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
Riemannian connection on a surface "g". The condition "QAQ=0" implies "Q" "g" "g" "Q" = 0 and hence that "Q" "g" "g" "Q" =0. There is another kinematic way of understanding parallel transport and geodesic curvature in terms of "rolling without slipping or twisting". Although well known to differential geometers since the early part of the twentieth century, it has also been applied to problems in engineering and robotics. Consider the 2-sphere as a rigid body in three-dimensional space rolling without slipping or twisting on a horizontal plane. The point of contact will describe a curve in the plane and on the surface. At each