A curve also known as the Gerono lemniscate. It is given by Cartesian coordinates
 | (1) |
 | (2) |
 |  |  | (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 curvature and tangential angle are
 |  | ![-(2sqrt(2)[2+cos(2t)]sint)/(a[2+cos(2t)+cos(4t)]^(3/2))](http://mathworld.wolfram.com/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))].](http://mathworld.wolfram.com/images/equations/EightCurve/Inline19.svg) | (7) |
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](http://mathworld.wolfram.com/images/equations/EightCurve/Inline34.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 few terms of which for 
, 1, ... are 1, 0, 4, 6, 36, 100, ... (OEIS A092765; M. Alekseyev, pers. comm., Apr. 19, 2006).
