# During a turn, do the rear wheels necessarily trace out the same arcs as the front wheels?

When a vehicle makes a turn, the two front wheels trace out two arcs as shown in the figure below. The wheel facing towards the inside of the turn has a steering angle that is greater than that of the outer wheel. This is necessary to ensure that both front wheels smoothly trace out two arcs, which have the same center, otherwise the front wheels will skid on the ground during the turn.

During a turn, do the rear wheels necessarily trace out the same arcs as the front wheels?

No, in general they do not. You can work this out from the geometry of Ackermann steering, discussed on my website article "Parallel Parking a Car".

In summary: look at the defining geometry for Ackermann steering, which I have sketched below:

Ackermann steering is defined by the intersection of the central unit normals to (axes of rotational symmetry of) all wheels at a common point $C$ in the diagram. It should therefore be very clear from the diagram that the radius of curvature for each wheel is different: the curvature radiusses for the forward wheel paths are a little bit bigger than those of the hinder wheel paths.

You can also see that if the forward and hinder wheels steer symmetrically as in some four wheel steering systems, corresponding wheel pairs do follow the same arc as the curvature radiusses would then be equal.

Note that the Argand plane co-ordinates are simply for my website, where I want to compute the group of transformations available to the driver and show that these transformations do indeed generate the whole of the proper Euclidean group $E^+(2)$- i.e. any simultaneous translation and planar orientation rotation can be realized as a finite sequence of steering operations.

Real cars always deviate from the Ackermann condition, however slightly. Indeed, if the tyre has a nonzero width, some part of it must either deform cyclically as it passes through the point of contact with the ground and practically there is always some slipping, which you can hear if you drive the car very slowly on a polished concrete surface (e.g. in a undercover carpark) and open your window.

Of course not. The rear wheels follow the trajectory of the front wheels only when the car is moving in a straight line.

At the turn, everything is completely different. The rear wheels tend to catch up with the front ones along a shorter path. They seem to "cut the corner", while the rear part of the car shifts closer to the center of the turn.

It's a common cause of accidents that they don't.

If you are turning a sharp corner (a good example would be turning into a narrow parking bay next to a pillar in a car park) any experienced driver knows that you need to let the front wheels get almost level with the centre before turning in as hard as you can. If you take a wide sweeping turn and assume the back wheels will follow the front you'll scrape the side of the car along the pillar (or whatever else is on the inside of the turn), as I am sure many people have done.