Cyclogon

In geometry, a cyclogon is the curve traced by a vertex of a regular polygon that rolls without slipping along a straight line.

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,


 * $$ A = P + 2C. \, $$

Cyclogons generated by quadrilaterals






Generalized cyclogons
A cyclogon is obtained when a polygon rolls over a straight line. Let it be assumed that the regular polygon rolls over the edge of another polygon. Let it also be assumed that the tracing point is not a point on the boundary of the polygon but possibly a point within the polygon or outside the polygon but lying in the plane of the polygon. In this more general situation, let a curve be traced by a point z on a regular polygonal disk with n sides rolling around another regular polygonal disk with m sides. The edges of the two regular polygons are assumed to have the same length. A point z attached rigidly to the n-gon traces out an arch consisting of n circular arcs before repeating the pattern periodically. This curve is called a trochogon — an epitrochogon if the n-gon rolls outside the m-gon, and a hypotrochogon if it rolls inside the m-gon. The trochogon is curtate if z is inside the n-gon, and prolate (with loops) if z is outside the n-gon. If z is at a vertex it traces an epicyclogon or a hypocyclogon.