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**Vlasev** Oh, no worries. The formula I have written is worked out like this: I assumed that the first point on the n-gon is the point $\displaystyle (1, 0)$. Then I've assumed the length of each side of the n-gon is L and that it intersects the ellipse at points $\displaystyle (1,0)$ and $\displaystyle (x_0,y_0)$. With the first equation $\displaystyle L^2 = (1-x_0)^2+y_0^2$ , I have used Pythagoras' theorem to show this relationship. With the next equation, I have used Pythagoras theorem to find where the next side should intersect the ellipse. However, since I have only 1 equation with 2 variables. To solve this problem, I need one more equation. This is not hard since we have the equation of the ellipse and the point $\displaystyle (x_1, y_1)$ must satisfy this equation also. This becomes a system of two equations which you can solve either by hand or using software.

Now for the general case, lets assume we have calculated the position of all points up to point n with coordinates $\displaystyle (x_n,y_n)$. From now on, we need to use the Pythagorean theorem to find where the next vertex of the polygon is. That's what the first equation shows, where I have replaced $\displaystyle L^2$ with the very first equation. Then the second equation makes sure that the point is on the ellipse.