# How small is the smallest circle a car can drive?

Lets say we have a model of a car with two fixed back wheels and two wheels in front that steer in the same angle:

• The wheelbase $w$ is the fixed distance of the two wheel axes.
• $\alpha_m$ is the maximum angle we can steer

What is the radius $r$ of the smallest left-circle we can drive with wheelbase $w$ and maximum left angle $\alpha_m$?

Clarification: You can suppose that the car has only one frontwheel and one backwheel.

(Does anybody know if there are books about "car physics" that deal with such questions? I think there has to be plenty of material about it, because this might be important for every car racing game. I was just thinking about how the position and orientation of a car changes when it drives for $t$ seconds with initial orientation $\alpha$ and position $(x,y)$, steering $\beta$ and velocity $v$. But I even can't answer the question above at the moment :-/ )

• I just found an excellent, easily readable paper on the subject. Any answer I could provide I would probably borrow from this. Enjoy. – J. W. Perry Oct 27 '13 at 19:00