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.



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:

\(\ds x^{2/3} + y^{2/3}\) \(=\) \(\ds a^{2/3} \paren {\cos^2 \theta + \sin^2 \theta}\)
\(\ds \) \(=\) \(\ds a^{2/3}\) Sum of Squares of Sine and Cosine

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Hypocycloid with Four Cusps: $11.8$
  • 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hypocycloid
  • 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): astroid
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 9$: Special Plane Curves: Hypocycloid with Four Cusps: $9.8.$
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