Fig. 1.

Visualization of the spherical and Hopf coordinates on SO(3) using angle and axis representation. This representation corresponds to a projection of the S³ onto the equatorial solid sphere which we draw in ${\mathbb{R}}^3$. (a) The full range of the spherical coordinate ψ ∈ [0,π/2] is shown while the coordinates (θ,ϕ) form a discretization of size 20 over S². (b) The half-spheres show the full ranges of the spherical coordinates θ ∈ [0,π], and ϕ ∈ [0,2π), while ψ takes four discrete values over [0,π/2]. (c) The full range of the Hopf coordinate ψ ∈ [0,2π) is shown while the coordinates (θ,ϕ) form a discretization of size 12 over S². (b) The spheres show the full ranges of the Hopf coordinates θ ∈ [0,π] and ϕ ∈ [0,2π), while ψ takes four discrete values over [0,2π).