Astroid

If the radius of the fixed circle is a then the equation is given by[4]

${\displaystyle x^{2/3}+y^{2/3}=a^{2/3}.\,}$

This implies that an astroid is also a superellipse.

Parametric equations are

${\displaystyle {\begin{aligned}&x=a\cos ^{3}t={a \over 4}(3\cos t+\cos 3t),\\[6pt]&y=a\sin ^{3}t={a \over 4}(3\sin t-\sin 3t).\end{aligned}}}$

The pedal equation with respect to the origin is

${\displaystyle r^{2}=a^{2}-3p^{2},}$

the Whewell equation is

${\displaystyle s={3a \over 4}\cos 2\varphi ,}$

and the Cesàro equation is

${\displaystyle R^{2}+4s^{2}={\frac {9a^{2}}{4}}.}$

The polar equation is[5]

${\displaystyle r={\frac {a}{(\cos ^{2/3}\theta +\sin ^{2/3}\theta )^{3/2}}}.}$

The astroid is a real locus of a plane algebraic curve of genus zero. It has the equation[6]

${\displaystyle (x^{2}+y^{2}-a^{2})^{3}+27a^{2}x^{2}y^{2}=0.\,}$

The astroid is, therefore, a real algebraic curve of degree six.

The polynomial equation may be derived from Leibniz's equation by elementary algebra:

${\displaystyle x^{2/3}+y^{2/3}=a^{2/3}.\,}$

Cube both sides:

${\displaystyle x^{6/3}+3x^{4/3}y^{2/3}+3x^{2/3}y^{4/3}+y^{6/3}=a^{6/3}\,}$

${\displaystyle x^{2}+3x^{2/3}y^{2/3}(x^{2/3}+y^{2/3})+y^{2}=a^{2}\,}$

${\displaystyle x^{2}+y^{2}-a^{2}=-3x^{2/3}y^{2/3}(x^{2/3}+y^{2/3})\,}$

Cube both sides again:

${\displaystyle (x^{2}+y^{2}-a^{2})^{3}=-27x^{2}y^{2}(x^{2/3}+y^{2/3})^{3}\,}$

But since:

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

It follows that

${\displaystyle (x^{2/3}+y^{2/3})^{3}=a^{2}.\,}$

Therefore:

${\displaystyle (x^{2}+y^{2}-a^{2})^{3}=-27x^{2}y^{2}a^{2}\,}$

or

${\displaystyle (x^{2}+y^{2}-a^{2})^{3}+27x^{2}y^{2}a^{2}=0.\,}$

Area enclosed[7]

${\displaystyle {\frac {3}{8}}\pi a^{2}}$

Length of curve

${\displaystyle 6a}$

Volume of the surface of revolution of the enclose area about the x-axis.

${\displaystyle {\frac {32}{105}}\pi a^{3}}$

Area of surface of revolution about the x-axis

${\displaystyle {\frac {12}{5}}\pi a^{2}}$

The astroid has four cusp singularities in the real plane, the points on the star. It has two more complex cusp singularities at infinity, and four complex double points, for a total of ten singularities.

The dual curve to the astroid is the cruciform curve with equation ${\displaystyle \textstyle x^{2}y^{2}=x^{2}+y^{2}.}$ The evolute of an astroid is an astroid twice as large.

The astroid has only one tangent line in each oriented direction, making it an example of a hedgehog.[8]

• Cardioid (epicycloid with one cusp)
• Nephroid (epicycloid with two cusps)
• Deltoid (hypocycloid with three cusps)
• Stoner–Wohlfarth astroid a use of this curve in magnetics.
• Spirograph

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 4–5, 34–35, 173–174. ISBN 0-486-60288-5.
• Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 10–11. ISBN 0-14-011813-6.
• R.C. Yates (1952). "Astroid". A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 1 ff.

Wikimedia Commons has media related to Astroid.

• "Astroid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Astroid". MathWorld.
• "Astroid" at The MacTutor History of Mathematics archive
• "Astroid" at The Encyclopedia of Remarkable Mathematical Forms
• Article on 2dcurves.com
• Visual Dictionary Of Special Plane Curves, Xah Lee
• Bars of an Astroid by Sándor Kabai, The Wolfram Demonstrations Project.