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Eight Curve


EightCurve

A curve also known as the Gerono lemniscate. It is given by Cartesian coordinates

 x^4=a^2(x^2-y^2),
(1)

polar coordinates,

 r^2=a^2sec^4thetacos(2theta),
(2)

and parametric equations

x=asint
(3)
y=asintcost.
(4)

It has vertical tangents at (+/-a,0) and horizontal tangents at (+/-1/2sqrt(2)a,+/-1/2a).

Setting x=0, z=x/2, and a^'=a/2 in the equation of the eight surface (i.e., scaling by half and relabeling the z-axis as the x-axis) gives the eight curve.

The area of the curve is

 A=4/3a^2.
(5)

The curvature and tangential angle are

kappa(t)=-(2sqrt(2)[2+cos(2t)]sint)/(a[2+cos(2t)+cos(4t)]^(3/2))
(6)
phi(t)=[tan^(-1)(sqrt(7)-4cost)+tan^(-1)(sqrt(7)+4cost)]-[tan^(-1)(4-sqrt(7))+tan^(-1)(4+sqrt(7))].
(7)

The arc length of the entire curve is given by

s=4aint_0^1sqrt((4x^4-5x^2+2)/(1-x^2))dx
(8)
=4aint_0^(pi/2)sqrt(4sin^4t-5sin^2t+2)dt
(9)
=8aint_0^(pi/2)sqrt((sin^2t-5/8)^2+7/(64))dt
(10)
=2sqrt(2)aint_0^(pi/2)sqrt(2+cos(2t)+cos(4t))dt
(11)
={4·2^(1/4)[E(k)-K(k)]+(3+2sqrt(2))2^(-1/4)×Pi(1/8(4-3sqrt(2)),k)}a
(12)
=6.09722...a
(13)

(OEIS A118178), where K(k) is a complete elliptic integral of the first kind, E(k) is a complete elliptic integral of the second kind, and Pi(x,k) is a complete elliptic integral of the third kind, all with elliptic modulus k=sqrt(2+2^(-1/2))/2 (D. W. Cantrell, pers. comm., Apr. 22, 2006). The arc length also has a surprising connection to 1-dimensional random walks via

 s=2piasum_(n=0)^infty4^(-n)(1/2; n)b_n,
(14)

where

b_n=sum_(k=0)^(n)(-3)^k(n; k)(4n-2k; 2n-k)
(15)
=(16^nsqrt(pi)_2F^~_1(-2n,-n;1/2-2n;3/4))/((2n)!)
(16)
=(4n; 2n)_2F_1(-2n,-n;1/2-2n;3/4)
(17)

and _2F^~_1(a,b;c;z) is a regularized hypergeometric function, the first few terms of which for n=0, 1, ... are 1, 0, 4, 6, 36, 100, ... (OEIS A092765; M. Alekseyev, pers. comm., Apr. 19, 2006).


See also

Butterfly Curve, Dumbbell Curve, Eight Surface, Lemniscate, Piriform Curve

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References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 71, 1989.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 124-126, 1972.MacTutor History of Mathematics Archive. "Eight Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Eight.html.Sloane, N. J. A. Sequences A092765 and A118178 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Eight Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EightCurve.html

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