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  "body": "\n## Summary\n\nThe complete elliptic integral of the second kind is the function $E(k) = \\int_0^{\\pi/2} \\sqrt{1 - k^2\\sin^2\\theta}\\,d\\theta$ of the modulus $k$.\nIt gives the perimeter of an ellipse: an ellipse with semi-major axis $a$ and eccentricity $e$ has perimeter $4aE(e)$.\nTogether with the [[complete integral of the first kind|RM54CPPE12]] it enters [[Legendre's relation|8GSKDNZ4TK]].\n\n## Explanations\n\n### L1 human\n\nA circle of radius $a$ has perimeter $2\\pi a$, but there is no such simple formula for an ellipse.\nThe perimeter of an ellipse is four times the length of a quarter arc, and the quarter arc is $a E(e)$.\nFor a circle $e = 0$, and $E(0) = \\pi/2$ gives back $2\\pi a$.\nFor a flat ellipse squeezed to a segment of length $2a$, $e = 1$ and $E(1) = 1$, so the perimeter is $4a$, the segment traced there and back.\n",
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