Roulette (Hypotrochogon) of a Polygon Rolling inside Another Polygon
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
2π
m
2. a stepwise rotation by a multiple of around the center of the base polygon

2π
n
3. a continuous rotation by around the vertex of the base polygon it was moved to by the previous rotation
ϕmod
2π(nm)
nm
n
m
ϕ