For comparison purposes, we note that a reciprocal shape change would be one in which we performed a sequence of maneuvers as follows: (1) steer angle θ by α degrees, (2) turn φ by β degrees, (3) turn φ by -β degrees, (4) steer θ by -α degrees. The transformation matrix for this sequence of maneuvers can be computed to be

The transformation matrix M does not move the car at all!