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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


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.


[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
[a2] E.A. Lockwood, "A book of curves" , Cambridge Univ. Press (1961)

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Astroid. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article