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I am not familiar with elliptical integrals on an academic level, but I came across them as solutions to certain integrals and did some general reading. Why would solving a seemingly 2 dimensional and nearly circular geometric problem yield solutions that have third degree polynomials? And how does a real ellipse relate to a complex Riemann surfaces?

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    $\begingroup$ A complex compact Riemann surface is a algebraic curve, and curves of genus $1$ are called elliptic curves. They're related to elliptic functions, some of these functions are involved in the computation of the length of an elliptic arc. $\endgroup$
    – Bernard
    Commented Jul 1, 2017 at 1:02
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    $\begingroup$ en.wikipedia.org/wiki/Elliptic_integral "elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse" Elliptic integrals being a function of two parameters only enforces that notion, since an ellipse has both a major and minor axis. If I looked an ellipse...where would obtain a third degree polynomial in the solution to any of its geometric properties? it doesn't seem very intuitive, I would think it still has something to do with quadratic or maybe quartic polynomials. $\endgroup$
    – RayOfHope
    Commented Jul 1, 2017 at 1:06

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Look at the answers to Birational Equivalence of Diophantine Equations and Elliptic Curves and https://mathoverflow.net/questions/239746/birationally-transforming-a-quartic-elliptic-curve.

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