Equation of Astroid

Theorem
Consider a circle $C_1$ of radius $b$ rolling without slipping around the inside of a circle $C_2$ of (larger) radius $a$ in a cartesian coordinate plane.

Consider a point $P$ on the circumference of $C_1$ where it is tangent to $C_2$ at point $A$ on the $x$-axis.

Consider the astroid $H$ traced out by the point $P$.

Let $\left({x, y}\right)$ be the coordinates of $P$ as it travels over the plane.

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

Proof

 * Astroid.png

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

By Equation of Hypocycloid, the equation of $H$ is given by:
 * $\begin{cases}

x & = \left({a - b}\right) \cos \theta + b \cos \left({\left({\dfrac {a - b} b}\right) \theta}\right)\\ y & = \left({a - b}\right) \sin \theta - b \sin \left({\left({\dfrac {a - b} b}\right) \theta}\right) \end{cases}$

From Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii, this can be generated by an inner circle $C_1$ of radius $\dfrac 1 4$ the radius of the outer circle.

Thus $a = 4 b$ and the equation of $H$ is now given by:
 * $\begin{cases}

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

From Triple Angle Formula for Cosine:
 * $\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$

and from Triple Angle Formula for Sine:
 * $\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$

Thus $H$ 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: