Cyclogon
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 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,
Examples
[edit]Cyclogons generated by an equilateral triangle and a square
[edit]
Prolate cyclogon generated by an equilateral triangle
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Prolate_cyclogon_generated_by_an_equilateral_triangle.gif/400px-Prolate_cyclogon_generated_by_an_equilateral_triangle.gif)
Curtate cyclogon generated by an equilateral triangle
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Curtate_cyclogon_generated_by_an_equilateral_triangle.gif/400px-Curtate_cyclogon_generated_by_an_equilateral_triangle.gif)
Cyclogons generated by quadrilaterals
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/39/Cyclogon_generated_by_a_non-convex_quadrilateral.png/400px-Cyclogon_generated_by_a_non-convex_quadrilateral.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/04/Cyclogon_generated_by_a_convex_quadrilateral.png/400px-Cyclogon_generated_by_a_convex_quadrilateral.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/12/Cyclogon_generated_by_a_star-like_quadrilateral.png/400px-Cyclogon_generated_by_a_star-like_quadrilateral.png)
Generalized cyclogons
[edit]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]
See also
[edit]References
[edit]- ^ Tom M. Apostol, Mamikon Mnatsakanian (2012). New Horizons in Geometry. Mathematical Association of America. p. 68. ISBN 9780883853542.
- ^ Ken Caviness. "Cyclogons". Wolfram Demonstrations Project. Retrieved 23 December 2015.
- ^ a b T. M. Apostol and M. A. Mnatsakanian (1999). "Cycloidal Areas without Calculus" (PDF). Math Horizons. 7 (1): 12–16. doi:10.1080/10724117.1999.12088451. Archived from the original (PDF) on 2005-01-30. Retrieved 23 December 2015.
- ^ Tom M. Apostopl and Mamikon A. Mnatsaknian (September 2002). "Generalized Cyclogons" (PDF). Math Horizons. Archived from the original (PDF) on 2005-01-30. Retrieved 23 December 2015.