Notion studied by Gregory in 1668, Steiner in 1846 and Habich in 1881. See this page by Alain Esculier, for all the animations he created, only a part of which is presented here.

Two curves and  form a wheel-road couple if  can roll without slipping on  so that a fixed point on the plane of  (the wheel hub) has a linear trajectory in the fixed plane. It is therefore a movement of a plane over a fixed plane the base of which is , the rolling curve of which is  and a roulette of which is linear.
The curves  and  can also be considered as two mating gear profiles, the hub of  being located at infinity (consider the movement in a frame linked to the wheel hub).

 If the wheel  is defined by its polar equation, when its hub is at O, and the road  by its Cartesian equation, the formulas linking the two equations are . From a wheel , we get the road  where F is a primitive of fg' (Grégory transformation). Conversely, from a road , we get the wheel  where H is a primitive of . If the wheel is defined by its pedal equation: , then the differential equation of the road is .

This notion was initially studied, not for a practical use of noncircular wheels, but because the calculations of the curvilinear abscissa are the same for the two curves (for the road, in Cartesian coordinates, and for the wheel, in polar coordinates), so that the rectification of one of them gives that of the other one.
There exist 2 theorems providing a geometrical definition of the road when the wheel is given.

 1) Steiner-Habich theorem: the road is the roulette with linear base of the negative pedal of the wheel; more precisely, if the negative pedal  of the wheel with respect to the hub rolls on a line, the locus of the hub is the road. In other words: given a curve  and a point, the pedal of this curve with respect to this point and its roulette with linear base form a wheel-road couple. When (C) rolls on (D), a point M of the plane linked to (C) describes a roulette (R) in the fixed plane. The pedal (P) of (C) with respect to M cuts (D) at the projection M' of M on (D). It can be proved that the curve (P'), symmetrical image of (P) about the perpendicular bisector of [MM'], rolls without slipping on the curve (R), which proves the Habich theorem, since M' describes (D).

2) Mannheim theorem:
Given a curve  and a point, the radial curve of this curve with respect to this point and its Mannheim curve form a wheel-road couple.

Examples:

If the wheel is circular with a centred hub, the road is a line parallel to the trajectory of the hub and it is the only case where this happens, but there is more than just this well-known case!