double cone | Kegel surface

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

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

2

ds^2 = a^2 + 1 du^2 + a^2 u^2 dv^2

dA = a sqrt(a^2 + 1) u du dv

x^_ = (0, 0, 0)

K(u, v) = 0

(for an infinite double-napped right cone with axis of symmetry along the z-axis and vertex located at at the origin)

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

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

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

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

left double bracketing bar x(u, v) right double bracketing bar = sqrt(a^2 + 1) abs(u)

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

algebraic surfaces | constant (Gaussian) curvature surfaces | developable surfaces | quadratic surfaces | ruled surfaces | surfaces of revolution