Epicycloid

Epicycloid

In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls without slipping around a fixed circle. It is a particular kind of roulette.

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).

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 between the larger circleand a circle of radius "R+2r".


The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid.

An epicycloid and its evolute are similar. [http://mathworld.wolfram.com/EpicycloidEvolute.html]

ee also

* Special cases: Cardioid, Nephroid
* Cycloid
* Hypocycloid
* Epitrochoid
* Hypotrochoid
* Spirograph
* Deferent and epicycle

References

*

External links

* [http://demonstrations.wolfram.com/Epicycloid/ Epicycloid] , MathWorld
* [http://demonstrations.wolfram.com/Epicycloid/ "Epicycloid"] by Michael Ford, The Wolfram Demonstrations Project, 2007


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Look at other dictionaries:

  • Epicycloid — Ep i*cy cloid, n. [Epicycle + oid: cf. F. [ e]picyclo[ i]de.] (Geom.) A curve traced by a point in the circumference of a circle which rolls on the convex side of a fixed circle. [1913 Webster] Note: Any point rigidly connected with the rolling… …   The Collaborative International Dictionary of English

  • epicycloid — [ep΄ə sī′kloid΄] n. [ EPICYCL(E) + OID] Geom. the curve traced by a point on the circumference of an epicycle that rolls around the outside of a fixed circle: cf. HYPOCYCLOID epicycloidal [ep΄ə sī′kloid′ l] adj …   English World dictionary

  • epicycloid — noun Date: circa 1755 a curve traced by a point on a circle that rolls on the outside of a fixed circle • epicycloidal adjective …   New Collegiate Dictionary

  • epicycloid — epicycloidal, adj. /ep euh suy kloyd/, n. Geom. a curve generated by the motion of a point on the circumference of a circle that rolls externally, without slipping, on a fixed circle. Equation: x = (a + b) cos(t) b cos [(a + b)t/b] and y = (a +… …   Universalium

  • epicycloid — noun The locus of a point on the circumference of a circle that rolls without slipping on the circumference of another circle …   Wiktionary

  • epicycloid — ep·i·cy·cloid || ‚epɪ saɪklɔɪd n. curve that is described by a point on the perimeter of a circle as the circle moves on the outside of the circumference of another circle (Mathematics) …   English contemporary dictionary

  • epicycloid — [ˌɛpɪ sʌɪklɔɪd] noun Mathematics a curve traced by a point on the circumference of a circle rolling on the exterior of another circle. Derivatives epicycloidal adjective …   English new terms dictionary

  • epicycloid — ep·i·cy·cloid …   English syllables

  • epicycloid — ep•i•cy•cloid [[t]ˌɛp əˈsaɪ klɔɪd[/t]] n. math. a curve generated by the motion of a point on the circumference of a circle that rolls externally, without slipping, on a fixed circle • Etymology: 1780–90 ep i•cy•cloi′dal, adj …   From formal English to slang

  • epicycloid — /ɛpiˈsaɪklɔɪd/ (say epee suykloyd) noun Geometry a curve generated by the motion of a point on the circumference of a circle which revolves, usually externally, on the circumference of a larger circle. –epicycloidal /ˌɛpisaɪˈklɔɪdl/ (say .epeesuy …  

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