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 the curve passes through the origin and write "y"="mx". Then "f" can be written If "b"+"mb" is not 0 then "f"=0 has a solution of multiplicity 1 at "x"=0 and the origin is a point of single contact with line "y"="mx". If "b"+"mb"=0 then "f"=0 has a solution of multiplicity 2 or higher and the line "y"="mx", or "b"x+"b"y=0, is tangent to the curve. In this case, if "c"+2"mc"+c"m" is not 0 then the curve has a point of double contact with "y"="mx". If the coefficient of "x", "c"+2"mc"+c"m", is 0 but the coefficient of "x" is not then the