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
Singular point of a curve smooth function "h" so that the curve has the form "y"="h"("x") near the origin. Similarly, if "b"≠0 then there is a smooth function "k" so that the curve has the form "x"="k"("y") near the origin. In either case, there is a smooth map from R to the plane which defines the curve in the neighborhood of the origin. Note that at the origin so the curve is non-singular or "regular" at the origin if at least one of the partial derivatives of "f" is non-zero. The singular points are those points on the curve where both partial derivatives vanish, Assume