What is "cyclogon"

In mathematics, in geometry, a cyclogon is the curve traced by a vertex of a polygon that rolls without slipping along a straight line. There are no restrictions on the nature of the polygon. It can be a regular polygon like an equilateral triangle or a square. The polygon need not even be convex: it could even be a star-shaped polygon. More generally the curves traced by points other than vertices have also been considered. In such cases it would be assumed that the tracing point is rigidly attached to the polygon. If the tracing point is outside the polygon, the curve is called a prolate cyclogon and if it is inside the polygon it is called a curtate cyclogon.

In the limit, as the number of sides increases to infinity, the cyclogon becomes a cycloid.

The cyclogon has an interesting property regarding its area. Let A denote the area of the region above the line and below one of the arches, let P denote the area of the rolling polygon, and let C denote the area of the disk that circumscribes the polygon. For every cyclogon generated by a regular polygon,