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


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References

[1] 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.

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



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