Stability of Polygons Inscribed in an Ellipse

number of sides n

6

length of semiminor axis a

0.25

show perpendiculars

show ellipse

This Demonstration concerns polygons with

n=4k+2

sides inscribed in an ellipse with semimajor axis 1 and semiminor axis

a

. 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.

Every convex polygon

P

can be defined by a function

R(α)

in a polar coordinate system with origin at the center of gravity

G

of an object with cross section

P

. On horizontal surfaces, all objects start rolling in a way that sends the center of gravity lower such that

R

decreases at the point of contact with the underlying surface. Equilibria occur if

dR

dα

=0

at this point. A balance point is stable at the minima of

R

, where

2

d

R

2

d

α

>0

.

The number of vertices

n=4k+2

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