The inverse trochoid problem

If a circle in the plane is to roll along a track so that a given point on the circle traces out another circle, what must the shape of the track be? In this paper we give a full answer to this question. This question arises in the design of internal combustion engines and other machines. It is a sort of inverse problem to the classical problem of the trajectories of points on circles rolling inside other circles; the problem whose solution is hypocycloids and hypotrochoids. We derive a system of nonlinear differential equations for the coordinates of the track regarded as a parametric curve. We express the solution of this differential equation in terms of an elliptic integral. We establish the uniqueness of this solution for a given choice of the relevant parameters, the radius of the circle to be traced out and the location on the rolling circle of the tracing point. We show that if we allow the rolling circle to intersect the track, there is at least one track for any circle. We find all possible tracks, and we derive formulas to determine, for a given radius of the circle to be traced out, how many different tracks produce circles of that radius; there is always a finite number, and the number increases roughly linearly with the radius. We characterize those tracks that are mathematical solutions but are not practical physical solutions; tracks that the rolling circle intersects in such a way that it would get stuck.