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

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An epitrochogon is the trace (roulette) of a point attached to a polygon rolling without slipping around the outside of another polygon [1].


The epitrochogon can be curtate if the tracing point is inside the rolling polygon, or prolate if it is outside.

If the tracing point is at a vertex of the rolling polygon, the trace becomes an epicyclogon.

In this Demonstration, the base polygon and the rolling polygon have the same edge length. The resulting epitrochogon is a sequence of circle arcs with the same subtending angle and the vertices of the base polygon as centers.

The rolling polygon and the attached tracing point are subject to a sequence of three geometric transformations:

2. a stepwise rotation by a multiple of around the center of the base polygon

3. a continuous rotation by around the vertex of the base polygon it was moved to by the previous rotation

is the number of vertices of the base polygon.

is the angular position of the rolling polygon around its centroid.


Contributed by: Erik Mahieu (April 2017)
Open content licensed under CC BY-NC-SA




[1] T. M. Apostol and M. A. Mnatsukanian, "Generalized Cyclogons," Math Horizons, 10(1), 2002, pp. 25–28.

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