11209

Roulette (Hypotrochogon) of a Polygon Rolling inside Another Polygon

A hypotrochogon is the trace (roulette) of a point attached to a regular polygon rolling without slipping inside another regular polygon [1].
The hypotrochogon 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 a hypocyclogon.
In this Demonstration, the base polygon and the rolling polygon have the same edge length. The resulting hypotrochogon 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 pole are subject to a sequence of three geometric transformations:
1. a stepwise rotation by a multiple of around its centroid
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 number of vertices of the rolling polygon.
is the angular position of the rolling polygon around its centroid.

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DETAILS

Reference
[1] T. M. Apostol and M. A. Mnatsakanian, "Generalized Cyclogons," Math Horizons, 10(1), 2002, pp. 25–28. www.mamikon.com/USArticles/GenCycloGons.pdf.

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