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    Ring Function

    Result

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

    Example plots

    Example plots

    Equations

    x(u, v) = cos(u) (a cos(v) + c) y(u, v) = sin(u) (a cos(v) + c) z(u, v) = a sin(v)

    (c - sqrt(x^2 + y^2))^2 + z^2 = a^2

    Surface properties

    4

    g = 1

    S = 4 π^2 a c

    ds^2 = (a cos(v) + c)^2 du^2 + a^2 dv^2

    dA = a abs(c + a cos(v)) du dv

    x^_ = (0, 0, 0)

    V = 2 π^2 a^2 c

    I = ((5 a^2)/8 + c^2/2 | 0 | 0 0 | (5 a^2)/8 + c^2/2 | 0 0 | 0 | (3 a^2)/4 + c^2)

    K(u, v) = cos(v)/(a (a cos(v) + c))

    (for a torus with center at the origin, rotational axis of symmetry about the z-axis, radius c, from the center of the hole to the center of the torus tube, and radius a of the tube; c>a corresponds to a ring torus, c = a to a horn torus (which is tangent to itself at the point at the origin), and c<a to a self-intersecting spindle torus (for which some of the properties given here are not well-defined or do not hold))

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