<< In playing with the [Siebeck] 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.

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If you can conformally map the interior of a circle to the interior of an ellipse, then there is a reasonable definition of an elliptical polygon. If such a map is constrained to take the circle's center to the ellipse's center, then what remains is a one-parameter family of such maps, determined by a rotation of the circle. And this circular family of conformal maps extends to such a family of homeomorphisms of the circle as a curve, to the ellipse as a curve. And this conformal map exists, by the Riemann mapping theorem. More information about it is in a paper by Gabor Szego, "Conformal Mapping of the Interior of an Ellipse onto a Circle", in the Aug.-Sept., 1950 American Mathematical Monthly (v. 57, no. 7). But I'm not sure there's a useful formula for the map of curves given there, though there are a number of Gosperesque formulas. The vertices of a regular n-gon inscribed in the circle will be mapped to n vertices with the same angular separation along the ellipse (with respect to the its center). These might make an appropriate definition for the vertices of an elliptical regular n-gon. Admittedly this answer a question different from the one asked. --Dan Those who sleep faster get more rest.