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Do you know of any `nice description' (say, the position of its vertices in polar coordinates) of the equilateral $n$-agons inscribed in a given ellipse?

What about if one of such vertices is supposed to be in the intersection of one of the (say major) axes, does this make any difference?

Thanks!

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    Metric and projective properties do not mix so well. It is clear by continuity that an inscribed equilateral $N$-agon with a vertex on a given point of the ellipse exists, but I won't expect finding its side length an easy algebraic task. Moreover, it is very likely that if $N$ is moderately large that cannot be done with straightedge and compass only. – 2017-02-08

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This is not a complete answer, but too long for a comment.

Before investigating the general problem, let's look at some simple specific example. Consider for example a pentagon inscribed into the ellipse $x^2+4y^2=1$. Pick the points as

$$A=(1,0)\quad B=(x_B,y_B)\quad C=(x_C,y_C)\quad D=(x_C,-y_C)\quad E=(x_B,-y_B)$$

Then solve for the variables in question and the resulting edge length $s$:

$$x_B\approx0.304\quad y_B\approx0.476\quad x_C\approx-0.537\quad y_C\approx0.422\quad s\approx0.843$$

Figure

But approximations are unsatisfactory here, so let's look at the underlying minimal polynomials of these algebraic numbers, which I computed in Sage like this:

PR. = QQ[] # declare multivariate polynomial ring
vs = ideal([              # formulate conditions for these variables
  xB^2 + 4*yB^2 - 1, xC^2 + 4*yC^2 - 1, # B and C on ellipse
  (xB - 1)^2 + yB^2 - s^2,              # |AB| = s
  (xB - xC)^2 + (yB - yC)^2 - s^2,      # |BC| = s
  2*yC - s                              # |CD| = s
]).variety(AA)            # compute all real algebraic solutions
v = [i for i in vs if i[s]>0 and i[yB]>0 and i[yC]>0 and i[xB]>i[xC]][0]
for g in PR.gens(): print v[g].minpoly().numerator()(g)

\begin{align*} 31293\,x_B^4 - 115344\,x_B^3 + 149974\,x_B^2 - 49648\,x_B + 4205 &= 0 \\ 979251849\,y_B^8 + 239544\,y_B^6 + 338986960\,y_B^4 - 211934720\,y_B^2 + 28037120 &= 0 \\ 10431\,x_C^4 - 20922\,x_C^3 + 3400\,x_C^2 + 8762\,x_C - 391 &= 0 \\ 108805761\,y_C^8 - 17105940\,y_C^6 - 4845200\,y_C^4 + 70720\,y_C^2 + 128000 &= 0 \\ 108805761\,s^8 - 68423760\,s^6 - 77523200\,s^4 + 4526080\,s^2 + 32768000 &= 0 \end{align*}

If you wanted to, you could turn these into radical expressions, but the results I got wouldn't fit the screen width here. So my message is that even for a fairly simple example, the results look very complicated. The degrees of the minimal polynomials are already pretty large, and the coefficients even more so. This doesn't get easier with more vertices or more complicated ellipses.

Also note that the Galois group of the polynomials for the $x$ coordinates is $S(4)$, so the coordinate is not something you can get from repeated quardatic extensions and therefore not a constructible number. For the $y$ coordinates $S(4)$ is included in the derived series as well. This confirms the assumption Jack D'Aurizio stated in a comment that a straightedge and compass construction of such a polygon would likely be impossible.

You might wonder whether using Cartesian coordinates instead of the polar ones you suggested are to blame for the uglyness of these results. But the solutions are algebraic in nature, so adding a bunch of trigonometric functions into the mix will likely only make things even more complicated.

This is no proof, of course, just a gut feeling that a nice description seems pretty unlikely to me.