How to generate two sets of distinct points on a sphere in julia language? - geometry

Question

I need to apply the PCA at different points of a spherical cap, but I don’t know how to build these sets of different points, I need at least 2 sets.
Here is a picture with the idea of what I need.
Spherical Cap

Answer 1

If I correctly understand, here is how I would do in R.
library(uniformly)
library(pracma)
library(rgl)
# sample points on a spherical cap
points_on_cap1 <- runif_on_sphericalCap(300, r = 2, h = 0.5)
# convert to spherical coordinates
sphcoords1 <- cart2sph(points_on_cap1)
# sample points on a spherical cap
points_on_cap2 <- runif_on_sphericalCap(300, r = 2, h = 0.5)
# rotate them, because this is the same spherical cap as before
points_on_cap2 <- rotate3d(points_on_cap2, 3*pi/4, 1, 1, 1)
# convert to spherical coordinates
sphcoords2 <- cart2sph(points_on_cap2)
# 3D plot
spheres3d(0, 0, 0, radius = 2, alpha = 0.5, color = "yellow")
points3d(points_on_cap1, color = "blue")
points3d(points_on_cap2, color = "red")
# 2D plot (of the spherical coordinates)
plot(
sphcoords1[, 1:2], xlim = c(-pi, pi), ylim = c(-pi/2, pi/2),
pch = 19, col = "blue"
)
points(sphcoords2[, 1:2], pch = 19, col = "red")
Do I understand?
Here is the function runif_on_sphericalCap:
function(n, r = 1, h){
stopifnot(h > 0, h < 2*r)
xy <- runif_in_sphere(n, 2L, 1)
k <- h * apply(xy, 1L, crossprod)
s <- sqrt(h * (2*r - k))
cbind(s*xy, r-k)
}
It always samples on a spherical cap with symmetry axis joining the center of the sphere to the North pole. That is why I do a rotation, to get another spherical cap.
Say me if I understand and I'll try to help you to convert the code to Julia.
EDIT: Julia code
using Random, Distributions, LinearAlgebra
function runif_in_sphere(n::I, d::I, r::R) where {I<:Integer, R<:Number}
G = Normal()
sims = rand(G, n, d)
norms = map(norm, eachrow(sims))
u = rand(n) .^ (1/d)
return r .* u .* broadcast(*, 1 ./ norms, sims)
end
function runif_on_sphericalCap(n::I, r::Number, h::Number) where {I<:Integer}
if h <= 0 || h >= 2*r
error("")
end
xy = runif_in_sphere(n, 2, 1.0)
k = h .* map(x -> dot(x,x), eachrow(xy))
s = sqrt.(h .* (2*r .- k))
return hcat(broadcast(*, s, xy), r .- k)
end