PROJECT:  SPECIAL PLANE CURVES

 
HISTORY

The epicycloid was named by Ole Roemer in 1674.  He discovered that cog-wheels with epicycloidal teeth turned with minimum friction.  These results were duplicated by Girard Desargues, Phillippe de la Hire and Charles Stephen.  The double generation theorem of the curve was first noted by Daniel Bernoulli in 1725.  Many others also worked with the epicycloid, such as Durer, Huygens, Leibniz, de L'Hospital, Jakob Bernoulli, Euler, Edmund Halley and Sir Isaac Newton.  The epicycloid curve is of special interest to astronomers, who find it in various coronas.  Sir Isaac Newton discussed the measuring of the curve length of the epicycloid in his Principia.  The more complicated spherical epicycloids, in which a moving circle is inclined at a constant angle to the plane of the fixed circle, were interests of the Bernoullis, Pierre Louis M. de Maupertuis, Francois Nicole, Alexis Claude Calirault and others.
 

WHAT IS AN EPICYCLOID?

An epicycloid is defined to be:

The path traced out by a point P on the edge of a Circle of Radius b rolling on the outside of a fixed Circle of Radius a.

PARAMETERIZATION OF THE CURVE:

Point P moves about circle of radius b:

x = b cos f
y = b sin f

The axis P''C' also translates at the angle t with respect to its original place P'C as the Circle B travels counterclockwise about Circle A :

x = b cos (f + t)
y = b sin (f + t)

 The arc length traced by point P is equal to the distance Circle B has traveled about Circle A:

s = a * t
s' = b * f

and,

       s = s'
 a *  t = b * f
       f = (a/b)t

so, replacing f  in the equations:

x = b cos [(a/b)t + t] = b cos {[(a + b)/b] t}
y = b sin [(a/b)t + t] = b sin {[(a + b)/b] t}

In addition the center of Circle B (the point, C) moves counterclockwise with respect to O by angle t, with a radius of (a + b).  This is added to the above translation:

x = (a + b) cos t + b cos {[(a + b)/b] t}
y = (a + b) sin t +  b sin {[(a + b)/b] t}

To get a curve with n cusps, then b = a/n, so that n rotations of b bring the point P back to its starting point.
 

THE LENGTH AND AREA OF THE CURVE

If a=(m-1)b and m is an integer, then,

length = 8mb

and,       area = b(m+m)

OTHER INTERESTING INFORMATION

Some special cases of epicycloids are:  when a=b, the curve formed is called a cardioid and has a single cusp;  a=2b is a nephroid which has two cusps; and, a=5b is a five-cusped epicycloid called a ranunculoid.

....................................
     Cardioid                               Nephroid                                    Ranunculoid

There are also three other curves which are closely related to the epicycloid:  the epitrochoid, the hypocycloid and the hypotrochoid.  All of these curves are traced by a point P on a circle of radius a, differing only in the location of the point P and whether circle "b" rolls on the inside or outside of circle "a".
.....................
             Epitrochoid                                Hypocycloid                               Hypotrochoid
 

The pedal curve of the epicycloid, providing the pedal point is the center, is called the rhodonea curve.  The vertexes of the epicycloid are located at the tips of  the roses' pedal.

The evolute of an epicycloid is scaled by 1/(1 + 2b), and rotated so that the old curves cusp is located at the new curves vertex.  The evolute is smaller than the original curve.

Evolute of the Epicycloid

EXPERIMENT WITH THE EPICYCLOID YOURSELF:

To experiment with the epicycloid yourself, simply use your mouse to click on and move points B (to change the radius of the big circle) and F (to change the radius of the small circle).  Then double click on the ANIMATE button to see the epicycloid form:

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REFERENCES

*, ENCYCLOPEDIA BRITANNICA, Eleventh Edition, Epicycloid.  University Press, Cambridge, England, 1910: p. 686.

http://www.best.com/~xah/SpecialPlaneCurves_dir/specialPlaneCurves.html

http://www.keypress.com/dgnewsletter/turning.html

http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/BrombacherAarnout/EMT669/EMT699.html

http://www.astro.virginia.edu/%7Eeww6n/math/Epicycloid.html

http://www.math.vt.edu/people/srirang/m1224s97/lec1224/lec1/lec1.html