Roulette (Epitrochogon) of a Disk Rolling around a Regular Polygon

This Demonstration shows the roulette drawn by a point attached to a disk rolling along a regular polygon.
For generalized cyclogons [1] and generalized trochoidal curves [2], these roulettes can be considered as trochogons with an infinite number of vertices of the rolling polygon.
In this Demonstration, the disk rolls on the outside of a polygon, and the trace of the point is called an epitrochogon.
These epitrochogons consist of sequences of two distinct curve types, each generated by a different type of motion: a cycloidal or rolling motion of the disk along the straight edges of the polygon, and a circular or rotating motion when turning around the vertices of the polygon.
The shape of these curves is determined by the ratio between the circumferences (or radii) of the polygon and the rolling disk. When this ratio is a rational number, after an integer number of loops, the epitrochogon is a closed curve.
In this Demonstration, the maximum number of loops is two. If the circumference ratio is an integer, the trace curve is closed after one loop around the polygon. If the ratio is a multiple of , the curve is closed after two loops.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


[1] T. M. Apostol and M. A. Mnatsukanian, "Generalized Cyclogons," Math Horizons, 2002 pp. 25–28. www.mamikon.com/USArticles/GenCycloGons.pdf.
[2] T. M. Apostol and M. A. Mnatsukanian, "Area & Arc Length of Trochogonal Arches," Math Horizons, 2003 pp. 24–30. www.mamikon.com/USArticles/TrochoGons.pdf.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+