# Evolute of Cycloid is Cycloid

## Theorem

The evolute of a cycloid is another cycloid.

## Proof

Let $C$ be the cycloid defined by the equations:

$\begin {cases} x = a \paren {\theta - \sin \theta} \\ y = a \paren {1 - \cos \theta} \end {cases}$
$(1): \quad \begin{cases} X = x - \dfrac {y' \paren {x'^2 + y'^2} } {x' y'' - y' x''} \\ Y = y + \dfrac {x' \paren {x'^2 + y'^2} } {x' y'' - y' x''} \end{cases}$

where:

$\tuple {x, y}$ denotes the Cartesian coordinates of a general point on $C$
$\tuple {X, Y}$ denotes the Cartesian coordinates of a general point on the evolute of $C$
$x'$ and $x''$ denote the derivative and second derivative respectively of $x$ with respect to $\theta$
$y'$ and $y''$ denote the derivative and second derivative respectively of $y$ with respect to $\theta$.

Thus we have:

 $\displaystyle x'$ $=$ $\displaystyle a \paren {1 - \cos \theta}$ $\displaystyle x''$ $=$ $\displaystyle a \sin \theta$

and:

 $\displaystyle y'$ $=$ $\displaystyle a \sin \theta$ $\displaystyle y''$ $=$ $\displaystyle a \cos \theta$

Thus:

 $\displaystyle \dfrac {x'^2 + y'^2} {x' y'' - y' x''}$ $=$ $\displaystyle \dfrac {a^2 \paren {1 - \cos \theta}^2 + a^2 \sin^2 \theta} {a \paren {1 - \cos \theta} a \cos \theta - a \sin \theta \, a \sin \theta}$ substituting for $x'$, $x''$, $y'$, $y''$ $\displaystyle$ $=$ $\displaystyle \dfrac {\paren {1 - 2 \cos \theta + \cos^2 \theta} + \sin^2 \theta} {\cos \theta - \cos^2 \theta - \sin^2 \theta}$ multiplying out, cancelling out $a^2$ $\displaystyle$ $=$ $\displaystyle \dfrac {2 - 2 \cos \theta} {\cos \theta - 1}$ Sum of Squares of Sine and Cosine $(2):\quad$ $\displaystyle$ $=$ $\displaystyle -2$ dividing top and bottom by $1 - \cos \theta$

and so:

 $\displaystyle X$ $=$ $\displaystyle a \paren {\theta - \sin \theta} - a \sin \theta \paren {-2}$ substituting for $x$ and $y'$ and from $(2)$ in $(1)$ $\displaystyle$ $=$ $\displaystyle a \paren {\theta + \sin \theta}$ simplifying

and:

 $\displaystyle Y$ $=$ $\displaystyle a \paren {1 - \cos \theta} + a \paren {1 - \cos \theta} \paren {-2}$ substituting for $y$ and $x'$ and from $(2)$ in $(1)$ $\displaystyle$ $=$ $\displaystyle -a \paren {1 - \cos \theta}$

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

$\blacksquare$