A curve also known as the Gerono lemniscate. It is given by Cartesian coordinates
 |
(1)
|
 |
(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)
|
The curvature and tangential angle are
 |  | ![-(2sqrt(2)[2+cos(2t)]sint)/(a[2+cos(2t)+cos(4t)]^(3/2))](/images/equations/EightCurve/Inline16.svg) |
(6)
|
 |  | ![[tan^(-1)(sqrt(7)-4cost)+tan^(-1)(sqrt(7)+4cost)]-[tan^(-1)(4-sqrt(7))+tan^(-1)(4+sqrt(7))].](/images/equations/EightCurve/Inline19.svg) |
(7)
|
The arc length of the entire curve is given by
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(8)
|
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(9)
|
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(10)
|
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(11)
|
 |  | ![{4·2^(1/4)[E(k)-K(k)]+(3+2sqrt(2))2^(-1/4)×Pi(1/8(4-3sqrt(2)),k)}a](/images/equations/EightCurve/Inline34.svg) |
(12)
|
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(13)
|
(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)
|
where
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(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).
