UFO surface

x(u, v) = a sinh(u) cos(v) y(u, v) = a sinh(u) sin(v) z(u, v) = (c u cosh(u))/sqrt(u^2)

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

2

ds^2 = 1/2 ((a^2 + c^2) cosh(2 u) + a^2 - c^2) du^2 + a^2 sinh^2(u) dv^2

dA = (a abs(sinh(u)) sqrt((a^2 + c^2) cosh(2 u) + a^2 - c^2))/sqrt(2) du dv

K(u, v) = (4 c^2)/((a^2 + c^2) cosh(2 u) + a^2 - c^2)^2

Γ | u | | | uu = ((a^2 + c^2) sinh(2 u))/((a^2 + c^2) cosh(2 u) + a^2 - c^2) Γ | u | | | vv = -(2 a^2 sinh(u) cosh(u))/((a^2 + c^2) cosh(2 u) + a^2 - c^2) Γ | v | | | uv = coth(u) Γ | v | | | vu = coth(u)

E(u, v) = 1/2 ((a^2 + c^2) cosh(2 u) + a^2 - c^2) F(u, v) = 0 G(u, v) = a^2 sinh^2(u)

e(u, v) = (sqrt(2) a c abs(u) sinh(u))/(u abs(sinh(u)) sqrt((a^2 + c^2) cosh(2 u) + a^2 - c^2)) f(u, v) = 0 g(u, v) = (sqrt(2) a c sgn(u) sinh(u) abs(sinh(u)))/sqrt((a^2 + c^2) cosh(2 u) + a^2 - c^2)

left double bracketing bar x(u, v) right double bracketing bar = sqrt((a^2 + c^2) cosh(2 u) - a^2 + c^2)/sqrt(2)

N^^(u, v) = ((sqrt(2) abs(u) abs(sinh(u)) cos(v) c)/(u sqrt(a^2 - c^2 + cosh(2 u) (a^2 + c^2))), (sqrt(2) abs(u) abs(sinh(u)) sin(v) c)/(u sqrt(a^2 - c^2 + cosh(2 u) (a^2 + c^2))), -(sinh(2 u) a)/(sqrt(2) sqrt(sinh^2(u) (a^2 - c^2 + cosh(2 u) (a^2 + c^2)))))

algebraic surfaces | quadratic surfaces | surfaces of revolution