Looking for the curve traced by a moving bicycle when its steering wheel is fully rotated I am looking for a curve traced by a moving bicycle when its steering wheel is fully rotated either clockwise or anti-clockwise.
How to model it mathematically?
Is the curve a circle?
My attempt is as follows:
Let $\vec{r}_1(t)$ and $\vec{r}_2(t)$ be the position vectors for the tangent points (between road and tires) on the rear and front tires, respectively. I know that $|\vec{r}_2(t)-\vec{r}_1(t)|$, $|\dot{\vec{r}}_1(t)|=|\dot{\vec{r}}_2(t)|$  and $\dot{\vec{r}}_1(t)\cdot\dot{\vec{r}}_2(t)$ are constants. $\dot{\vec{r}}_1(t)$ is in the direction of $\vec{r}_2(t)-\vec{r}_1(t)$.
Assuming the tires rolls without slipping then their linear velocity is the same.

It almost has to be a circle, as if the steering wheel isn't moving then the curvature of the track has to be constant.
One way to model it is to imagine just two wheels (like a bicycle).  If you draw line for the direction of travel of each wheel you can find a circle that is tangent to both lines.  This would allow you to get a relation between the angle of the front wheels and the radius of curvature.  The next step would be to handle the width of the car, but then you need the wheels to slip or to rotate different amounts.
