Astroid

Not to be confused with asteroid.
Astroid
The construction of the astroid.

An astroid is a particular mathematical curve: a hypocycloid with four cusps. Specifically, it the locus of a point on a circle as it rolls inside a fixed circle with four times the radius.[1] By double generation, it is also the locus of a point on a circle as it rolls inside a fixed circle with 4/3 times the radius. If the radius of the fixed circle is a then the equation is given by

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

This implies that an astroid is also a superellipse.

Its modern name comes from the Greek word for "star". The curve had a variety of names, including tetracuspid (still used), cubocycloid, and paracycle. It is nearly identical in form to the evolute of an ellipse.

A circle of radius 1/4 rolls around inside a circles]], so that it is tangent to the astroid (which is therefore an envelope). The polar equation is

$r=\frac{|\sec(\theta)|}{(1+\tan^{2/3}(\theta))^{3/2}}$

and the parametric equations are

$x=\cos^3\theta,\qquad y=\sin^3\theta.$

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

$(x^2+y^2-1)^3+27x^2y^2=0. \,$

The astroid is therefore of degree six, and 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 $\textstyle x^2 y^2 = x^2 + y^2.$ The evolute of an astroid is an astroid twice as large.

An astroid created by a circle rolling inside a circle of radius $a$ will have an area of $\frac{3}{8} \pi a^2$ and a perimeter of 6a.