Is $\vert\sin{x}\vert$ a cycloid? Forgive this seemingly basic question; I recently found out about cycloids and cannot find any answers on the web. My guess is that it’s not, due to some part of the definition of a cycloid, but I can’t really find something about cycloids which explicitly helps me answer my question/I am not rounded enough to decipher the necessary information. 

One possible explanation is using the parametrization of a cycloid. It is given by 
$$
x = r(t-\sin t)\\
y = r(1-\cos t)
$$
for some $r>0$. If $y = |\sin x|$ is a cycloid, or more generally, homothetic to cycloid, then it will be possible to find some constant $a\in \mathbb{R}$ such that 
$$
|\sin(r(t-\sin t))| = ar(1-\cos t)
$$
for all $t\in \mathbb{R}$. However, it is not true. 
