Equation of Astroid/Cartesian Form

Theorem
Let $H$ be the astroid generated by the epicycle $C_1$ of radius $b$ rolling without slipping around the inside of a deferent $C_2$ of radius $a = 4 b$.

Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.

Let $P$ be a point on the circumference of $C_1$.

Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis.

Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.

The point $P = \tuple {x, y}$ is described by the equation:
 * $x^{2/3} + y^{2/3} = a^{2/3}$

Proof
By definition, an astroid is a hypocycloid with $4$ cusps.


 * Astroid.png

From the parametric form of the equation of an astroid, $H$ can be expressed as:


 * $\begin{cases}

x & = 4 b \cos^3 \theta = a \cos^3 \theta \\ y & = 4 b \sin^3 \theta = a \sin^3 \theta \end{cases}$

Squaring, taking cube roots and adding: