Length of Arc of Astroid

Theorem
The total length of the arcs of an astroid constructed within a stator of radius $a$ is given by:
 * $\mathcal L = 6 a$

Proof
Let $H$ be embedded in a cartesian coordinate plane with its center at the origin and its cusps positioned on the axes.


 * Astroid.png

We have that $\mathcal L$ is $4$ times the length of one arc of the astroid.

From Arc Length for Parametric Equations:


 * $\displaystyle \mathcal L = 4 \int_{\theta \mathop = 0}^{\theta \mathop = \pi/2} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$

where, from Equation of Astroid:
 * $\begin{cases}

x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$

We have:

Thus:

Thus: