# How do you compute for the center of a rotation of a bicycle?

I know how to find the center of a rotation, like where you connect the side and make bisector with the help of your compass. But in a situation like this how do I get the $a_2$? I'm doing an advance study with steering dynamics and lack the value of $a_2$ I know this might be a little silly, I just want to understand like you know when I ride my bike. This is for the Ackermann geometry steering in cars. The center of rotation is where the line of reaction forces intersect. In the diagram above it is clear that knowning $\ell$ and $\delta$ will yield

$$R_1 = \frac{\ell}{\sin \delta} \, .$$

If you want to find the dynamic center of rotation, given a rigid body and lets say it's center of mass $C$, if you know the position of the center of mass $\vec{r}_C$, the linear velocity of the center of mass $\vec{v}_C$ and the rotational velocity of the rigid body $\vec\omega$ then the instant center of rotation is

$$\vec{r}_O = \vec{r}_C + \frac{\vec \omega \times \vec{v}_C}{\vec\omega \cdot \vec \omega}$$

where $\cdot$ is the vector dot product, and $\times$ the vector cross product. The above is true for any point on the rigid body, not only $C$.

In 2D the above simplifies to

$$(x_O,y_O) = (x_C,y_C) + \left(-\frac{\dot{y}_C}{\omega}, -\frac{\dot{x}_C}{\omega} \right) \, .$$