Clifford torus handle

CliffordTorusHandle.R

library(rgl)
library(cgalMeshes)

p <- function(theta, phi) {
  c(
    cos(theta)*cos(phi), cos(theta)*sin(phi), 
    sin(theta)*cos(phi), sin(theta)*sin(phi)
  )
}

s <- Vectorize(function(theta, phi) {
  q <- p(theta, phi)
  q[1L:3L] / (1 - q[4L])
})

# with blue and yellow ribbons
open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.95)
for(i in c(1:5, 11:15)) {
  a <- 2 * (i-1) * pi/30
  mesh <- parametricMesh(
    s, c(0, 2*pi), c(a, a + pi/30), periodic = c(TRUE, FALSE), 
    nu = 300, nv = 20
  )
  shade3d(addNormals(mesh), col = "navy")
  mesh <- parametricMesh(
    s, c(0, 2*pi), c(a + pi/30, a + pi/15), periodic = c(TRUE, FALSE), 
    nu = 300, nv = 20
  )
  shade3d(addNormals(mesh), col = "yellow")
}


# tubular 
theta_ <- seq(0, 2*pi, length.out = 300L)
open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.9)
k <- 1L
for(i in c(1L, 1L, 2L, 2L, 3L, 3L, 4L, 4L, 5L, 5L)) {
  k <- -k
  phi <- pi/2 + k * (pi/9 + i * pi/15)
  pts <- t(s(theta_, phi))
  mesh <- cylinder3d(pts, radius = 0.07, sides = 30, closed = TRUE)
  shade3d(addNormals(mesh), col = "navy")
}


# # -- if you want an animation 
M <- par3d("userMatrix")
movie3d(
  par3dinterp(
    time = seq(0, 1, len = 9),
    userMatrix = list(
      M,
      rotate3d(M, pi, 1, 0, 0),
      rotate3d(M, pi, 1, 1, 0),
      rotate3d(M, pi, 1, 1, 1),
      rotate3d(M, pi, 0, 1, 1),
      rotate3d(M, pi, 0, 1, 0),
      rotate3d(M, pi, 1, 0, 1),
      rotate3d(M, pi, 0, 0, 1),
      M
    )
  ),
  fps = 100,
  duration = 1,
  dir = ".",
  movie = "zzpic",
  convert = FALSE, webshot = FALSE
)

library(gifski)
gifski(
  png_files = Sys.glob("zzpic*.png"),
  gif_file = "CliffordTorusHandle_tubular.gif",
  width = 512,
  height = 512,
  delay = 1/8
)
file.remove(Sys.glob("zzpic*.png"))