Editing Astroid (section)

{{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.