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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

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


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

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