spaghetti bundle | Spindel surface

x(u, v) = a sqrt(u^2 + 1) cos(v) y(u, v) = a sqrt(u^2 + 1) sin(v) z(u, v) = c u

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

2

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

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

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

g_(uu) = (a^2 u^2)/(u^2 + 1) + c^2 g_(vv) = a^2 (u^2 + 1)

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

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

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

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

N^^(u, v) = (cos(v) c sqrt((1 + u^2)/(c^2 + u^2 (a^2 + c^2))), sin(v) c sqrt((1 + u^2)/(c^2 + u^2 (a^2 + c^2))), -(u a)/sqrt(c^2 + u^2 (a^2 + c^2)))

algebraic surfaces | doubly ruled surfaces | ruled surfaces | quadratic surfaces | surfaces of revolution