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# 19.4: Variations

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

In Sections 19.1,2,3, we imagined that the cycloid was generated by a circle that was rolling counterclockwise along the line $$y = 2a$$. We can also imagine variations such as the circle rolling clockwise along $$y = 0$$, or we can start with P at the top of the circle rather than at the bottom. I summarise in this section four variations. The distinction between $$\psi$$ and $$\theta$$ is as follows. The angle that the tangent to the cycloid makes with the positively-directed $$x$$-axis is $$\psi$$; that is to say, $$dx/dy = tan \psi$$.The circle rolls through an angle $$2 \theta$$. There is a simple relation between $$\psi$$ and $$\theta$$, which is different for each case.

In each figure, $$x$$ and $$y$$ are plotted in units of $$a$$. The vertical height between vertices and cusps is $$2a$$, the horizontal distance between a cusp and the next vertex is $$\pi a$$, and the arc length between a cusp and the next vertex is $$4a$$.

I. Circle rolls counterclockwise along $$y = 2a$$. P starts at the bottom. The cusps are up. A vertex is at the origin.

\begin{align*} x &= a(2 \theta + \sin 2 \theta ) \\[4pt] y &= 2 a \sin^2 \theta \label{19.4.2}\tag{19.4.2} \\[4pt] s &= 4a \sin \theta \label{19.4.3}\tag{19.4.3} \\[4pt] ^2 &= 8 ay \label{19.4.4}\tag{19.4.4} \\[4pt] \psi &= \theta . \label{19.4.5}\tag{19.4.5} \end{align*} II. Circle rolls clockwise along $$y = 0$$. P starts at the bottom. The cusps are down. A cusp is at the origin.

$x = a(2\theta - sin 2\theta ) \label{19.4.6}\tag{19.4.6}$

$y = 2a sin^2 \theta \label{19.4.7}\tag{19.4.7}$

$s = 4a(1-cos \theta) \label{19.4.8}\tag{19.4.8}$

$s^2 = 8a(y-s) \label{19.4.9}\tag{19.4.9}$

$\psi = 90 \circ - \theta. \label{19.4.10}\tag{19.4.10}$ III. Circle rolls clockwise along $$y = 0$$. P starts at the top. The cusps are down. A vertex is at $$x = 0$$.

$x = a( 2 \theta + \sin 2 \theta \label{19.4.11}\tag{19.4.11}$

$y = 2 a \cos^2 \theta \label{19.4.12}\tag{19.4.12}$

$s = 4 a \sin \theta \label{19.4.13}\tag{19.4.13}$

$s^2 = 8a(2a-y) \label{19.4.14}\tag{19.4.14}$

$\psi = 180 \circ - \theta . \label{19.4.15}\tag{19.4.15}$ IV. Circle rolls counterclockwise along $$y = 2a$$. P starts at the top. The cusps are up. A cusp is at $$x = 0$$.

$x = a (2 \theta - \sin 2 \theta) \label{19.4.16}\tag{19.4.16}$

$y = 2a \cos^2 \theta \label{19.4.17}\tag{19.4.17}$

$s = 4a (1-\cos \theta) \label{19.4.18}\tag{19.4.18}$

$s^2 - 8as + 8a(2a - y) = 0 \label{19.4.19}\tag{19.4.19}$

$\psi = 90 \circ + \theta \label{19.4.20}\tag{19.4.20}$ 