I just occured to me after I sent this message that one might conceivably draw a regular elliptical n-gon using a modification of the classical "2 pins and a string" method. Consider a cyclic chain (like a bicycle chain, except imagine that the links are longer) consisting of m links. Place two pins k links apart. Then place the chain taut against the two pins & stretch it straight & choose one of the joints between two links & draw a dot. Then move to the next joint along the chain & keep the chain taut. In order to handle the case where the perimeter point is collinear with the line of the two foci, the total number of links m should be an even number. It would appear that if the ellipse foci are an integer number (k) of link-lengths apart, then the dots drawn should be a pretty close approximation to a regular elliptical n-gon ?? At 08:41 PM 11/17/2010, Henry Baker 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.