11453

Roulette (Hypotrochogon) of a Circle Rolling inside a Regular Polygon

This Demonstration shows the roulette drawn by a point attached to a circle rolling without slipping inside a regular polygon.
For generalized cyclogons [1] and generalized trochoidal curves [2], these roulettes can be considered as limiting cases of hypotrochogons with an infinite number of vertices of the rolling polygon.
The shape of these curves depends on the radius of the rolling disk. When the ratio between the circumferences of the polygon and the rolling disk is the integer , the traced curve is closed, with leaves.
Use the button bar to select the right disk radius to get a closed trace with between 3 and 10 leaves.

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DETAILS

For a closed curve, after one loop around the polygon with circumradius equal to , the polygon perimeter of one loop is and the circumference of the circle is .
For a closed curve, , with , where the number of leaves of the closed curve is in the case of one complete loop inside the polygon, is the number of vertices of the polygon, is the radius of the disk and is the number of revolutions of the disk around its center.
References
[1] T. M. Apostol and M. A. Mnatsakanian, "Generalized Cyclogons," Math Horizons, 2002 pp. 25–28. www.mamikon.com/USArticles/GenCycloGons.pdf.
[2] T. M. Apostol and M. A. Mnatsakanian, "Area & Arc Length of Trochogonal Arches," Math Horizons, 2003 pp. 24–30. www.mamikon.com/USArticles/TrochoGons.pdf.

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