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    Cross-cap

    Example plot

    Example plot

    Equations

    x(u, v) = 1/2 a sin(u) sin(2 v) y(u, v) = a sin(2 u) sin^2(v) z(u, v) = a cos(2 u) sin^2(v)

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

    Surface properties

    4

    ds^2 = 1/4 a^2 sin^2(v) ((cos(2 u) - 7) cos(2 v) + cos(2 u) + 9) du^2 + 1/4 a^2 sin(2 u) sin(4 v) du dv + -1/4 a^2 (2 cos^2(u) cos(4 v) + cos(2 u) - 3) dv^2

    dA = 1/8 a^2 sqrt(-sin^2(2 u) sin^2(4 v) - 4 sin^2(v) ((cos(2 u) - 7) cos(2 v) + cos(2 u) + 9) (2 cos^2(u) cos(4 v) + cos(2 u) - 3)) du dv

    V = (π a^3)/2

    I = ((7 a^2)/16 | 0 | 0 0 | (59 a^2)/240 | 0 0 | 0 | (11 a^2)/40)

    K(u, v) = -(256 (cos(2 u) + csc^2(v) - 2))/(a^2 (cos^2(u) (6 cos(4 v) - 8 cos(2 v)) + cos(2 u) - 15)^2)

    Metric properties

    g_(uu) = 1/4 a^2 sin^2(v) ((cos(2 u) - 7) cos(2 v) + cos(2 u) + 9) g_(uv) = 1/8 a^2 sin(2 u) sin(4 v) g_(vu) = 1/8 a^2 sin(2 u) sin(4 v) g_(vv) = -1/4 a^2 (2 cos^2(u) cos(4 v) + cos(2 u) - 3)

    Γ | u | | | uu = -(4 sin(2 u) cos^2(v))/(cos(2 u - 2 v) + cos(2 (u + v)) + 2 cos(2 u) - 14 cos(2 v) + 18) Γ | u | | | uv = (2 cot(v) (cos(2 u - 2 v) + cos(2 (u + v)) - 14 cos(2 v) + 16))/(cos(2 u - 2 v) + cos(2 (u + v)) + 2 cos(2 u) - 14 cos(2 v) + 18) Γ | u | | | vu = (2 cot(v) (cos(2 u - 2 v) + cos(2 (u + v)) - 14 cos(2 v) + 16))/(cos(2 u - 2 v) + cos(2 (u + v)) + 2 cos(2 u) - 14 cos(2 v) + 18) Γ | u | | | vv = -(16 sin(2 u) cos^2(v))/(cos(2 u - 2 v) + cos(2 (u + v)) + 2 cos(2 u) - 14 cos(2 v) + 18) Γ | v | | | uu = -(sin(2 v) (cos(2 u - 2 v) + cos(2 (u + v)) + 14 cos(2 v) - 16))/(cos(2 u - 4 v) + cos(2 u + 4 v) + 2 cos(2 u) + 2 cos(4 v) - 6) Γ | v | | | uv = -(4 sin(2 u) cos^2(2 v))/(cos(2 u - 4 v) + cos(2 u + 4 v) + 2 cos(2 u) + 2 cos(4 v) - 6) Γ | v | | | vu = -(4 sin(2 u) cos^2(2 v))/(cos(2 u - 4 v) + cos(2 u + 4 v) + 2 cos(2 u) + 2 cos(4 v) - 6) Γ | v | | | vv = -(8 cos^2(u) sin(4 v))/(cos(2 u - 4 v) + cos(2 u + 4 v) + 2 cos(2 u) + 2 cos(4 v) - 6)

    E(u, v) = 1/4 a^2 sin^2(v) ((cos(2 u) - 7) cos(2 v) + cos(2 u) + 9) F(u, v) = 1/8 a^2 sin(2 u) sin(4 v) G(u, v) = -1/4 a^2 (2 cos^2(u) cos(4 v) + cos(2 u) - 3)

    e(u, v) = (4 a sin(u) sin^2(v) (3 cos(2 v) - 1))/sqrt(4 cos(2 u) sin^2(v) (3 cos(2 v) + 1) + 4 cos(2 v) - 3 cos(4 v) + 15) f(u, v) = -(8 a cos(u) sin(v) cos(v))/sqrt(4 cos(2 u) sin^2(v) (3 cos(2 v) + 1) + 4 cos(2 v) - 3 cos(4 v) + 15) g(u, v) = -(8 a sin(u))/sqrt(4 cos(2 u) sin^2(v) (3 cos(2 v) + 1) + 4 cos(2 v) - 3 cos(4 v) + 15)

    Vector properties

    left double bracketing bar x(u, v) right double bracketing bar = 1/2 a sqrt(sin^2(u) sin^2(2 v) + 4 sin^4(v))

    N^^(u, v) = (-(8 cos(v) sin(v))/sqrt(15 + 4 cos(2 v) - 3 cos(4 v) + 4 cos(2 u) (1 + 3 cos(2 v)) sin^2(v)), (cos(u) + 3 cos(u) cos(2 v) + 2 cos(3 u) sin^2(v))/sqrt(15 + 4 cos(2 v) - 3 cos(4 v) + 4 cos(2 u) (1 + 3 cos(2 v)) sin^2(v)), -(4 sin(u) (cos^2(u) + cos(2 v) sin^2(u)))/sqrt(15 + 4 cos(2 v) - 3 cos(4 v) + 4 cos(2 u) (1 + 3 cos(2 v)) sin^2(v)))

    Properties

    algebraic surfaces | closed surfaces | quartic surfaces | Steiner surfaces

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