Gupf surface

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

z = (x^2 + y^2)/a^2

2

ds^2 = a^2 + 4 u^2 du^2 + a^2 u^2 dv^2

dA = a u sqrt(a^2 + 4 u^2) du dv

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

g_(uu) = a^2 + 4 u^2 g_(vv) = a^2 u^2

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

E(u, v) = a^2 + 4 u^2 F(u, v) = 0 G(u, v) = a^2 u^2

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

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

N^^(u, v) = ((2 u cos(v))/sqrt(4 u^2 + a^2), (2 u sin(v))/sqrt(4 u^2 + a^2), -a/sqrt(4 u^2 + a^2))

algebraic surfaces | quadratic surfaces | surfaces of revolution