Curve generated by rolling a circle inside another circle with 4x or (4/3)x the radius
In mathematics, an astroid is a particular type of roulette 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.[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. 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.[2][3] 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.
Equations
If the radius of the fixed circle is a then the equation is given by[4]This implies that an astroid is also a superellipse.
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[7]
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.