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| {{Distinguish|asteroid}}
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| [[Image:Astroid.svg|240px|thumb|right| Astroid]]
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| [[Image:HypotrochoidOn4.gif|500px|thumb|The construction of the astroid.]]
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| An '''astroid''' (sometimes spelled '''asteroid''') is a particular mathematical [[curve]]: a [[hypocycloid]] with four [[cusp (singularity)|cusp]]s. Astroids are also [[superellipse]]s: all astroids are scaled versions of the curve specified by the equation
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| :<math>x^{2/3} + y^{2/3} = 1. \,</math>
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| Its modern name comes from the [[Greek language|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.
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| 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 (mathematics)#Example 4|envelope]]). The [[polar coordinate system|polar equation]] is
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| :<math>r=\frac{|\sec(\theta)|}{(1+\tan^{2/3}(\theta))^{3/2}}</math>
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| and the [[parametric equation]]s are
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| :<math>x=\cos^3\theta,\qquad y=\sin^3\theta.</math>
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| The astroid is a real locus of a [[algebraic curve|plane algebraic curve]] of [[geometric genus|genus]] zero. It has the equation
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| :<math>(x^2+y^2-1)^3+27x^2y^2=0. \,</math>
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| 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.
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| The [[Plücker formula|dual curve]] to the astroid is the [[cruciform curve]] with equation <math>\textstyle x^2 y^2 = x^2 + y^2.</math>
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| The [[evolute]] of an astroid is an astroid twice as large.
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| An astroid created by a circle rolling inside a circle of radius <math>a</math> will have an area of <math>\frac{3}{8} \pi a^2</math> and a perimeter of 6a.
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| ==See also==
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| * [[Cycloid]]
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| * [[Deltoid curve|Deltoid]] a curve with three cusps.
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| * [[Stoner–Wohlfarth astroid]] a use of this curve in magnetics.
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| ==References==
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| * {{Cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=4–5,34–35,173–174 }}
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| * {{Cite book | author = Wells D | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = 10–11}}
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| * {{Cite book | author=R.C. Yates | title=A Handbook on Curves and Their Properties | location=Ann Arbor, MI | publisher=J. W. Edwards | pages=1 ff.|chapter=Astroid| year=1952 }}
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| == External links ==
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| {{commonscat|Astroid}}
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| * {{springer|title=Astroid|id=p/a013540}}
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| * {{MathWorld | urlname=Astroid | title=Astroid}}
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| * [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Astroid.html "Astroid" at The MacTutor History of Mathematics archive]
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| * [http://www.mathcurve.com/courbes2d/astroid/astroid.shtml "Astroïde" at Encyclopédie des Formes Mathématiques Remarquables] (in French)
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| * [http://www.2dcurves.com/roulette/roulettea.html Article on 2dcurves.com]
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| * [http://xahlee.org/SpecialPlaneCurves_dir/Astroid_dir/astroid.html Visual Dictionary Of Special Plane Curves, Xah Lee]
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| * [http://demonstrations.wolfram.com/BarsOfAnAstroid/ Bars of an Astroid] by Sándor Kabai, [[The Wolfram Demonstrations Project]].
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| [[Category:Sextic curves]]
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