ABSTRACT
In Chapter 12 we showed that where a line £ is rolled, without slipping, on a given curve T, a fixed point on the line traces out an involute to the curve T; moreover, all involutes to the curve T are obtained in this way. We now consider the more general situation where a general curve T\ rolls without slipping on a fixed curve 1^, and determine the curve traced out by a point which is rigidly attached (by rods for example) to Ti. Such a curve is called a roulette. Examples of roulettes are cycloids, limagons, the nephroid, rose curves, the deltoid, and the astroid; we consider these and other specific roulettes. As the curve Ti rolls on T2, the plane in which Ti is fixed performs a rigid motion. We show conversely, and perhaps surprisingly, that any rigid motion of the plane is given by rolling some curve in the plane on a second curve in a fixed plane. Because of this the general theory of roulettes has importance in geometry and kinematics.
