Equation of Cycloid

Curve
Consider a circle radius $a$ rolling without slipping along the x-axis of a cartesian coordinate plane.

Consider the point on the circumference of this circle which is at the origin when its center is on the y-axis.

Let $\left({x, y}\right)$ be the coordinates of this point as it travels over the plane.

The point $\left({x, y}\right)$ is described by the equations:
 * $x = a \left({\theta - \sin \theta}\right)$
 * $y = a \left({1 - \cos \theta}\right)$

The curve this traces out is called the cycloid.

Arc
An arc of a cycloid is defined as being a part of the cycloid traced out when $\theta$ goes from $2 n \pi$ to $2 \left({n + 1}\right) \pi$ for $n \in \Z$.

Cusp
Any of the points of the cycloid where $\theta = 2 n \pi$ where $n \in \Z$ is a cusp.

Proof

 * Cycloid.png

Let the circle have rolled so that the radius to the point $\left({x, y}\right)$ is at angle $\theta$ to the vertical.

The center of the circle is at $\left({a \theta, a}\right)$.

Then it follows from the definition of sine and cosine that:
 * $x = a \theta - a \sin \theta$
 * $y = a - a \cos \theta$

whence the result.