## Epicycloid

The path traced out by a point on the Edge of a Circle of Radius rolling on the outside of a Circle of Radius .

It is given by the equations

 (1) (2)

 (3) (4)

 (5)
But

 (6)

so
 (7)

Note that is the parameter here, not the polar angle. The polar angle from the center is
 (8)

To get Cusps in the epicycloid, , because then rotations of bring the point on the edge back to its starting position.

 (9)

so
 (10)

An epicycloid with one cusp is called a Cardioid, one with two cusps is called a Nephroid, and one with five cusps is called a Ranunculoid.

-epicycloids can also be constructed by beginning with the Diameter of a Circle, offsetting one end by a series of steps while at the same time offsetting the other end by steps times as large. After traveling around the Circle once, an -cusped epicycloid is produced, as illustrated above (Madachy 1979).

Epicycloids have Torsion

 (11)

and satisfy
 (12)

where is the Radius of Curvature ().

References

Bogomolny, A. Cycloids.'' http://www.cut-the-knot.com/pythagoras/cycloids.html.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 160-164 and 169, 1972.

Lee, X. Epicycloid and Hypocycloid.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/EpiHypocycloid_dir/epiHypocycloid.html.

MacTutor History of Mathematics Archive. Epicycloid.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Epicycloid.html.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 219-225, 1979.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 50-52, 1991.

Yates, R. C. Epi- and Hypo-Cycloids.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 81-85, 1952.