View source for Astroid

A plane algebraic curve of order six, described by a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135401.png" /> on a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135402.png" /> rolling on the inside of a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135403.png" />; a [[Hypocycloid|hypocycloid]] with module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135404.png" />. Its equation in orthogonal Cartesian coordinates is

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and a parametric representation is

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<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/a013540a.gif" />

Figure: a013540a

There are four cusps (see Fig.). The length of the arc from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135407.png" /> is

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The length of the entire curve is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135409.png" />. The radius of curvature is

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The area bounded by the curve is

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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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,   "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,   "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.A. Lockwood,   "A book of curves" , Cambridge Univ. Press  (1961)</TD></TR></table>