11348

Tracing a Cyclogon: Roulette of a Polygon Rolling along a Line

This Demonstration traces the path of a point attached to a regular polygon rolling without slipping along a straight line. The point is called the pole or tracing point.
If the pole is a vertex of the polygon (Snapshot 1), the traced curve is called a cyclogon.
If the pole is inside the polygon, the curve is a curtate cyclogon (Snapshot 3). If outside the polygon, it is a prolate cyclogon (Snapshot 4).

SNAPSHOTS

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

DETAILS

The rolling polygon and the pole are subject to a sequence of three geometric transformations:
1. A stepwise rotation by a multiple of around the centroid of the polygon.
2. A stepwise translation along the axis by an edge length: .
3. A continuous rotation by around a point on the axis where it was moved by the previous translation.
The variable is the angular position of the polygon around its centroid.
The variable is the number of vertices of the polygon.
The resulting cyclogon is a sequence of circular arcs with the same subtending angle: . The centers of the arcs are on the axis and are each an edge length apart.
    • 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.









 
RELATED RESOURCES
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+