details Curves of constant diameter and inscribed polygons Curves of constant diameter and inscribed polygons

2005-04-14

arxiv.org/abs/math/0504300v2

Elemente der Mathematik, 2009, vol. 64, no. 3, p. 102-108

Mathematics

Metric Geometry

6 pages, 10 figures. Aimed at an undergraduate audiance. V2 includes the published version, with many suggestions from the ref

Scientific paper

A simple closed curve in the Euclidean plane is said to have property C_n(R) if at each point we can inscribe a unique regular $n$-gon with edges length $R$. C_2(R) is equivalent to having constant diameter. We show that smooth curves satisfying C_n(R) other than the circle do exist for all n, and that the circle is the only $C^2$ regular curve satisfying C_2(R) and C_4(R') where $R'=R/\sqrt{2}$. In an addendum, we show that the last assertion holds for any R and R'. The proofs use only elementary differential calculus and geometry.

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