disk2square function

Disk to square

Conformal map from the unit disk to the square $[-1,1] \times [-1,1]$. The function is vectorized.

disk2square(z)

Arguments

• z: a complex number in the unit disk

Returns

A complex number in the square $[-1,1] \times [-1,1]$.

Examples

n <- 70L
r <- seq(0, 1, length.out = n)
theta <- seq(0, 2*pi, length.out = n+1L)[-1L]
Grid <- transform(
  expand.grid(R = r, Theta = theta),
  Z = R*exp(1i*Theta)
)
s <- vapply(Grid$Z, disk2square, complex(1L))
plot(Re(s), Im(s), pch = ".", asp = 1, cex = 2)
#
# a more insightful plot ####
r_ <- seq(0, 1, length.out = 10L)
theta_ <- seq(0, 2*pi, length.out = 33)[-1L]
plot(
  NULL, xlim = c(-1, 1), ylim = c(-1, 1), asp = 1, xlab = "x", ylab = "y"
)
for(r in r_) {
  theta <- sort(
    c(seq(0, 2, length.out = 200L), c(1/4, 3/4, 5/4, 7/4))
  )
  z <- r*(cospi(theta) + 1i*sinpi(theta))
  s <- vapply(z, disk2square, complex(1L))
  lines(Re(s), Im(s), col = "blue", lwd = 2)
}
for(theta in theta_) {
  r <- seq(0, 1, length.out = 30L)
  z <- r*exp(1i*theta)
  s <- vapply(z, disk2square, complex(1L))
  lines(Re(s), Im(s), col = "green", lwd = 2)
}