# Hypocycloid

In

geometry , a**hypocycloid**is a specialplane curve generated by the trace of a fixed point on a smallcircle that rolls within a larger circle. It is comparable to thecycloid but instead of the circle rolling along a line, it rolls within a circle.[

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The red curve is a hypocycloid traced as the smaller black circle rolls around inside the larger blue circle (parameters are R=3.0, r=1.0, and so k=3), giving adeltoid .]If the smaller circle has radius "r", and the larger circle has radius "R" = "kr", then the

parametric equations for the curve can be given by:$x(\; heta)\; =\; r\; (k-1)\; left(\; cos\; heta\; +\; frac\{cos((k-1)\; heta)\}\{k-1\}\; ight),$

:$y(\; heta)\; =\; r\; (k-1)\; left(\; sin\; heta\; -\; frac\{sin((k-1)\; heta)\}\{k-1\}\; ight).$

If "k" is an integer, then the curve is closed, and has "k" cusps (i.e., sharp corners, where the curve is not

differentiable ). Specially for k=2 the curve is a straight line and the circles are called Cardano circles.Girolamo Cardano was the first to describe these hypocycloids, which had applications in the technology of high-speedprinting press .If "k" is a

rational number , say "k" = "p"/"q" expressed in simplest terms, then the curve has "p" cusps.If "k" is an

irrational number , then the curve never closes, and fills the space within the larger circle except for a disk of radius "R" − "r" in the center of the larger circle.

deltoidastroid The hypocycloid is a special kind of

hypotrochoid , which are a particular kind of roulette.A hypocycloid with three cusps is known as a deltoid.

A hypocycloid curve with four cusps is known as an

astroid .**Derived curves**The

evolute of a hypocycloid is an enlarged version of the hypocycloid itself, whiletheinvolute of a hypocycloid is a reduced copy of itself. [*http://mathworld.wolfram.com/HypocycloidEvolute.html*]The pedal of a hypocycloid with pole at the center of the hypocycloid is a

rose curve .The

isoptic of a hypocycloid is a hypocycloid.**Hypocycloids in popular culture**Curves similar to hypocyloids can be drawn with the "

Spirograph " toy. Specifically, the Spirograph can drawhypotrochoid s andepitrochoid s.The

Pittsburgh Steelers ' logo includes threeastroid s (hypocycloids of fourcusp s). In his weekly NFL.com column "Tuesday Morning Quarterback,"Gregg Easterbrook often refers to the Steelers as the Hypocycloids.Portland, Oregon's flag features a hypocycloid; an

astroid .The 2007 redesign of "

The Price is Right "'s set features astroids on the three main doors and the turntable area [*http://www.tvsquad.com/2007/08/21/a-glimpse-at-drew-careys-price-is-right/*] .**ee also*** Special cases:

Astroid , Deltoid

*Cycloid

*Epicycloid

*Hypotrochoid

*Epitrochoid

*Spirograph **References***

**External links*** [

*http://www.portlandonline.com/auditor/index.cfm?a=jbgg&c=cheid City of Portland Flag*]

* [*http://www.carloslabs.com/node/21 A free Javascript tool for generating Hypocyloid curves*]

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Hypocycloid**— Hy po*cy cloid, n. [Pref. hypo + cycloid: cf. F. hypocyclo[ i]de.] (Geom.) A curve traced by a point in the circumference of a circle which rolls on the concave side in the fixed circle. Cf. {Epicycloid}, and {Trochoid}. [1913 Webster] … The Collaborative International Dictionary of English**hypocycloid**— [hī΄pō sī′kloid΄, hī΄pəsī′kloid΄; ] also [ hip΄ōsī′kloid΄, hip΄əsī′kloid΄] n. [ HYPO + CYCLOID] Geom. the curve traced by a point on the circumference of an epicycle that rolls around the inside of a fixed circle: cf. EPICYCLOID … English World dictionary**hypocycloid**— noun Date: 1843 a curve traced by a point on the circumference of a circle rolling internally on the circumference of a fixed circle … New Collegiate Dictionary**hypocycloid**— hypocycloidal, adj. /huy peuh suy kloyd/, n. Geom. a curve generated by the motion of a point on the circumference of a circle that rolls internally, without slipping, on a given circle. [1835 45; HYPO + CYCLOID] * * * … Universalium**hypocycloid**— noun The locus of a point on the circumference of a circle that rolls without slipping inside the circumference of another circle … Wiktionary**hypocycloid**— n. geometric figure formed by tracing the movement of a set point on the circumference of a smaller circle that is rolling within a larger circle (Geometry) … English contemporary dictionary**hypocycloid**— [ˌhʌɪpə(ʊ) sʌɪklɔɪd] noun Mathematics the curve traced by a point on the circumference of a circle which is rolling on the interior of another circle. Derivatives hypocycloidal adjective … English new terms dictionary**hypocycloid**— hy·po·cycloid … English syllables**hypocycloid**— hy•po•cy•cloid [[t]ˌhaɪ pəˈsaɪ klɔɪd[/t]] n. math. a curve generated by the motion of a point on the circumference of a circle that rolls internally, without slipping, on a fixed circle • Etymology: 1835–45 hy po•cy•cloi′dal, adj … From formal English to slang**hypocycloid**— /haɪpəˈsaɪklɔɪd/ (say huypuh suykloyd) noun a curve generated by the motion of a point on the circumference of a circle which rolls internally, without slipping, on a given circle. –hypocycloidal /haɪpəsaɪˈklɔɪdl/ (say huypuhsuy kloydl),… … Australian English dictionary