doubled sphere

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

(-a^2 + x^2 + y^2 + z^2)^2 = 0

4

S = 4 π a^2

ds^2 = a^2 sin^2(v) du^2 + a^2 dv^2

dA = a^2 sin(v) du dv

x^_ = (0, 0, 0)

V = (4 π a^3)/3

I = ((2 a^2)/5 | 0 | 0 0 | (2 a^2)/5 | 0 0 | 0 | (2 a^2)/5)

K(u, v) = 1/a^2

(for a double sphere centered at at the origin and of radius a)

g_(uu) = a^2 cos^2(v) g_(vv) = a^2

Γ | u | | | uv = cot(v) Γ | u | | | vu = cot(v) Γ | v | | | uu = sin(v) (-cos(v))

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

e(u, v) = a^2 sin^2(v) f(u, v) = 0 g(u, v) = a^2

left double bracketing bar x(u, v) right double bracketing bar = a

N^^(u, v) = (cos(u) sin(v), sin(u) sin(v), cos(v))

algebraic surfaces | closed surfaces | constant (Gaussian) curvature surfaces | minimal surfaces | quartic surfaces | surfaces of revolution