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?
 A: 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.
A: To me it looks like this. The tracks are two concentric circles. Back wheel turns in a circle radius $b$, frame length is constant = $a$, (instead of 1) tangent to this circle. Front wheel turns on a circle radius $\sqrt{a^2 + b^2}$. You turned handle bar by angle $\alpha = \arctan \frac{a}{b}$. If $\alpha = 90\,^{\circ}$, $b=0$, an extreme special case when back wheel does not move on ground.
A: After the which way did bicycle go book, there has been some systematic development of theory related to the bicycle problem. Much of that is either done or cited in papers by Tabachnikov and his coauthors, available online: 
http://arxiv.org/find/all/1/all:+AND+bicycle+tracks/0/1/0/all/0/1
http://arxiv.org/abs/math/0405445
A: Take a figure eight curve, made of two tangent circles, not necesarily of the same size. 
Or an arbitrary chain of tangent circles. 
A: Would you take an (acrobatic) exercise of riding the bike backwards with the front wheel turned back? If you succeed, and managed to keep the front wheel going along a straight line, then the rear wheel would go along a tractrix – despite which direction you chose to ride.
