19.4: Variations

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 $x = 0$.

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

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

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

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

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