The tooth profiles of two meshing spur gears are considered. A given profile may envelop another gear profile. The paper presents an explicit parametrization of the gear profile. It is also shown that the corresponding conjugate rack is explicitly determined.
Analogously, the line of action is described. Afterwards, the inverse problem is studied. If the gear center, the gear ratio, and a line of action are given then conjugate gear profiles are determined by the solution of an ordinary differential equation of first-order. An example of application is shown. This differential equation is geometrically interpreted by introducing the pole distance. At any instant the rate of change of the pole distance relates to the inclination of the rack profile. It is shown that for a specific linear relation, corresponding conjugate profiles are trochoids of second- or third-order.