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:
\(\ds x'\) | \(=\) | \(\ds a \paren {1 - \cos \theta}\) | ||||||||||||
\(\ds x''\) | \(=\) | \(\ds a \sin \theta\) |
and:
\(\ds y'\) | \(=\) | \(\ds a \sin \theta\) | ||||||||||||
\(\ds y''\) | \(=\) | \(\ds a \cos \theta\) |
Thus:
\(\ds \dfrac {x'^2 + y'^2} {x' y'' - y' x''}\) | \(=\) | \(\ds \dfrac {a^2 \paren {1 - \cos \theta}^2 + a^2 \sin^2 \theta} {a \paren {1 - \cos \theta} a \cos \theta - a \sin \theta \, a \sin \theta}\) | substituting for $x'$, $x''$, $y'$, $y''$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {1 - 2 \cos \theta + \cos^2 \theta} + \sin^2 \theta} {\cos \theta - \cos^2 \theta - \sin^2 \theta}\) | multiplying out, cancelling out $a^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 - 2 \cos \theta} {\cos \theta - 1}\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \) | \(=\) | \(\ds -2\) | dividing top and bottom by $1 - \cos \theta$ |
and so:
\(\ds X\) | \(=\) | \(\ds a \paren {\theta - \sin \theta} - a \sin \theta \paren {-2}\) | substituting for $x$ and $y'$ and from $(2)$ in $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds a \paren {\theta + \sin \theta}\) | simplifying |
and:
\(\ds Y\) | \(=\) | \(\ds a \paren {1 - \cos \theta} + a \paren {1 - \cos \theta} \paren {-2}\) | substituting for $y$ and $x'$ and from $(2)$ in $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -a \paren {1 - \cos \theta}\) |
The cycloid $C$ (blue) and its evolute (red) are illustrated below:
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.23$: Evolutes and Involutes. The Evolute of a Cycloid: Example $2$