Ambiguous Curve: can you follow the bicycle?

Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance between the two wheels is $1$ then we can describe the front track by

$$\tau(t)=\alpha(t)+\alpha'(t)\;.$$

Suppose we know the two (back and front) trace of a bicycle. Can you determine the orientation of the curves? For example if $\alpha$ was a circle the answer is no.

More precisely the question is:

Is there a smooth closed curve parameterized by the arc length $\alpha$ such that

$$\tau([0,1])=\gamma([0,1])$$

where $\gamma(t)=\alpha(1-t)-\alpha'(1-t)$?

If trace of $\alpha$ is a circle we have $\tau([0,1])=\gamma([0,1])$. Is there another?

Solution 1:

Yes, Franz Wegner constructed pairs of smooth closed curves that are not circles and can serve as pairs of bicycle tracks traversed in either direction. They can be expressed analytically in terms of Weierstrass's $\sigma$ and $\zeta$ functions. Interestingly enough such curves also describes shapes that can float in any position, and trajectories of electrons moving in a parabolic magnetic field.

Short description and a picture are here http://www.tphys.uni-heidelberg.de/~wegner/Fl2mvs/Movies.html#animations, mathematical details and more pictures here http://arxiv.org/pdf/physics/0701241v3.pdf.