Epicycloid

Summary

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

The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).

EquationsEdit

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 either:

 
 

or:

 
 

in a more concise and complex form[1]


 


where

  • angle   is in turns:  
  • smaller circle has radius r
  • the larger circle has radius kr

AreaEdit

(Assuming the initial point lies on the larger circle.) When k is a positive integer, the area of this epicycloid is

 

If k is a positive integer, then the curve is closed, and has k cusps (i.e., sharp corners).

If k is a rational number, say k = p / q expressed as irreducible fraction, then the curve has p cusps.

To close the curve and
complete the 1st repeating pattern :
θ = 0 to q rotations
α = 0 to p rotations
total rotations of outer rolling circle = p + q rotations

Count the animation rotations to see p and q .

If k is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius R + 2r.

The distance OP from (x=0,y=0) origin to (the point   on the small circle) varies up and down as

R <= OP <= (R + 2r)

R = radius of large circle and

2r = diameter of small circle

The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid, two cusps is a nephroid.

An epicycloid and its evolute are similar.[2]

ProofEdit

 
sketch for proof

We assume that the position of   is what we want to solve,   is the angle from the tangential point to the moving point  , and   is the angle from the starting point to the tangential point.

Since there is no sliding between the two cycles, then we have that

 

By the definition of angle (which is the rate arc over radius), then we have that

 

and

 .

From these two conditions, we get the identity

 .

By calculating, we get the relation between   and  , which is

 .

From the figure, we see the position of the point   on the small circle clearly.

 
 

See alsoEdit

 
Animated gif with turtle in MSWLogo (Cardioid)[3]

ReferencesEdit

  • J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 161, 168–170, 175. ISBN 978-0-486-60288-2.
  1. ^ Epicycloids and Blaschke products by Chunlei Cao, Alastair Fletcher, Zhuan Ye
  2. ^ Epicycloid Evolute - from Wolfram MathWorld
  3. ^ Pietrocola, Giorgio (2005). "Tartapelago". Maecla.

External linksEdit