| A
Sampler for the Student . . . .
column will repeat twice. |
These are very large
files. Be patient! |
Cycloid
|
 |
Trochoid
 |
Epicycloid
"Epi"
implies the trace of a point on a circle rolling outside another circle .
. . . .
x
x
Hypocycloid
. . . .while "Hypo" implies the trace
of a point on a circle rolling inside another circle . .
. .
Epitrochoid
....................
and "tro" implies looping rather
than a cusp.
Hypotrochoid
|
Dr. Chabot's Maple Work Sheets can be altered to graph any curve in the cycloid family on any domain. However, you must own a copy of the Maple software. |
Arguably, the Cycloid Family of curves features the most distinguished group of investigators in mathematics. Galileo and Father Mersenne are credited with being the first to name and discuss its special properties (1599). They were followed by Torricelli, Fermat, Descartes, Roberval, Wren, Huygens, Desargues, Johann Bernoulli, Leibniz, Newton, Jakob Bernoulli, L'Hôpital and others. This is probably too brief a list. One might assert that a fascination with the motion of the cycloidal curves led a century of civilization's greatest mathematicians into modern mathematics. Certainly, the birth of the calculus, especially the calculus of variations, flourished among these remarkable men who were determined to understand its many special qualities. Because of the frequency of disputes among mathematicians in the 17th century, the cycloid became known as the "Helen of Geometers." The name is appropriately based on Greek mythology. Helen was the most beautiful woman in the world. The Trojan war that followed her capture was one of the fiercest conflicts in ancient times. |
| Shikin, Eugene V., Handbook
and Atlas of Curves, CRC Press, 1995.
Yates, R. C., Curves and their Properties, NCTM, 1952. Also in A Handbook on Curves and their Properties, various publishers including the NCTM. Weisstein, Eric. W., CRC Concise Encyclopedia of MATHEMATICS, Chapman & Hall/ CRC, 2nd ed., 2003. |
| For Mathematica® code that will
create many of these graphs: Gray, A., MODERN DIFFERENTIAL GEOMETRY of Curves and Surfaces with Mathematica®, 2nd. ed., CRC Press, 1998. |