Length of Arc of Nephroid

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:

Thus:

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: