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).

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.