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.
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.