Equation of Astroid

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.

Parametric Form

The point $P = \tuple {x, y}$ is described by the parametric equation:

$\begin{cases}

x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$ where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$.

Cartesian Form

The point $P = \tuple {x, y}$ is described by the equation:

$x^{2/3} + y^{2/3} = a^{2/3}$