Roulette (curve) application of this is that a square wheel could roll without bouncing on a road that is a matched series of catenary arcs. Roulette (curve) In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes. Roughly speaking, a roulette is the curve described by a point (called the "generator" or "pole") attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. More precisely, given a curve attached to a plane which is moving so that the curve rolls, without slipping, along
Curve resistance (railroad) is zero cant deficiency and results in a frictionless banked turn. But deviate from this speed (either higher or lower) and the curve resistance increases due to the unbalance in forces which tends to pull the vehicle sideways (and would be felt by a passenger in a passenger train). Note that for empty rail cars (low wheel loads) the specific curve resistance is higher, similar to the phenomena of higher rolling resistance for empty cars on a straight track. However, these experiments did not provide usable formulas for curve resistance, because the experiments were, unfortunately, all done on a test