Astroid


An astroid is a particular mathematical curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside a fixed circle with four times the radius. 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. It can also be defined as the envelope of a line segment of fixed length that moves while keeping an end point on each of the axes. It is therefore the envelope of the moving bar in the Trammel of Archimedes.
Its modern name comes from the Greek word for "star". It was proposed, originally in the form of "Astrois", by Joseph Johann von Littrow in 1838. The curve had a variety of names, including tetracuspid, cubocycloid, and paracycle. It is nearly identical in form to the evolute of an ellipse.

Equations

If the radius of the fixed circle is a then the equation is given by
This implies that an astroid is also a superellipse.
Parametric equations are
The pedal equation with respect to the origin is
the Whewell equation is
and the Cesàro equation is
The polar equation is
The astroid is a real locus of a plane algebraic curve of genus zero. It has the equation
The astroid is, therefore, a real algebraic curve of degree six.

Derivation of the polynomial equation

The polynomial equation may be derived from Leibniz's equation by elementary algebra:
Cube both sides:
Cube both sides again:
But since:
It follows that
Therefore:
or

Metric properties

;Area enclosed
;Length of curve
;Volume of the surface of revolution of the enclose area about the x-axis.
;Area of surface of revolution about the x-axis

Properties

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
The evolute of an astroid is an astroid twice as large.