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 or on a parametrised curve, "g"("t") = ("t","t"). Both definitions give a singular point at the origin. However, a node such as that of "y"−"x"−"x" = 0 at the origin is a singularity of the curve considered as an algebraic curve, but if we parameterize it as "g"("t") = ("t"−1,"t"("t"−1)), then "g"′("t") never vanishes, and hence the node is "not" a singularity of the parameterized curve as defined above. Care needs to be taken when choosing a parameterization. For instance the straight line "y" = 0 can be parameterised by "g"("t") = ("t",0) which has a singularity at the origin.