Parametric Equation of Involute of Circle

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Theorem

Let $C$ be a circle of radius $a$ whose center is at the origin of a cartesian plane.

The involute $V$ of $C$ can be described by the parametric equation:

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


Proof

By definition the involute of $C$ is described by the endpoint of a string unwinding from $C$.

Let that endpoint start at $\tuple {a, 0}$ on the circumference of $C$.


Involute-of-Circle.png


Let $P = \tuple {x, y}$ be an arbitrary point on $V$.

Let $Q$ be the point at which the cord is tangent to $C$.

Then $PQ$ equals the arc of $C$ from which the cord has unwound.

Thus:

$PQ = a \theta$

where $\theta$ is the angle of $OQ$ to the $x$-axis.

Thus:

\(\displaystyle x\) \(=\) \(\displaystyle OQ \cos \theta + PQ \, \map \cos {\theta - \dfrac \pi 2}\)
\(\displaystyle \) \(=\) \(\displaystyle a \cos \theta + a \theta \paren {\cos \theta \cos \dfrac \pi 2 + \sin \theta \sin \dfrac \pi 2}\) Cosine of Difference
\(\displaystyle \) \(=\) \(\displaystyle a \paren {\cos \theta + \theta \sin \theta}\) various trigonometrical identities


\(\displaystyle y\) \(=\) \(\displaystyle OQ \sin \theta + PQ \, \map \sin {\theta - \dfrac \pi 2}\)
\(\displaystyle \) \(=\) \(\displaystyle a \sin \theta + a \theta \paren {\sin \theta \cos \dfrac \pi 2 - \cos \theta \sin \dfrac \pi 2}\) Sine of Difference
\(\displaystyle \) \(=\) \(\displaystyle a \paren {\sin \theta - \theta \cos \theta}\) various trigonometrical identities

Hence the result.

$\blacksquare$


Sources