Equation of Astroid/Parametric 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 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 $C_1$.


Proof

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


By Equation of Hypocycloid, the equation of $H$ is given by:

$\begin{cases} x & = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta} \\ y & = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta} \end{cases}$

From Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii, this can be generated by a epicycle $C_1$ of radius $\dfrac 1 4$ the radius of the deferent.

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:

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

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Hypocycloid with Four Cusps: $11.9$
  • 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): astroid
  • 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: $9.9.$
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): astroid
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): astroid
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