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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!

Cycloid
Definitions

Explanations

Continuous Animation

Trochoid
Definitions

Explanations

Continuous Animation

Definitions

Explanations

Continuous Animation

Epicycloid
Definitions

Explanations

Continuous Animation
 

"Epi" implies the trace of a point on a circle rolling outside another circle . . . . .

Hypocycloid
. . . .while "Hypo" implies the trace of a point on a circle rolling inside another circle . . . .
 
 

Definitions

Explanations

Continuous Animation

Epitrochoid
and  "tro" implies looping rather than a cusp.
 
 
 
 
 

Definitions

Explanations

Continuous Animation

Hypotrochoid
Definitions

Explanations

Continuous Animation

Definitions

Explanations

Continuous Animation

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  >