On the discrete bicycle transformation

The motivation for this paper comes from the study of a simple model of bicycle motion. The bicycle is modeled as an oriented segment in the plane of fixed length `, the wheelbase of the bicycle. The motion is constrained so that the segment is always tangent to the path of the rear wheel; this non-holonomic constraint is due to the fact that the rear wheel is fixed on the frame, whereas the front wheel can steer. See [8, 10, 13, 15] and the references therein. If the rear wheel path γ is prescribed, and the direction of motion is chosen, the front wheel path Γ is constructed by drawing the tangent segments of length ` to γ. Note that the rear track may have cusp: they occur when the steering angle equals 90◦. Changing the direction of motion to the opposite yields another front track, say, Γ′. We say that the curves Γ and Γ′ are in the bicycle correspondence. See Figure 1. If the front wheel path Γ is prescribed then the rear wheel follows a constant-distance pursuit curve, and its trajectory is uniquely determined, once the initial position of the bicycle is chosen. A monodromy map MΓ,` arises that assigns to every initial position of the bicycle its terminal position. If Γ is a closed curve then MΓ,` is a self-map of a circle of radius `, uniquely defined up to conjugation. The bicycle monodromy MΓ,` is a Möbius transformation [9, 10, 13].