Editing Astroid (section)

==Equations==
If the radius of the fixed circle is ''a'' then the equation is given by<ref>Yates, for section</ref>
<math display="block">x^{2/3} + y^{2/3} = a^{2/3}. </math>
This implies that an astroid is also a [[superellipse]].

[[Parametric equation]]s are
<math display="block">
\begin{align}
 x = a\cos^3 t &= \frac{a}{4} \left( 3\cos \left(t\right) + \cos \left(3t\right)\right), \\[2ex]
 y = a\sin^3 t &= \frac{a}{4} \left( 3\sin \left(t\right) - \sin \left(3t\right) \right).
\end{align}
</math>

The [[pedal equation]] with respect to the origin is 
<math display="block">r^2 = a^2 - 3p^2,</math>

the [[Whewell equation]] is 
<math display="block">s = {3a \over 4} \cos 2\varphi,</math>
and the [[Cesàro equation]] is
<math display="block">R^2 + 4s^2 = \frac{9a^2}{4}.</math>

The [[polar coordinate system|polar equation]] is<ref>{{MathWorld | urlname=Astroid | title=Astroid}}</ref>
<math display="block">r = \frac{a}{\left(\cos^{2/3}\theta + \sin^{2/3}\theta\right)^{3/2}}.</math>

The astroid is a real locus of a [[algebraic curve|plane algebraic curve]] of [[geometric genus|genus]] zero. It has the equation<ref>A derivation of this equation is given on p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf</ref>
<math display="block">\left(x^2 + y^2 - a^2\right)^3 + 27 a^2 x^2 y^2 = 0. </math>

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