View source for Astroid

{{short description|Curve generated by rolling a circle inside another circle with 4x or (4/3)x the radius}}
{{Distinguish|Asteroid
}}
[[File:Astroid.svg|thumb|Astroid]]
[[File:HypotrochoidOn4.gif|thumb|The hypocycloid construction of the astroid.]]
[[File:Astroid created with Elipses with a plus b const.svg|thumb|Astroid {{math|1= ''x''{{sup|{{frac|2|3}}}} + ''y''{{sup|{{frac|2|3}}}} = ''r''{{sup|{{frac|2|3}}}}}} as the common [[Envelope (mathematics)|envelope]] of a family of [[ellipse]]s of equation {{math|1= ({{frac|''x''|''a''}}){{sup|2}} + ({{frac|''y''|''b''}}){{sup|2}} = ''r''{{sup|2}}}}, where {{math|1= ''a'' + ''b'' = 1}}.]]
[[File:sliding_ladder_in_astroid.svg|thumb|link={{filepath:sliding_ladder_in_astroid.svg}}|The envelope of a ladder (coloured lines in the top-right quadrant) sliding down a vertical wall, and its reflections (other quadrants) is an astroid. The midpoints trace out a circle while other points trace out ellipses similar to the previous figure. [{{filepath:sliding_ladder_in_astroid.svg}} {{nowrap|In the SVG file,}}] hover over a ladder to highlight it.]]
[[File:Normal lines to the ellipse.svg|thumb|right|Astroid as an evolute of ellipse]]

In [[mathematics]], an '''astroid''' is a particular type of [[roulette curve]]: a [[hypocycloid]] with four [[cusp (singularity)|cusp]]s. Specifically, it is the [[Locus (mathematics)|locus]] of a point on a circle as it [[Rolling|rolls]] inside a fixed circle with four times the [[radius]].<ref>Yates</ref> 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 (mathematics)|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 (mathematics)|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.<ref>{{cite book|author=J. J. v. Littrow|title=Kurze Anleitung zur gesammten Mathematik|chapter=§99. Die Astrois|year=1838|location=Wien|pages=299|chapter-url=https://books.google.com/books?id=AERmAAAAcAAJ&pg=PA299}}</ref><ref>{{cite book|author=Loria, Gino|title=Spezielle algebraische und transscendente ebene kurven. Theorie und Geschichte|url=https://archive.org/details/speziellealgebr00lorigoog|year=1902|location=Leipzig|pages=[https://archive.org/details/speziellealgebr00lorigoog/page/n250 224]}}</ref> 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.

==Equations==
If the radius of the fixed circle is ''a'' then the equation is given by<ref>Yates, for section</ref>
<math display="block">x^{2/3} + y^{2/3} = a^{2/3}. </math>
This implies that an astroid is also a [[superellipse]].

[[Parametric equation]]s are
<math display="block">
\begin{align}
 x = a\cos^3 t &= \frac{a}{4} \left( 3\cos \left(t\right) + \cos \left(3t\right)\right), \\[2ex]
 y = a\sin^3 t &= \frac{a}{4} \left( 3\sin \left(t\right) - \sin \left(3t\right) \right).
\end{align}
</math>

The [[pedal equation]] with respect to the origin is 
<math display="block">r^2 = a^2 - 3p^2,</math>

the [[Whewell equation]] is 
<math display="block">s = {3a \over 4} \cos 2\varphi,</math>
and the [[Cesàro equation]] is
<math display="block">R^2 + 4s^2 = \frac{9a^2}{4}.</math>

The [[polar coordinate system|polar equation]] is<ref>{{MathWorld | urlname=Astroid | title=Astroid}}</ref>
<math display="block">r = \frac{a}{\left(\cos^{2/3}\theta + \sin^{2/3}\theta\right)^{3/2}}.</math>

The astroid is a real locus of a [[algebraic curve|plane algebraic curve]] of [[geometric genus|genus]] zero. It has the equation<ref>A derivation of this equation is given on p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf</ref>
<math display="block">\left(x^2 + y^2 - a^2\right)^3 + 27 a^2 x^2 y^2 = 0. </math>

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:
<math display="block">x^{2/3} + y^{2/3} = a^{2/3}. </math>

Cube both sides:
<math display="block">\begin{align}
x^{6/3} + 3x^{4/3}y^{2/3} + 3x^{2/3}y^{4/3} + y^{6/3} &= a^{6/3} \\[1.5ex]
x^2 + 3x^{2/3}y^{2/3} \left(x^{2/3} + y^{2/3}\right) + y^2 &= a^2 \\[1ex]
x^2 + y^2 - a^2 &= -3x^{2/3}y^{2/3} \left(x^{2/3} + y^{2/3}\right)
 \end{align}</math>

Cube both sides again:
<math display="block">\left(x^2 + y^2 - a^2\right)^3 = -27 x^2 y^2 \left(x^{2/3} + y^{2/3}\right)^3</math>

But since:
<math display="block">x^{2/3} + y^{2/3} = a^{2/3} \,</math>

It follows that
<math display="block">\left(x^{2/3} + y^{2/3}\right)^3 = a^2.</math>

Therefore:
<math display="block">\left(x^2 + y^2 - a^2\right)^3 = -27 x^2 y^2 a^2</math>
or
<math display="block">\left(x^2 + y^2 - a^2\right)^3 + 27 x^2 y^2 a^2 = 0. </math>

==Metric properties==
;Area enclosed<ref>Yates, for section</ref> :<math>\frac{3}{8} \pi a^2</math>
;Length of curve :<math>6a</math>
;Volume of the surface of revolution of the enclose area about the ''x''-axis. :<math>\frac{32}{105}\pi a^3</math>
;Area of surface of revolution about the ''x''-axis :<math>\frac{12}{5}\pi a^2</math>

==Properties==
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 <math display="inline"> x^2 y^2 = x^2 + y^2.</math>
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 (geometry)|hedgehog]].<ref>{{cite journal
 | last1 = Nishimura | first1 = Takashi
 | last2 = Sakemi | first2 = Yu
 | doi = 10.14492/hokmj/1319595861
 | issue = 3
 | journal = Hokkaido Mathematical Journal
 | mr = 2883496
 | pages = 361–373
 | title = View from inside
 | volume = 40
 | year = 2011| doi-access = free
 }}</ref>

==See also==
* [[Cardioid]] – an epicycloid with one cusp
* [[Nephroid]] – an epicycloid with two cusps
* [[Deltoid curve|Deltoid]] – a hypocycloid with three cusps
* [[Stoner–Wohlfarth astroid]] – a use of this curve in magnetics
* [[Spirograph]]

==References==
{{Reflist}}
* {{Cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | url=https://archive.org/details/catalogspecialpl00lawr | url-access=limited | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=[https://archive.org/details/catalogspecialpl00lawr/page/n20 4]–5,34–35,173–174 }}
* {{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&ndash;11}}
* {{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 }}

== External links ==
{{commons category|Astroid}}
* {{springer|title=Astroid|id=p/a013540}}
* [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Astroid.html "Astroid" at The MacTutor History of Mathematics archive]
* [http://www.mathcurve.com/courbes2d.gb/astroid/astroid.shtml "Astroid" at The Encyclopedia of Remarkable Mathematical Forms]
* [https://web.archive.org/web/20170824133613/http://www.2dcurves.com/roulette/roulettea.html Article on 2dcurves.com]
* [http://xahlee.org/SpecialPlaneCurves_dir/Astroid_dir/astroid.html Visual Dictionary Of Special Plane Curves, Xah Lee]
* [http://demonstrations.wolfram.com/BarsOfAnAstroid/ Bars of an Astroid] by Sándor Kabai, [[The Wolfram Demonstrations Project]].

[[Category:Sextic curves]]
[[Category:Roulettes (curve)]]