How to find the parametric equation of a cycloid? "A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia !cycloid animation In many calculus books I have, the cycloid, in parametric form, is used in examples to find arc length of parametric equations. This is the parametric equation for the cycloid: $$\begin{align*}x &= r(t - \sin t)\\\ y &= r(1 - \cos t)\end{align*}$$ How are these equations found in the first place?

$t$ measures the angle through which the wheel has rotated, starting with your point in the "down" position. Since the wheel is rolling, the distance it has rolled is the distance along the circumference of the wheel from your point to the "down" position, which (since the wheel has radius $r$) is $rt$. So the centre of the wheel, which was initially at $(0,r)$, is now at $(rt,r)$. Your point is displaced from this by $-r\sin(t)$ horizontally and $-r\cos(t)$ vertically, so it is at $(rt - r\sin(t), r - r\cos(t))$.