# Inscribed $n$-gons of maximum perimeter for an ellipse

It appears that the max area inscribed $$n$$-gon for an ellipse is not unique - if one inscribes a regular $$n$$-gon in a circle with radius a (there are infinitely many orientations for this inscribed regular $$n$$-gon) and then compresses the circle along one direction into an ellipse with minor axis $$b$$, the inscribed regular $$n$$-gon gets pressed into an $$n$$-gon that seems to be a max area inscribed $$n$$-gon for the $$(a,b)$$ ellipse.

1. Given an integer $$n$$ and an ellipse with major and minor axes $$a$$ and $$b$$, how does one find and characterize inscribed $$n$$-gon(s) of maximum perimeter?

Guess: the max perimeter inscribed $$n$$-gons for an ellipse are probably polygonal closed billiard paths (such polygons for $$n \ge 6$$ are shown here). If this guess is right, does it generalize to cases where a closed convex curve allows such polygonal billiard paths with m edges? The guess also implies a unique answer for an ellipse and $$n$$; that brings up the issue: given any point $$P$$ on the ellipse boundary, to find the inscribed $$n$$-gon that necessarily contains $$P$$ and has maximum perimeter.

1. When are the circumscribed $$n$$-gons of least perimeter and least area for an ellipse identical?

Note: one can also ask about inscribed and circumscribed $$n$$-gons in curves of constant width.

• Ellipses and circles are the same, up to linear automorphisms. Your description captures thus the situation: In cercles you get maximal area for regular inscribed polygons and in ellipses you get maximal area for polygons which correspond to regular polygons of the associated circle. Commented Oct 28, 2021 at 20:18

Consider allowing one vertex $$B$$ to vary on your curve, with neighbouring vertices $$A$$ and $$C$$ fixed. You want $$B$$ to be the point on the curve between $$A$$ and $$C$$ that maximizes total distance from $$A$$ and $$C$$. The curves of constant total distance from $$A$$ and $$C$$ are ellipses with foci at $$A$$ and $$C$$, so this total distance will be maximized at a point where your curve is tangent to one of those ellipses. As is well known, a light ray from one focus of an ellipse reflected off the ellipse goes to the other focus. If your curve is tangent to the ellipse at $$B$$, that means a light ray from $$A$$ will reflect off your curve at $$B$$ and go to $$C$$.
Standard methods of numerical optimization ought to work to find inscribed $$n$$-gons of maximum perimeter.
Here's one for $$n=7$$ with an ellipse of major and minor axes $$2$$ and $$1$$, produced using Maple.