Rate of Change of Cartesian Coordinates of Cycloid

Theorem
Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$.

Consider a point $P$ on the circumference of this circle.

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

Then the rate of change of $x$ and $y$ can be expresssed as:

where $\theta$ is the angle turned by $C$ after time $t$.

Proof
Let the center of $C$ be $O$.

, let $P$ be at the origin at time $t = t_0$.

By definition, $P$ traces out a cycloid.

From Equation of Cycloid, $P = \tuple {x, y}$ is described by:
 * $(1): \quad \begin{cases} x & = a \paren {\theta - \sin \theta} \\

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

Let $\tuple {x_c, y_c}$ be the coordinates of $O$ at time $t$.

We have that:
 * $y_c$ is constant: $y_c = a$

From Body under Constant Acceleration: Velocity after Time
 * $x_c = \mathbf v_0 t$

as the acceleration of $O$ is zero.

But $x_c$ is equal to the length of the arc of $C$ that has rolled along the $x$-axis.

By Arc Length of Sector:
 * $x_c = a \theta$

and so:

Thus:

and: