19.8: Contracted and Extended Cycloids

As in Section 19.1, we consider a circle of radius a rolling to the right on the line $y = 2a$. The point P is initially below the centre of the circle, but, instead of being on the rim of the circle, its distance from the centre of the circle is $r$. If $r < a$, the path described by P will be a contracted cycloid; if $r > a$, the path is an extended cycloid. (I think there’s a case for using this nomenclature the other way round, but most authors seem to use “contracted” for $r < a$ and “extended” for $r > a$.) It should not take long to be convinced, by arguments similar to those in Section 19.1, that the parametric equations to a contracted or extended cycloid are

$$x = 2 a \theta + r \sin 2 \theta \label{19.8.1}\tag{19.8.1}$$

and

$$y = a - r \cos 2 \theta \label{19.8.2}\tag{19.8.2}$$

These are illustrated in Figures XIX.8 and XIX.9 for a contracted cycloid with $r = 0.5a$ and an extended cycloid with $r = 1.5a$.