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Hyperbolic Lemniscate Function

Result

x(t) = (a cos(t))/(sin^2(t) + 1)
y(t) = (a sin(t) cos(t))/(sin^2(t) + 1)

Plot

Equations

a^2 x^2 - a^2 y^2 - x^4 - 2 x^2 y^2 - y^4 = 0

r(θ) = a sqrt(cos(2 θ))

Parametric properties

s = (a Γ(1/4)^2)/sqrt(2 π)

s(t) = a F(t|-1)

κ(t) = (3 sqrt(2) cos(t))/(a sqrt(3 - cos(2 t)))

m(t) = -(2 (csc(t) - 3 sin(t)))/(cos(2 t) + 5)

ϕ(t) = 3 tan^(-1)(sin(t))

left double bracketing bar x(t) right double bracketing bar = (a abs(cos(t)))/sqrt(sin^2(t) + 1)