Random Points on a Sphere

randomness method

evenly distributed

spherical coordinates

normalized cartesian

number of points

random seed

There are several ways to pick random points on a sphere. Using points of the form

cos(θ)

1-

2

u

,sin(θ)

1-

2

u

,u

, where

0≤θ≤2π

and

-1≤u≤1

, gives an evenly distributed set of points. Using spherical coordinates,

(cos(θ)sin(ϕ),sin(θ)sin(ϕ),cos(ϕ))

, causes too many points to cluster at the poles. Picking points at random in a cube around the sphere and normalizing,

x

2

x

+

2

y

+

2

z

,

y

2

x

+

2

y

+

2

z

,

z

2

x

+

2

y

+

2

z

, creates too many points that come from the corners of the cube.