Tractrices, Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old Conjecture

The geometry of the tracks left by a bicycle has received much attention recently [9, 10, 11, 18, 20, 27]. In this paper we discuss the connection between the motion of a bicycle and that of a curious device known as a hatchet planimeter, and we will prove a conjecture about this planimeter that was made in 1906. Bicycle. We use a very simple model of a bicycle as a moving segment in the plane. The segment has fixed length `, the wheelbase of the bicycle. We denote the endpoints of the segment by F and R for the front and rear wheels. The motion is constrained so that the segment is always tangent to the path of the rear wheel. We will refer to this as the “bicycle constraint”. This non-holonomic constraint is due to the fact that the rear wheel is fixed on the frame, whereas the front wheel can steer. The configuration space of a segment of fixed length is 3-dimensional, and the bicycle constraint defines a completely non-integrable 2-dimensional distribution on it. This is an example of a contact structure, see, e.g., [2, 14]; we shall not dwell on this connection with contact geometry. If the path of the front wheel F is prescribed then the rear wheel R follows a constant-distance pursuit curve. The trajectory of the rear wheel is uniquely determined once the initial position of the bicycle is chosen. For example, when F follows a straight line, R describes the classical tractrix, see Figure 1. More generally, one may call the trajectory of the rear wheel R the tractrix of the trajectory of the front wheel F .