Haar Measure of distance Ball SO(3)

so3.jl

# https://math.stackexchange.com/questions/1049788/haar-measure-of-an-angle-distance-ball-in-so3

using StatsBase
using LinearAlgebra
using StatPlots
using Plots
using Rotations

function sample_angles(N)
    map(1:N) do _
        rot = rand(RotMatrix{3})
        AngleAxis(rot).theta
    end
end


N_bins    = 100
N_samples = 10^5
wts = weights(fill(1.,N_samples))
edges = range(0,stop=pi,length=N_bins)
h = fit(Histogram, sample_angles(N_samples), wts, edges, closed=:left)

pdf(angle) = 1 - cos(angle)
plot(title="Pdf angle of rotation Haar Measure")
plot!(rad2deg.(edges)[2:end], N_bins*normalize!(h.weights,1), label="simulation")
plot!(rad2deg.(edges), pdf.(edges), label="pdf")