Length of Arc of Nephroid

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Theorem

The total length of the arcs of a nephroid constructed around a stator of radius $a$ is given by:

$\mathcal L = 12 a$


Proof

Let a nephroid $H$ be embedded in a cartesian coordinate plane with its center at the origin and its cusps positioned at $\left({\pm a, 0}\right)$.


Nephroid.png


We have that $\mathcal L$ is $2$ times the length of one arc of the nephroid.

From Arc Length for Parametric Equations:

$\displaystyle \mathcal L = 2 \int_{\theta \mathop = 0}^{\theta \mathop = \pi} \sqrt {\left({\frac{\mathrm d x} {\mathrm d \theta}}\right)^2 + \left({\frac{\mathrm d y} {\mathrm d \theta}}\right)^2} \mathrm d \theta$

where, from Equation of Nephroid:

$\begin{cases} x & = 3 b \cos \theta - b \cos 3 \theta \\ y & = 3 b \sin \theta - b \sin 3 \theta \end{cases}$


We have:

\(\displaystyle \frac {\mathrm d x} {\mathrm d \theta}\) \(=\) \(\displaystyle -3 b \sin \theta + 3 b \sin 3 \theta\)
\(\displaystyle \frac {\mathrm d y} {\mathrm d \theta}\) \(=\) \(\displaystyle 3 b \cos \theta - 3 b \cos 3 \theta\)


Thus:

\(\displaystyle \) \(\) \(\displaystyle \left({\frac {\mathrm d x} {\mathrm d \theta} }\right)^2 + \left({\frac {\mathrm d y} {\mathrm d \theta} }\right)^2\)
\(\displaystyle \) \(=\) \(\displaystyle \left({-3 b \sin \theta + 3 b \sin 3 \theta}\right)^2 + \left({3 b \cos \theta - 3 b \cos 3 \theta}\right)^2\)
\(\displaystyle \) \(=\) \(\displaystyle 9 b^2 \left({\left({-\sin \theta + \sin 3 \theta}\right)^2 + \left({\cos \theta - \cos 3 \theta}\right)^2}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 9 b^2 \left({\sin^2 \theta - 2 \sin \theta \sin 3 \theta + \sin^2 3 \theta + \cos^2 \theta - 2 \cos \theta \cos 3 \theta + \cos^2 3 \theta}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 9 b^2 \left({2 - 2 \sin \theta \sin 3 \theta - 2 \cos \theta \cos 3 \theta}\right)\) Sum of Squares of Sine and Cosine
\(\displaystyle \) \(=\) \(\displaystyle 18 b^2 \left({1 - \sin \theta \sin 3 \theta - \cos \theta \cos 3 \theta}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 18 b^2 \left({1 - \sin \theta \left({3 \sin \theta - 4 \sin^3 \theta}\right) - \cos \theta \cos 2 \theta}\right)\) Triple Angle Formula for Sine
\(\displaystyle \) \(=\) \(\displaystyle 18 b^2 \left({1 - \sin \theta \left({3 \sin \theta - 4 \sin^3 \theta}\right) - \cos \theta \left({4 \cos^3 \theta - 3 \cos \theta}\right)}\right)\) Triple Angle Formula for Cosine
\(\displaystyle \) \(=\) \(\displaystyle 18 b^2 \left({1 - 3 \sin^2 \theta + 4 \sin^4 \theta - 4 \cos^4 \theta + 3 \cos^2 \theta}\right)\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle 18 b^2 \left({1 + 3 \cos 2 \theta + 4 \sin^4 \theta - 4 \cos^4 \theta}\right)\) Double Angle Formula for Cosine
\(\displaystyle \) \(=\) \(\displaystyle 18 b^2 \left({1 + 3 \cos 2 \theta + \dfrac {3 - 4 \cos 2 x + \cos 4 x} 2 - 4 \cos^4 \theta}\right)\) Power Reduction Formula for $\sin^4$
\(\displaystyle \) \(=\) \(\displaystyle 18 b^2 \left({1 + 3 \cos 2 \theta + \dfrac {3 - 4 \cos 2 \theta + \cos 4 \theta} 2 - \dfrac {3 + 4 \cos 2 \theta + \cos 4 \theta} 2}\right)\) Power Reduction Formula for $\cos^4$
\(\displaystyle \) \(=\) \(\displaystyle 18 b^2 \left({1 - \cos 2 \theta}\right)\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle 18 b^2 \left({2 \sin^2 \theta}\right)\) Square of Sine
\(\displaystyle \) \(=\) \(\displaystyle 36 b^2 \sin^2 \theta\) simplifying

Thus:

$\sqrt {\left({\dfrac {\mathrm d x} {\mathrm d \theta} }\right)^2 + \left({\dfrac {\mathrm d y} {\mathrm d \theta} }\right)^2} = 6 b \sin \theta$


So:

\(\displaystyle \mathcal L\) \(=\) \(\displaystyle 2 \int_0^\pi 6 b \sin \theta \, \mathrm d \theta\)
\(\displaystyle \) \(=\) \(\displaystyle 12 b \int_0^\pi \sin \theta \, \mathrm d \theta\)
\(\displaystyle \) \(=\) \(\displaystyle 12 b \Big[{-\cos \theta}\Big]_0^\pi\)
\(\displaystyle \) \(=\) \(\displaystyle 12 b \left({-\cos \pi - \left({-\cos 0}\right)}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 12 b \left({- \left({-1}\right) - \left({-1}\right)}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 24 b\)
\(\displaystyle \) \(=\) \(\displaystyle 12 a\)

$\blacksquare$


Sources