Bicycling mathematics

(Under construction. Last modified: .)

Definitions. A pair of parametrized curves $(\mathbf{b}(t), \mathbf{f}(t))$ in $\mathbb R^2$ is a bicycle path if (1) $\|\mathbf{b}(t)- \mathbf{f}(t)\|=\ell$ (a constant, the 'bicycle length'), and (2) $\dot{\mathbf{b}} \| (\mathbf{b}-\mathbf{f})$ at all times (the 'no skid' condition). For a given (smooth) $\mathbf{f}(t)$ one can write $\mathbf{b}=\mathbf{f}+\ell\mathbf{r}$, with $\mathbf{r}(t)\in S^1$, then the above is equivalent to $\ell\dot{\mathbf{r}}=-\dot{\mathbf{f}}+(\mathbf{r}\cdot\dot{\mathbf{f}}) \mathbf{r}$ (the 'bicycle eqn'). The flow of this ODE, over some segment of the front track $\mathbf{f}(t)$, defines a circle map, which turns out to be an element of $\mathrm{PSL}_2(\mathbb{R})$ (a Mobius transformation), called the bicycling monodromy of the front track (esp when it is closed).

The bicycling eqn makes sense also in $\mathbb{R}^{n}$, where $\mathbf{r}\in S^{n-1}$ and where the monodromy $S^{n-1}\to S^{n-1}$ is a Mobius (or conformal) transformation. In case $n=3$ the Mobius group is $\mathrm{PSL}_2(\mathbb{C}).$

Bicycling geodesics. Define the length of a bike path to be the length of the front track. What are the bicycling geodesics? (paths of shortest length, between some two fixed configurations). Answer: the front track is a non-inflectional Euler's elasicae.

Gallery

Which way will the the bicycle move if we pull on the pedal? (solution here, but think 1st!)	Non intuitive behavior	Which way did the bicycle go?
Bicycle correspondence	The tractrix	Prytz planimeter
Parabolic monodromy	Hyperbolic monodromy	A 4-fold circle with trivial monodromy
3D bicycling	A 'wide' bicycling geodesic	A 'narrow' bicycling geodesic
Euler's soliton	Flipping a narrow geodesic	Flipping a wide geodesic