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    Witch of Agnesi

    Plot

    Plot

    Alternate names

    Agnésienne | averisera | avversiera | cubique d'Agnesi | versed sine curve | witch curve

    Equations

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

    8 a^3 = 4 a^2 y + x^2 y

    r(θ) = a sqrt((sqrt(81 cot^2(θ) + 12) + 9 cot(θ))^(2/3)/(3 2^(2/3) 3^(1/3)) + 1/((sqrt(81 cot^2(θ) + 12) + 9 cot(θ))^(2/3)/(3 2^(2/3) 3^(1/3)) + (2/3)^(2/3)/(sqrt(81 cot^2(θ) + 12) + 9 cot(θ))^(2/3) + 1/3)^2 + (2/3)^(2/3)/(sqrt(81 cot^2(θ) + 12) + 9 cot(θ))^(2/3) - 2/3)

    (for the curve based on a circle of radius a and center (0, a))

    Properties

    algebraic | cubic

    Basic properties

    A = 4 π a^2

    d = 3

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