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?

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
