Cars supported on axles

When the car rests on axles and the wheel is not circular, it has to travel on an irregular road to avoid it going up and down.
It is intended that during the movement the wheel and the road remain in contact, that the length traveled on the edge of the wheel is the same as that described on the road, and furthermore that the geometric center of the wheel travels without oscillations.

We will consider two cases:

Remark: if you move the mouse over the figures marked with , you will see an animated gif.

In the exhibition Matemática Viva there was also a module in which wheels of different shapes match the perfect road.