Take the tour ×  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.  share|improve
this question add comment  2 Answers  If by "fully rotated" you
mean that the front wheel is turned 90 degrees, then there is no
solution in which the wheels roll without slipping. The only
time the no-slip condition is satisfied while the wheels are
orthogonal is when the bike is stationary.  share|improve this
answer