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:
the wheels are squares and, therefore, with non-constant width. It can be proved  that the road corresponding to a smooth movement for such wheels is made of inverted catenary arcs.
Remark: if you move the mouse over the figures marked with , you will see an animated gif.
if the wheels have the shape of a Reuleaux triangle and, therefore, of constant width, the perfect road is of the type indicated in the figure:
In the exhibition Matemática Viva there was also a module in which wheels of different shapes match the perfect road.