# Astroid

A plane algebraic curve of order six, described by a point $M$ on a circle of radius $r$ rolling on the inside of a circle of radius $R=4r$; a hypocycloid with module $m=4$. Its equation in orthogonal Cartesian coordinates is

$$x^{2/3}+y^{2/3}=R^{2/3};$$

and a parametric representation is

$$x=R\cos^3\frac t4,\quad y=R\sin^3\frac t4.$$ Figure: a013540a

There are four cusps (see Fig.). The length of the arc from the point $A$ is

$$l=\frac32R\sin^2\frac t4.$$

The length of the entire curve is $6R$. The radius of curvature is

$$r_k=\frac32R\sin\frac t2.$$

The area bounded by the curve is

$$S=\frac38\pi R^2.$$

The astroid is the envelope of a family of segments of constant length, the ends of which are located on two mutually perpendicular straight lines. This property of the astroid is connected with one of its generalizations — the so-called oblique astroid, which is the envelope of the segments of constant length with their ends located on two straight lines intersecting at an arbitrary angle.

How to Cite This Entry:
Astroid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Astroid&oldid=31592
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article