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.