An astroid (sometimes spelled asteroid) is a particular mathematical curve: a hypocycloid with four cusps. Astroids are also superellipses: all astroids are scaled versions of the curve specified by the equation
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.
A circle of radius 1/4 rolls around inside a circle of radius 1 and a point on its circumference traces an astroid. A line segment of length 1 slides with one end on the x-axis and the other on the y-axis, so that it is tangent to the astroid (which is therefore an envelope). The polar equation is
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.