Dullo surface

x(u, v) = a cos(u) (cos(v) + 1) y(u, v) = a sin(u) (cos(v) + 1) z(u, v) = a sin(v)

(x^2 + y^2 + z^2)^2 = 4 a^2 (x^2 + y^2)

4

S = 4 π^2 a^2

ds^2 = 4 a^2 cos^4(v/2) du^2 + a^2 dv^2

dA = a^2 (cos(v) + 1) du dv

x^_ = (0, 0, 0)

V = 2 π^2 a^3

I = ((9 a^2)/8 | 0 | 0 0 | (9 a^2)/8 | 0 0 | 0 | (7 a^2)/4)

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

g_(uu) = 4 a^2 cos^4(v/2) g_(vv) = a^2

Γ | u | | | uv = -tan(v/2) Γ | u | | | vu = -tan(v/2) Γ | v | | | uu = 4 sin(v/2) cos^3(v/2)

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

e(u, v) = -2 a cos^2(v/2) cos(v) f(u, v) = 0 g(u, v) = -a

left double bracketing bar x(u, v) right double bracketing bar = 2 a abs(cos(v/2))

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

algebraic surfaces | closed surfaces | quartic surfaces | surfaces of revolution