An eight curve is a curve also known as the Gerono lemniscate. For 
,
it is given by Cartesian coordinates
 |
(1)
|
 |
(2)
|
and parametric equations, with real 
,
 |  |  |
(3)
|
 |  |  |
(4)
|
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 signed curvature and tangential angle are
 |  | ![-(2sqrt(2)[2+cos(2t)]sint)/(a[2+cos(2t)+cos(4t)]^(3/2))](/images/equations/EightCurve/Inline18.svg) |
(6)
|
 |  |  |
(7)
|
where the complex argument is chosen continuously with 
, giving the direction of the tangent
vector.
The arc length of the entire curve is given by
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(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/Inline37.svg) |
(12)
|
 |  |  |
(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
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
and 
is a regularized
hypergeometric function. The first values of 
for 
, 1, ... are 1, 0, 4, 6, 36,
100, ... (OEIS A092765; M. Alekseyev,
pers. comm., Apr. 19, 2006).
