Bicycle Curves

In this paper we examine bicycle curves from the point of view of Fourier series. We analyze necessary and sufficient conditions for a circle to admit an infinitesimal deformation as a bicycle curve with rotation number ρ and derive conditions equivalent to those obtained by Tabachnikov. We then show that whenever ρ ∈ Q is not 1 2 , that there is no such deformation. Introduction Two closed curves Γ(t) and γ(t) in R are called bicycle curves if, under some parameterization, the distance from Γ(t) and γ(t) remains constant as t varies. These curves are called ambiguous bicycle curves if there is both a positively and negatively oriented direction in which the two can be traced such that the distance between them remains constant. A result due to Tabachnikov established that given Γ(t), there exists γ(t) for which the pair is ambiguous if and only if the arc length parameterization of Γ has the property that for some 0<ρ<1, |Γ(t+ρ)-Γ(t)| is constant for all t; in this case Γ(t) is called a bicycle curve with rotation number ρ. Trivially, it is clear that a circle is an example of a bicycle curve with arbitrary rotation number. Interest in this paper, therefore, lies in finding noncircluar bicycle curves. As these turn out not to be obvious, it is interesting to determine when a circle can be infinitesimally deformed into a bicycle curve with a given rotation number. This was originally examined by Tabachnikov, who showed that a circle admits an infinitesimal deformation as a bicycle curve with rotation number ρ if and only if ∃ m∈Z≥2 such that m tan(πρ) = tan(mπρ). Bicycle Curves as Fourier Series Let Γ(t) be a bicycle curve, then it satisfies: | Γ′(t) |= 1,∀t (1)