# Turning radius of a vehicle

What's the minimum turning radius of a vehicle, rectangular in shape, with length l units and width w units? One key point to consider, would be that, the inclination of the front wheels can be varied continuously(from 0 to 45 degrees - in both directions) while taking a turn(expert drivers do that all the time and can turn a vehicle in a very short space.).

This is not a homework problem. It's a problem that popped in my head, due to my own parking woes! As such, I am not sure, that the data given would be enough to solve it. So readers are welcome to improve the question(i.e they can suggest additional constraints or data that needs to be given).

While I've previously taken an analytic geometry course in high school, I've never attempted a problem like this. In college,(being an electrical engineering student) my analytical geometry education was not as comprehensive as that of a mechanical or civil engineering student. So I've no idea how to proceed.

This is the first in a series of parking problems, that popped in my head! Some of the other problems require more careful wording, so I've delayed their posting.

PS: When I say turning radius, I am referring to the wall to wall turning radius, as defined in the wikipedia entry on 'Turning Radius'. Also see this - http://firechief.com/apparatus/firefighting_turning_circles

EDIT: In case you can solve this question for any particular set of costraints, do proceed. Just make sure that you mention what those constraints are. Hopefully, someone else might be able to generalize. In case you need additional data, you can assume that data - again just state your data assumptions, so that others can understand.

-
I think you would need to bound the number of 'phases' (i.e. the left-right switches) to turn your car, otherwise the limit would be the circumcircle of vehicle shape I guess. – dtldarek Feb 26 '13 at 12:36
@dtldarek, the angle of the front wheels with respect to the wehicle body is limited to 45º – vonbrand Feb 26 '13 at 12:57
@vonbrand, what dtltarek was saying is that the turn of 45 degrees can be done in any number of switches(2, 3, 4 etc). So perhaps the number of switches need to be bound. I am not in a position to answer whether this is necessary, but I think it would be an important point to consider. – Nikhil Panikkar Feb 26 '13 at 13:44
Something important to consider is that both wheels do not turn to the same inclination... we use Ackermann steering for better turns... – apnorton Feb 26 '13 at 13:54
Regarding praking problems: I know that for a school level competition a young German researcher wrote a computer program to construct parking trajectories using clothoids. See this PDF starting at page 9 if you understand German or want a look at the pictures. Might be of interest, although slightly off topic. – MvG Feb 27 '13 at 21:59