Equation of Cycloid

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

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

Consider the cycloid traced out by the point $P$.

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

The point $P = \tuple {x, y}$ is described by the equations:
 * $x = a \paren {\theta - \sin \theta}$
 * $y = a \paren {1 - \cos \theta}$

Proof

 * Cycloid.png

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

The center of the circle is at $\tuple {a \theta, a}$.

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.

Also see

 * Length of Arc of Cycloid
 * Area under Arc of Cycloid