# 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?

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We already have a plane-curves tag. –  Rahul Jan 10 '13 at 1:17
@RahulNarain ok! –  user52188 Jan 10 '13 at 1:18
There's something wrong with your definition of $\gamma$, since you're applying it to $[0,1]$ but it's only defined on $[-1,0]$ (since $\alpha$ is only defined on $[0,1]$). Don't you mean simply $\gamma(t)=\alpha(t)-\alpha'(t)$? –  joriki Jan 10 '13 at 1:38
–  Michael E2 Jan 10 '13 at 2:58
See Exercise 27 on pp. 22-23 of my differential geometry notes math.uga.edu/~shifrin/ShifrinDiffGeo.pdf ... In general the differential equation cannot be solved explicitly and it is highly unlikely that $\tau$ will be a closed curve. –  Ted Shifrin May 6 '13 at 18:43

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.

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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

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