# Looking for the curve traced by a moving bicycle when its steering bar is fully rotated

I am looking for a curve traced by a moving bicycle when its steering bar 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.

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