Pseudo Hopf torus with holes

pht.R

# https://mathmod.deviantart.com/art/Pseudo-Hopf-Tori-565531249
library(misc3d)
library(rgl)

cu <- cv <- 0.0000000001
N <- 3
Fx <- function(u,v){ -cos(u+v)/(sqrt(2)+cos(v-u)) }
DFx <- function(u,v){
  DFxu = (Fx(u,v)-Fx(u+cu,v))/cu
  DFxv = (Fx(u,v)-Fx(u,v+cv))/cv
  c(DFxu, DFxv)
}
Fy <- function(u,v){ sin(v-u)/(sqrt(2)+cos(v-u)) }
DFy <- function(u,v){
  DFyu = (Fy(u,v)-Fy(u+cu,v))/cu
  DFyv = (Fy(u,v)-Fy(u,v+cv))/cv
  c(DFyu, DFyv)
}
Fz <- function(u,v){ sin(u+v)/(sqrt(2)+cos(v-u)) }
DFz <- function(u,v){
  DFzu = (Fz(u,v)-Fz(u+cu,v))/cu
  DFzv = (Fz(u,v)-Fz(u,v+cv))/cv
  c(DFzu, DFzv)
}
n1 <- function(u,v){
  DFy(u,v)[1]*DFz(u,v)[2]-DFz(u,v)[1]*DFy(u,v)[2]
}
n2 <- function(u,v){
  DFz(u,v)[1]*DFx(u,v)[2]-DFx(u,v)[1]*DFz(u,v)[2]
}
n3 <- function(u,v){
  DFx(u,v)[1]*DFy(u,v)[2]-DFy(u,v)[1]*DFx(u,v)[2]
}
R <- function(u,v){
  sqrt(n1(u,v)^2+n2(u,v)^2+n3(u,v)^2)
}
fx <- function(u,v){
  Thickness = (abs(sin(15*u)*cos(15*v)))^7 + .4*((sin(2*N*u)))
  Fx(u,v)+Thickness*n1(u,v)/R(u,v)
}
fy <- function(u,v){
  Thickness = (abs(sin(15*u)*cos(15*v)))^7 + .4*((sin(2*N*u)))
  Fy(u,v)+Thickness*n2(u,v)/R(u,v)
}
fz <- function(u,v){
  Thickness = (abs(sin(15*u)*cos(15*v)))^7 + .4*((sin(2*N*u)))
  Fz(u,v)+Thickness*n3(u,v)/R(u,v)
}

open3d(windowRect = c(50,50,562,562))
parametric3d(Vectorize(fx), Vectorize(fy), Vectorize(fz), 
             umin=0, umax=pi, vmin=0, vmax=2*pi, n=300, 
             color="gold", engine="rgl", smooth = TRUE)

# # -- 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 = "PseudoHopfTorus.gif",
  width = 512,
  height = 512,
  delay = 1/10
)
file.remove(Sys.glob("zzpic*.png"))