ABSTRACT
Every point x ∈ R3 \ {0} given in Cartesian coordinates by the vector (x1, x2, x3)
T can be described in spherical coordinates by a vector (r, ϑ, ϕ)T with r > 0, ϑ ∈ [0, π] and ϕ ∈ [0, 2π) (see Figure 1). We have
(x1, x2, x3) T = (r sinϑ cosϕ, r sinϑ sinϕ, r cosϑ)T ,
r = √
x21 + x 2 2 + x
We denote by S2 the unit sphere embedded into R3; i.e.,
S 2 :=
{ x ∈ R3 : ‖x‖2 = 1
} and identify ξ ∈ S2 with the vector (ϑ, ϕ)T. Let ξ = (ϑ, ϕ)T, η = (ϑ′, ϕ′)T ∈ S2 and α be the angle spanned by the origin, ξ and η. Then the standard inner product ξ · η = cosα is given by
cosα = cosϑ cosϑ′ + sinϑ sinϑ′ cos(ϕ− ϕ′).
