In geometry, a cyclogon is the curve traced by a vertex of a regular polygon that rolls without slipping along a straight line.[1] [2]
In the limit, as the number of sides increases to infinity, the cyclogon becomes a cycloid.[3]
The cyclogon has an interesting property regarding its area.[3] Let denote the area of the region above the line and below one of the arches, let denote the area of the rolling polygon, and let denote the area of the disk that circumscribes thepolygon. For every cyclogon generated by a regular polygon,
A=P+2C.
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.[4]