An astroid is a particular mathematical curve: a hypocycloid with four cusps. Specifically, it the locus of a point on a circle as it rolls inside a fixed circle with four times the radius. By double generation, it is also the locus of a point on a circle as it rolls inside a fixed circle with 4/3 times the radius. If the radius of the fixed circle is a then the equation is given by
This implies that an astroid is also a superellipse.
Its modern name comes from the Greek word for "star". The curve had a variety of names, including tetracuspid (still used), cubocycloid, and paracycle. It is nearly identical in form to the evolute of an ellipse.
and the parametric equations are
The astroid is therefore of degree six, and has four cusp singularities in the real plane, the points on the star. It has two more complex cusp singularities at infinity, and four complex double points, for a total of ten singularities.
An astroid created by a circle rolling inside a circle of radius will have an area of and a perimeter of 6a.
- Deltoid a curve with three cusps.
- Stoner–Wohlfarth astroid a use of this curve in magnetics.
- J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 4–5,34–35,173–174. ISBN 0-486-60288-5.
- Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 10–11. ISBN 0-14-011813-6.
- R.C. Yates (1952). "Astroid". A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 1 ff.
|Wikimedia Commons has media related to Astroid.|
- Hazewinkel, Michiel, ed. (2001), "Astroid", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Astroid", MathWorld.
- "Astroid" at The MacTutor History of Mathematics archive
- "Astroïde" at Encyclopédie des Formes Mathématiques Remarquables (in French)
- Article on 2dcurves.com
- Visual Dictionary Of Special Plane Curves, Xah Lee
- Bars of an Astroid by Sándor Kabai, The Wolfram Demonstrations Project.