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Sphere

Example plot

Equations

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

x^2 + y^2 + z^2 = a^2

Surface properties

g = 0

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 sphere with center at the origin and radius a)

Distance properties

s^_ = (4 a)/3

(where lengths and areas refer to line segments and triangles picked at random from points on the surface)

Metric properties

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

Vector properties

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))

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

Properties

algebraic surfaces | closed surfaces | constant (Gaussian) curvature surfaces | quadratic surfaces | surfaces of revolution