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.[1][2] 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.[3]
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,
Contents
Examples
Cyclogons generated by an equilateral triangle and a square
|
File:Cyclogon generated by triangle.gif
Animation showing the generation of one arch of a cyclogon by an equilateral triangle as the triangle rolls over a straight line without slipping.
|
File:Cyclogon generated by a square.gif
Animation showing the generation of one arch of a cyclogon by a square as the square rolls over a straight line without slipping.
|
Prolate cyclogon generated by an equilateral triangle
Curtate cyclogon generated by an equilateral triangle
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.[4]
