Roulette (Hypotrochogon) of a Polygon Rolling inside Another Polygon

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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 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. Mnatsakanian, "Generalized Cyclogons," Math Horizons, 10(1), 2002, pp. 25–28.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.