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

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

2

g = 0

S = 2 π (a^2 + c^2 2F1(1/2, 1, 3/2, 1 - c^2/a^2))

ds^2 = a^2 sin^2(v) du^2 + 1/2 (a^2 + (a - c) (a + c) cos(2 v) + c^2) dv^2

dA = (a sin(v) sqrt(a^2 + (a - c) (a + c) cos(2 v) + c^2))/sqrt(2) du dv

x^_ = (0, 0, 0)

V = 4/3 π a^2 c

I = (1/5 (a^2 + c^2) | 0 | 0 0 | 1/5 (a^2 + c^2) | 0 0 | 0 | 1/5 (a^2 + c^2))

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

(for a spheroid with center at the origin, semiaxis a along the x- and y-axes, and semiaxis c along the z-axis; if a>c, the spheroid is called oblate, if a

g_(uu) = a^2 sin^2(v) g_(vv) = 1/2 (a^2 + (a - c) (a + c) cos(2 v) + c^2)

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

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

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

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

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

N^^(x, y, z) = ((x c^2)/sqrt(z^2 a^4 + (x^2 + y^2) c^4), (y c^2)/sqrt(z^2 a^4 + (x^2 + y^2) c^4), (z a^2)/sqrt(z^2 a^4 + (x^2 + y^2) c^4))

algebraic surfaces | closed surfaces | quadratic surfaces | surfaces of revolution