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Curve Bank Home
Dr. Paul Chabot
Department of Mathematics
California State Univ.,
Los Angeles

The Cycloid Family of Curves
Cycloid, Trochoid, Epicycloid, Hypocycloid,
Epitrochoid and Hypotrochoid
Create Your Own Animations Using Maple!

The graphics in this deposit were created using Maple software.

NCB Deposit # 30

A Sampler for
the Student . . . .
Each animation in the left
column will repeat twice.


These are very
large files.
Be patient! 
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.
< http://mathworld.wolfram.com/Cycloid.html >

