# Astroid

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

and a parametric representation is

Figure: a013540a

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

The length of the entire curve is . The radius of curvature is

The area bounded by the curve is

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) |

#### Comments

#### References

[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) |

**How to Cite This Entry:**

Astroid.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Astroid&oldid=13095