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) |
(6) |
(7) |
(8) |
To get Cusps in the epicycloid, , because then rotations of bring the point on
the edge back to its starting position.
(9) |
(10) |
-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) |
(12) |
See also Cardioid, Cyclide, Cycloid, Epicycloid--1-Cusped, Hypocycloid, Nephroid, Ranunculoid
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.
© 1996-9 Eric W. Weisstein