A curve also known as the Gerono lemniscate. It is given by Cartesian coordinates
(1)
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(2)
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(3)
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(4)
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It has vertical tangents at and horizontal tangents at .
Setting , , and in the equation of the eight surface (i.e., scaling by half and relabeling the -axis as the -axis) gives the eight curve.
The area of the curve is
(5)
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The curvature and tangential angle are
(6)
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(7)
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The arc length of the entire curve is given by
(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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(OEIS A118178), where is a complete elliptic integral of the first kind, is a complete elliptic integral of the second kind, and is a complete elliptic integral of the third kind, all with elliptic modulus (D. W. Cantrell, pers. comm., Apr. 22, 2006). The arc length also has a surprising connection to 1-dimensional random walks via
(14)
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where
(15)
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(16)
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(17)
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and is a regularized hypergeometric function, the first few terms of which for , 1, ... are 1, 0, 4, 6, 36, 100, ... (OEIS A092765; M. Alekseyev, pers. comm., Apr. 19, 2006).