 # Stability of Polygons Inscribed in an Ellipse

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This Demonstration concerns polygons with sides inscribed in an ellipse with semimajor axis 1 and semiminor axis . If there exists a perpendicular line from a side that intersects the center of gravity, then the side is stable. The stable sides are shown in green.

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Every convex polygon can be defined by a function in a polar coordinate system with origin at the center of gravity of an object with cross section . On horizontal surfaces, all objects start rolling in a way that sends the center of gravity lower such that decreases at the point of contact with the underlying surface. Equilibria occur if at this point. A balance point is stable at the minima of , where .

The number of vertices because in these cases, the center of gravity of the polygon and the ellipse are equal.

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Contributed by: Tímea Deichler (June 2015)
Suggested by: Gabor Domokos
Open content licensed under CC BY-NC-SA

## Snapshots   ## Details

References

 G. Domokos, Z. Lángi, and T. Szabó, "On the Equilibria of Finely Discretized Curves and Surfaces," Monatshefte für Mathematik, 168(3–4), 2012 pp. 321–345.

 "The Gömböc." (Jun 25, 2015) www.gomboc.eu.

## Permanent Citation

Tímea Deichler

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