Equation of Cycloid in Cartesian Coordinates

Theorem
Consider a circle of radius $a$ rolling without slipping along the $x$-axis of a cartesian 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 $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.

The point $P = \tuple {x, y}$ is described by the equation:
 * $a \sin^{-1} \paren {\dfrac {\sqrt {2 a y - y^2} } a} = \sqrt {2 a y - y^2} + x$

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

Expressing $\theta$ and $\sin \theta$ in terms of $y$:

Substituting for $\theta$ and $\sin \theta$ in the expression for $x$: