Bingo! Yes, this is exactly the sort of thing I had in mind. So the positions of the points on these ellipses can be calculated as the roots of polynomials? (Cool!) But it does appear that there are integer constraints governing various ratios about the possible ellipses constructed in this fashion. At 09:05 PM 11/17/2010, James Buddenhagen wrote:

Rita the dog asked a more specific but related question on Y!A a while back, with links to a couple of pictures in the additional details. See here: http://answers.yahoo.com/question/index;_ylt=As6rCggttnM1wrVEcxQE_VXty6IX;_y...

On Wed, Nov 17, 2010 at 10:39 PM, Henry Baker <hbaker1@pipeline.com> wrote:

...

In playing with the Marden Theorem, I started thinking about approximations to ellipses, and was wondering about regular elliptical n-gons. I'm not talking about a regular n-gon stretched/squashed or squashed in the X or Y direction, because for stretched/squashed n-gons the lengths of those sides are no longer equal.

Since the perimeter of an ellipse isn't easily calculated, I would imagine that this will be a difficult problem.

What I'm really looking for is a kind of inverse to the ellipse perimeter problem: given n sides of length 1 and a major/minor radius ratio (alternately an eccentricity), how to draw the ellipse itself. Which major/minor ratios and/or eccentricities can actually be represented this way?

I would assume that someone has looked into this sort of problem, but I wouldn't even know where to begin to look.