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Cycloids - Zykloide  (Selfmade OPL Programs for Psion5)

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Written by u.hornstein


Program to draw animated graphs of cycloids, epicycloids and hypocycloids.

Programm zum animierten Zeichnen von Zykloiden, Epizykloiden und Hypozykloiden.

General info on cycloids can easily be found in Wikipedia:

  • "A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line. 
  • If we further allow the line on which the circle rolls to be an arbitrary circle then we get the epicycloid (circle rolling on outside of another circle, point on the rim)"
  • If we further allow the line on which the circle rolls to be an arbitrary circle with the moving circle on the inside, point on the rim, then we get the hypocycloid.

This program was originally quickly written for my nephew Joel Krause. We studied the mechanism of high and low tides of the oceans. One of the factors is the centrifugal force of the earth while rotating around an axis which is not going through the earth's center. This program can be used to show the movement of the center of the earth (or moon) when looking at the earth-moon system.
The common center of gravity of this system is 4671 km away from the center of the earth. If we simplify the ellipse of the path of the earth-moon system around the sun to a circle, then the center of the earth walks on a epicycloid line around the sun, showing about 12 bumps on its path, 1 per month. While travelling on this epicycloid path, it rotates around itself, producing day and night. Altogether a not very simple movement. The moon, of course, travels on another epicycloid with larger 'bumps' in opposition to the earth.

"Cycloids.opo" can calculate and display these routes, and many more, but not to scale, and only one at a time.

This picture, taken from the German Wikipedia, shows the meaning of the variable lambda as used in Cycloids.opo:
cycloids with different lambda values

Some sample epicycloid outputs, computed and animated by Cycloids.opo:

The dots are equi-distant in time, i. e. closely set dots mean slow speed.

The above mentioned path of the center of the earth looks about like this (bumps grossly enlarged compared to the reality, sun in the center):

The looks of hypocycloids are left for you to discover.

The program is published including source code. Just put the Cycloids.opo file in any folder of your choice and run it (mark it and press enter).
If you want to play with the code also, put the Cycloids.opl file into the same folder, run it as above (the program editor will open), translate and execute.
Operate the program using Enter, Escape, q, p, s and r keypresses as applicable.

Thanks to U. Krinzner for his free workshop on Sprite programming. The use of sprites was a nice exercise for me: I could not only draw the cycloids, but also the motion of the moving circle.
Every user is welcome to experiment with and enhance the code wherever (s)he wishes. Feel free to send me your contributions.


Be sure to know the Disclaimer, valid for all of my software.

Executable on Psion machines (ER5) like Psion 5,  5mx, 5mxPro, Ericsson MC218, Netbook. Tested on MC218 only.

Download Cycloids.zip  (Version 1.01 with source code)   Downloads: [off] (since 19.4.2008)  




Temporary files in dedicated folder C:\System\Temp\cycloid\.



First version.

relevant terms: epicycle, cycloid, epicycloid, hypocycloid, epitrochoid, cardioid, spirograph, Zykloid, Epizykloid, Hypozykloid, Psion.

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Stand / Last Revision: 23.4.2008

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