In mathematics, an astroid is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is 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. It can also be defined as the envelope of a line segment of fixed length that moves while keeping an end point on each of the axes. It is therefore the envelope of the moving bar in the Trammel of Archimedes.
Its modern name comes from the Greek word for "star". It was proposed, originally in the form of "Astrois", by Joseph Johann von Littrow in 1838. 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.
If the radius of the fixed circle is a then the equation is given by
This implies that an astroid is also a superellipse.
The pedal equation with respect to the origin is
the Whewell equation is
and the Cesàro equation is
The polar equation is
The astroid is a real locus of a plane algebraic curve of genus zero. It has the equation
The astroid is, therefore, a real algebraic curve of degree six.
Derivation of the polynomial equation
The polynomial equation may be derived from Leibniz's equation by elementary algebra:
Cube both sides:
Cube both sides again:
It follows that
- Area enclosed
- Length of curve
- Volume of the surface of revolution of the enclose area about the x-axis.
- Area of surface of revolution about the x-axis
The astroid 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.
The dual curve to the astroid is the cruciform curve with equation The evolute of an astroid is an astroid twice as large.
The astroid has only one tangent line in each oriented direction, making it an example of a hedgehog.
- Cardioid (epicycloid with one cusp)
- Nephroid (epicycloid with two cusps)
- Deltoid (hypocycloid with three cusps)
- Stoner–Wohlfarth astroid a use of this curve in magnetics.
- ^ Yates
- ^ J. J. v. Littrow (1838). "§99. Die Astrois". Kurze Anleitung zur gesammten Mathematik. Wien. p. 299.
- ^ Loria, Gino (1902). Spezielle algebraische und transscendente ebene kurven. Theorie und Geschichte. Leipzig. pp. 224.
- ^ Yates, for section
- ^ Mathworld
- ^ A derivation of this equation is given on p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf
- ^ Yates, for section
- ^ Nishimura, Takashi; Sakemi, Yu (2011). "View from inside". Hokkaido Mathematical Journal. 40 (3): 361–373. doi:10.14492/hokmj/1319595861. MR 2883496.
- 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.
- "Astroid", Encyclopedia of Mathematics, EMS Press, 2001 
- Weisstein, Eric W. "Astroid". MathWorld.
- "Astroid" at The MacTutor History of Mathematics archive
- "Astroid" at The Encyclopedia of Remarkable Mathematical Forms
- Article on 2dcurves.com
- Visual Dictionary Of Special Plane Curves, Xah Lee
- Bars of an Astroid by Sándor Kabai, The Wolfram Demonstrations Project.