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.

References

 [1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)