Evolute of Cycloid is Cycloid

Theorem
The evolute of a cycloid is another cycloid.

Proof
Let $C$ be the cycloid defined by the equations:
 * $\begin{cases}

x = a \left({\theta - \sin \theta}\right) \\ y = a \left({1 - \cos \theta}\right) \end{cases}$

From Parametric Equations for Evolute: Formulation 2:


 * $(1): \quad \begin{cases}

X = x - \dfrac {y' \left({x'^2 + y'^2}\right)} {x' y - y' x} \\ Y = y + \dfrac {x' \left({x'^2 + y'^2}\right)} {x' y - y' x} \end{cases}$

where:
 * $\left({x, y}\right)$ denotes the Cartesian coordinates of a general point on $C$
 * $\left({X, Y}\right)$ denotes the Cartesian coordinates of a general point on the evolute of $C$
 * $x'$ and $x''$ denote the derivative and second derivative respectively of $x$ $\theta$
 * $y'$ and $y''$ denote the derivative and second derivative respectively of $y$ $\theta$.

Thus we have:

and:

Thus:

and so:

and:

The cycloid $C$ (blue) and its evolute (red) are illustrated below: