8-dong surface | octdong surface

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

a^2 (x^2 + y^2 - 4 z^2) + 4 z^4 = 0

4

S = 1/128 π a^2 (240 + 136 sqrt(5) + 31 log(17 + 8 sqrt(5)))

ds^2 = a^2 sin^2(2 v) du^2 + 1/2 a^2 (cos(2 v) + 4 cos(4 v) + 5) dv^2

dA = (a^2 sqrt(cos(2 v) + 4 cos(4 v) + 5) abs(sin(2 v)))/sqrt(2) du dv

x^_ = (0, 0, 0)

V = (16 π a^3)/15

I = ((13 a^2)/21 | 0 | 0 0 | (13 a^2)/21 | 0 0 | 0 | (8 a^2)/21)

K(u, v) = (4 (cos(2 v) + 2))/(a^2 (cos(2 v) + 4 cos(4 v) + 5)^2)

g_(uu) = a^2 sin^2(2 v) g_(vv) = 1/2 a^2 (cos(2 v) + 4 cos(4 v) + 5)

Γ | u | | | uv = 2 cot(2 v) Γ | u | | | vu = 2 cot(2 v) Γ | v | | | uu = -(2 sin(4 v))/(cos(2 v) + 4 cos(4 v) + 5) Γ | v | | | vv = -(sin(2 v) + 8 sin(4 v))/(cos(2 v) + 4 cos(4 v) + 5)

E(u, v) = a^2 sin^2(2 v) F(u, v) = 0 G(u, v) = 1/2 a^2 (cos(2 v) + 4 cos(4 v) + 5)

e(u, v) = -(4 sqrt(2) a sin^2(v) cos^3(v))/sqrt(sin^2(2 v) (cos(2 v) + 4 cos(4 v) + 5)) f(u, v) = 0 g(u, v) = -(2 sqrt(2) a sin^2(v) (5 cos(v) + cos(3 v)))/sqrt(sin^2(2 v) (cos(2 v) + 4 cos(4 v) + 5))

left double bracketing bar x(u, v) right double bracketing bar = a sqrt(2 cos(2 v) + 3) abs(sin(v))

N^^(u, v) = (-(2 sqrt(2) cos(u) cos^2(v) sin(v))/(abs(sin(2 v)) sqrt(5 + cos(2 v) + 4 cos(4 v))), -(2 sqrt(2) cos^2(v) sin(u) sin(v))/(abs(sin(2 v)) sqrt(5 + cos(2 v) + 4 cos(4 v))), (sqrt(2) sin(4 v))/(abs(sin(2 v)) sqrt(5 + cos(2 v) + 4 cos(4 v))))

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