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    Pseudosphere

    Example plot

    Example plot

    Alternate names

    antisphere | tractricoid | tractrisoid | tractroid

    Equations

    x(u, v) = a sech(u) cos(v) y(u, v) = a sech(u) sin(v) z(u, v) = a (u - tanh(u))

    z^2 = (a sech^(-1)(sqrt(x^2 + y^2)/a) - sqrt(a^2 - x^2 - y^2))^2

    Surface properties

    S = 4 π a^2

    ds^2 = a^2 tanh^2(u) du^2 + a^2 sech^2(u) dv^2

    dA = a^2 tanh(u) sech(u) du dv

    x^_ = (0, 0, 0)

    V = (2 π a^3)/3

    K(u, v) = -1/a^2

    Metric properties

    g_(uu) = a^2 tanh^2(u) g_(vv) = a^2 sech^2(u)

    Γ | u | | | uu = csch(u) sech(u) Γ | u | | | vv = csch(u) sech(u) Γ | v | | | uv = -tanh(u) Γ | v | | | vu = -tanh(u)

    E(u, v) = a^2 tanh^2(u) F(u, v) = 0 G(u, v) = a^2 sech^2(u)

    e(u, v) = a tanh(u) (-sech(u)) f(u, v) = 0 g(u, v) = a tanh(u) sech(u)

    Vector properties

    left double bracketing bar x(u, v) right double bracketing bar = a sqrt(u^2 - 2 u tanh(u) + 1)

    N^^(u, v) = (abs(tanh(u)) cos(v), abs(tanh(u)) sin(v), sech(u) sgn(tanh(u)))

    Properties

    constant (Gaussian) curvature surfaces | surfaces of revolution

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