x(u, v) = u y(u, v) = v z(u, v) = v^2/b^2 - u^2/a^2

z = y^2/b^2 - x^2/a^2

2

ds^2 = (4 u^2)/a^4 + 1 du^2 + -(8 u v)/(a^2 b^2) du dv + (4 v^2)/b^4 + 1 dv^2

dA = sqrt((4 u^2)/a^4 + (4 v^2)/b^4 + 1) du dv

K(u, v) = -(4 a^6 b^6)/(a^4 (b^4 + 4 v^2) + 4 b^4 u^2)^2

g_(uu) = (4 u^2)/a^4 + 1 g_(uv) = -(4 u v)/(a^2 b^2) g_(vu) = -(4 u v)/(a^2 b^2) g_(vv) = (4 v^2)/b^4 + 1

Γ | u | | | uu = (4 u)/(a^4 + 4 u^2) Γ | u | | | vv = -(4 a^2 u)/(b^2 (a^4 + 4 u^2)) Γ | v | | | uu = -(4 b^2 v)/(a^2 (b^4 + 4 v^2)) Γ | v | | | vv = (4 v)/(b^4 + 4 v^2)

E(u, v) = (4 u^2)/a^4 + 1 F(u, v) = -(4 u v)/(a^2 b^2) G(u, v) = (4 v^2)/b^4 + 1

e(u, v) = -2/(a^2 sqrt((4 u^2)/a^4 + (4 v^2)/b^4 + 1)) f(u, v) = 0 g(u, v) = 2/(b^2 sqrt((4 u^2)/a^4 + (4 v^2)/b^4 + 1))

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

N^^(u, v) = (-(2 u)/(a^2 sqrt(1 + (4 u^2)/a^4 + (4 v^2)/b^4)), (2 v)/(sqrt(1 + (4 u^2)/a^4 + (4 v^2)/b^4) b^2), -1/sqrt(1 + (4 u^2)/a^4 + (4 v^2)/b^4))

algebraic surfaces | doubly ruled surfaces | quadratic surfaces | ruled surfaces