Roulette (Hypotrochogon) of a Polygon Rolling inside Another Polygon

base polygon

rolling polygon

square

pentagon

hexagon

heptagon

octagon

triangle

square

pentagon

hexagon

heptagon

pole offset from centroid

prolate hypotrochogon

pole in center

pole at vertex

rolling angle

-0.3

zoom control

2.5

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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

2π

m

around its centroid

2. a stepwise rotation by a multiple of

-

2π

n

around the center of the base polygon

3. a continuous rotation by

ϕmod-

2π(n-m)

nm

around the vertex of the base polygon it was moved to by the previous rotation

n

is the number of vertices of the base polygon.

m

is the number of vertices of the rolling polygon.

ϕ

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