Epicycloid
From Wikipedia, the free encyclopedia
In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls around 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:
- <math>x(\theta) = r (k+1) \left( \cos \theta - \frac{\cos((k+1)\theta)}{k+1} \right) </math>
- <math>y(\theta) = r (k+1) \left( \sin \theta - \frac{\sin((k+1)\theta)}{k+1} \right) </math>
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 circle and a circle of radius R+2r.
| epicycloid examples | |||
k=1 |
k=2 |
k=3 |
k=4 |
k=2.1=21/10 |
k=3.8=19/5 |
k=5.5=11/2 |
k=7.2=36/5 |
The epicycloid is a special kind of epitrochoid.
An epicycle with one cusp is a cardioid.
An epicycloid and its evolute are similar.[1]
See also: cycloid, hypocycloid, deferent and epicycle.
af:Episikloïed bg:Епициклоида de:Epizykloide es:Epicicloide fr:épicycloïde it:epicicloide nl:Cycloïdes#Epicycloïde pl:Epicykloida ru:Эпициклоида zh:外摆线
