# Stability of Polygons Inscribed in an Ellipse

Stability of Polygons Inscribed in an Ellipse

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.

n=4k+2

a

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 =0 at this point. A balance point is stable at the minima of , where Rα>0.

P

R(α)

G

P

R

dR

dα

R

2

d

2

d

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

n=4k+2