Roulette (Epitrochogon) of a Disk Rolling around a Regular Polygon

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This Demonstration simulates a circle rolling without slipping on the outside of a stationary regular polygon of circumradius 1. A point is attached to the circle; its trace is called an epitrochogon.


For generalized cyclogons [1] and generalized trochoidal curves [2], these roulettes can be considered a limiting case of epitrochogons with an infinite number of vertices of the rolling polygon.

These epitrochogons consist of sequences of two distinct curve types, each generated by a different type of motion:

1. a cycloidal or rolling motion of the circle along the straight edges of the polygon.

2. a circular motion when turning around the vertices of the polygon.


Contributed by: Erik Mahieu (December 2016)
Open content licensed under CC BY-NC-SA




[1] T. M. Apostol and M. A. Mnatsukanian, "Generalized Cyclogons," Math Horizons, 2002 pp. 25–28.

[2] T. M. Apostol and M. A. Mnatsukanian, "Area & Arc Length of Trochogonal Arches," Math Horizons, 2003 pp. 24–30.

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