Evolute of Cycloid is Cycloid

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Theorem

The evolute of a cycloid is another cycloid.


Proof

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

$\begin {cases} x = a \paren {\theta - \sin \theta} \\ y = a \paren {1 - \cos \theta} \end {cases}$


From Parametric Equations for Evolute: Formulation 2:

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

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:


CycloidAndEvolute.png

$\blacksquare$


Sources