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 $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.

The point $P = \tuple {x, y}$ is described by the equations:

$x = a \paren {\theta - \sin \theta}$

$y = a \paren {1 - \cos \theta}$

Let the circle have rolled so that the radius to the point $P = \tuple {x, y}$ 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.

$\blacksquare$