Spherical triangle with rgl - https://i.imgur.com/05qwdir.gifv

sphericalTriangle.R

library(rgl)
library(abind) # to use the abind function
library(plyr) # to use the alply function

# normalize a vector
normalize <- function(vec){
  vecnorm <- sqrt(sum(vec^2))
  vec/vecnorm
}

# split a triangle into 4 triangles
# vertices of the triangle are assumed to lie on the unit sphere
splitTriangle1 <- function(triangle){
  a <- triangle[1,]
  b <- triangle[2,]
  c <- triangle[3,]
  mab <- normalize((a+b)/2)
  mac <- normalize((a+c)/2)
  mbc <- normalize((b+c)/2)
  abind(
    rbind(a,mab,mac),
    rbind(b,mbc,mab),
    rbind(c,mac,mbc),
    rbind(mab,mbc,mac),
    along = 3
  )
}

# example of splitTriangle1
triangle <- rbind(
  c(0,0,1),
  c(0,1,0),
  c(1,0,0)
)
splitTriangle1(triangle)

# iterate the splitTriangle1 function
splitTriangle <- function(triangle, n){
  triangles <- splitTriangle1(triangle)
  for(i in seq_len(n-1)){
    triangles <- abind(alply(triangles,3, splitTriangle1), along=3)
  }
  unname(triangles)
}

# example of splitTriangle
triangle <- rbind(
  c(1,0,0),
  c(0,1,0),
  c(0,0,1)
)
triangles <- splitTriangle(triangle, 3)
for(i in seq_len(dim(triangles)[3])){
  triangles3d(triangles[,,i], color="green")
}

# main function: draw a spherical triangle
drawSphericalTriangle <- function(triangle, n=5, col="green"){
  triangles <- splitTriangle(triangle, n)
  for(i in seq_len(dim(triangles)[3])){
    triangles3d(triangles[,,i], color=col)
  }
}

#### example: triangulated sphere ####
randomOnSphere <- function(n){
  theta <- runif(n, 0, 2*pi)
  phi <- runif(n, 0, pi)
  cbind(cos(theta)*sin(phi), sin(theta)*sin(phi), cos(phi))
}
points <- randomOnSphere(50)
library(geometry)
hull <- convhulln(points, options="Qt")
triangles <- alply(hull, 1, 
                   function(ijk){
                     rbind(points[ijk[1],],points[ijk[2],],points[ijk[3],])
                   })
colors <- randomcoloR::distinctColorPalette(length(triangles))
lapply(seq_along(triangles), 
       function(i) drawSphericalTriangle(triangles[[i]], n=5, col=colors[i]))

sphericalTriangle2.R

library(rgl)
library(abind) # to use the abind function
library(plyr) # to use the alply function

# normalize a vector
normalize <- function(vec){
  vecnorm <- sqrt(sum(vec^2))
  vec/vecnorm
}

# split a triangle into 4 triangles
# vertices of the triangle are assumed to lie on the unit sphere
splitTriangle1 <- function(triangle){
  a <- triangle[1,]
  b <- triangle[2,]
  c <- triangle[3,]
  mab <- normalize((a+b)/2)
  mac <- normalize((a+c)/2)
  mbc <- normalize((b+c)/2)
  abind(
    rbind(a,mab,mac),
    rbind(b,mbc,mab),
    rbind(c,mac,mbc),
    rbind(mab,mbc,mac),
    along = 3
  )
}

# example of splitTriangle1
triangle <- rbind(
  c(0,0,1),
  c(0,1,0),
  c(1,0,0)
)
splitTriangle1(triangle)

# iterate the splitTriangle1 function to do the mesh
sphtriMesh <- function(triangle, n){
  normal <- rgl:::xprod(triangle[2,]-triangle[1,],triangle[3,]-triangle[1,])
  center <- colMeans(triangle)
  dot <- crossprod(normal, center)
  if(dot < 0){
    triangle <- triangle[c(1,3,2),]
  }
  triangles <- splitTriangle1(triangle)
  for(i in seq_len(n-1)){
    triangles <- abind(alply(triangles,3, splitTriangle1), along=3)
  }
  vertices <- do.call(cbind, alply(triangles, 3, t))
  tris <- matrix(1:(3*4^n), nrow=3)
  out <- list(
    vb = rbind(vertices,1),
    it = tris,
    primitivetype = "triangle",
    material = list(),
    normals = rbind(vertices,1)
  )
  class(out) <- c("mesh3d", "shape3d")
  out
}

# example 
triangle <- rbind(
  c(1,0,0),
  c(0,1,0),
  c(0,0,1)
)
mesh <- sphtriMesh(triangle, 4)
shade3d(mesh, color="red")

triangle <- rbind(
  c(1,0,0),
  c(0,0,1),
  c(0,1,0)
)
mesh <- sphtriMesh(triangle, 4)
shade3d(mesh, color="red")

#### example: triangulated sphere #### 
points <- uniformly::runif_on_sphere(50, 3)
library(geometry)
hull <- convhulln(points, options="Qt")
triangles <- alply(hull, 1, 
                   function(ijk){
                     rbind(points[ijk[1],],points[ijk[2],],points[ijk[3],])
                   })
colors <- randomcoloR::distinctColorPalette(length(triangles))
rgl.material(back="culled")
invisible(lapply(seq_along(triangles), 
                 function(i){
                   mesh <- sphtriMesh(triangles[[i]], n=3)
                   shade3d(mesh, color=colors[i])
                 }))