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Przemyslaw Kajetanowicz
Institute of Mathematics and Computer Science
Wroclaw University of Technology
Wroclaw, Poland

Przemyslaw.Kajetanowicz@pwr.wroc.pl

The Cycloid Family of Curves
continued . . .

Hypotrocoid

 The graphics in this deposit were created using GeoGebra.

NCB Deposit  # 135

Study this work sheet.  Then click on the work sheet or this link to see the animation.
You must have Java to see the animations.
Be sure to move the sliders at the top of the animated image.

WARNING:  Be patient!  You may have to download a newer version of Java.   With Java present, your computer should download the Java driven animation files.

 Kajetanowicz's Work Sheets can be altered to graph any curve in the cycloid family on any domain.  If you have GeoGebra on your computer, open the following files to see how nicely everything behaves. <  http://www.geogebratube.org/material/show/id/81877  >

 Now you may enjoy seeing other animations of the classic cycloid.  Go to . . . . Arguably, the Cycloid Family of curves features the most distinguished group of investigators in all 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.

 References 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 also create many of these graphs:      Gray, A.,  MODERN DIFFERENTIAL GEOMETRY of Curves and Surfaces with Mathematica®,   2nd. ed., CRC Press, 1998. For Maplesoft code to create several members of the cycloid family see < http://curvebank.calstatela.edu/cycloidmaple/cycloid.htm >
 2014