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lemniscate constants

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1: 19.20 Special Cases
R F ( 0 , 0 , z ) = .
The first lemniscate constant is given by …
19.20.21 R D ( x , x , z ) = 3 z - x ( R C ( z , x ) - 1 z ) , x z , x z 0 .
The second lemniscate constant is given by …
2: Bibliography T
  • J. Todd (1975) The lemniscate constants. Comm. ACM 18 (1), pp. 14–19.
  • 3: 19.3 Graphics
    See accompanying text
    Figure 19.3.6: Π ( ϕ , 2 , k ) as a function of k 2 and sin 2 ϕ for - 1 k 2 3 , 0 sin 2 ϕ < 1 . …Its value tends to - as k 2 1 + by (19.6.6), and to the negative of the second lemniscate constant (see (19.20.22)) as k 2 ( = csc 2 ϕ ) 2 - . Magnify 3D Help
    4: 19.21 Connection Formulas
    The case z = 1 shows that the product of the two lemniscate constants, (19.20.2) and (19.20.22), is π / 4 . …
    5: 22.18 Mathematical Applications
    Lemniscate
    In polar coordinates, x = r cos ϕ , y = r sin ϕ , the lemniscate is given by r 2 = cos ( 2 ϕ ) , 0 ϕ 2 π . …
    22.18.7 a x 2 y 2 + b ( x 2 y + x y 2 ) + c ( x 2 + y 2 ) + 2 d x y + e ( x + y ) + f = 0 ,
    in which a , b , c , d , e , f are real constants, can be achieved in terms of single-valued functions. …