# On a Möbius transformation

Posted on June 21, 2022 by Stéphane Laurent

Consider a complex number $$\gamma$$ such that $$|\gamma| < 1$$ and the following matrix: $M = \begin{pmatrix} i & \gamma \\ \bar\gamma & -i \end{pmatrix}.$

Then the Möbius transformation associated to this matrix is nice. Why? Because:

• it maps the unit disk to itself;

• it is of order $$2$$;

• its fractional powers have a closed form.

For these reasons, I often use this Möbius transformation in my shaders.

Let us derive the fractional powers of $$M$$. We set $$h = \sqrt{1-|\gamma|^2}$$.

The eigenvalues of $$M$$ are \begin{align} \lambda_1 & = -ih \\ \lambda_2 & = ih = \bar{\lambda_1} \end{align} with corresponding eigen vectors \begin{align} v_1 & = \begin{pmatrix} (1-h)\dfrac{i\gamma}{|\gamma|^2} \\ 1 \end{pmatrix} \\ v_2 & = \begin{pmatrix} (1+h)\dfrac{i\gamma}{|\gamma|^2} \\ 1 \end{pmatrix}. \end{align} Let $$P = \begin{pmatrix} v_1 & v_2 \end{pmatrix}$$. Then $\frac{1}{\det(P)} = \frac{i\bar\gamma}{2h}$ and for any complex numbers $$d_1$$ and $$d_2$$, $P \begin{pmatrix} d_1 & 0 \\ 0 & d_2 \end{pmatrix} P^{-1} = \frac{1}{2h} \begin{pmatrix} d_2(1+h)-d_1(1-h) & i(d_1-d_2)\gamma \\ i(d_1-d_2)\bar\gamma & d_1(1+h)-d_2(1-h) \end{pmatrix}.$

In particular, $$M^t$$ is given by $\begin{pmatrix} a & b \\ \bar b & \bar a \end{pmatrix}$ where \begin{align} a & = \Re(d_1) - i \dfrac{\Im(d_1)}{h}, \\ b & = \gamma \dfrac{\Im(d_2)}{h}, \\ d_1 & = \bar{d_2}, \\ d_2 & = h^t \exp\left(i\dfrac{t\pi}{2}\right). \end{align}

M_power_t <- function(gamma, t){
h <- sqrt(1-Mod(gamma)^2)
d2 <- h^t * (cos(t*pi/2) + 1i*sin(t*pi/2))
d1 <- Conj(d2)
a <- Re(d1) - 1i*Im(d1)/h
b <- gamma * Im(d2)/h
c <- Conj(b)
d <- Conj(a)
c(a = a, b = b, c = c, d = d)
}

Let’s apply this Möbius transformation now. Here is a visualization of the Dedekind eta function, a complex function availale in the jacobi package:

# background color
bkgcol <- rgb(21, 25, 30, maxColorValue = 255)

modulo <- function(a, p) {
a - p * ifelse(a > 0, floor(a/p), ceiling(a/p))
}

colormap <- function(z){
if(is.na(z)){
return(bkgcol)
}
if(is.infinite(z) || is.nan(z)){
return("#000000")
}
x <- Re(z)
y <- Im(z)
r <- modulo(Mod(z), 1)
g <- 2 * abs(modulo(atan2(y, x), 0.5))
b <- abs(modulo(x*y, 1))
if(is.nan(b)){
return("#000000")
}
rgb(
8 * (1 - cos(r-0.5)),
8 * (1 - cos(g-0.5)),
8 * (1 - cos(b-0.5)),
maxColorValue = 1
)
}

library(jacobi)
f <- Vectorize(function(x, y){
q <- x + 1i*y
if(Mod(q) > 0.9999 || (Im(q) == 0 && Re(q) <= 0)){
return(bkgcol)
}
tau <- -1i * log(q) / pi
z <- eta(tau)
colormap(z)
})

x <- y <- seq(-1, 1, len = 2000)
image <- outer(x, y, f)

opar <- par(mar = c(0,0,0,0), bg = bkgcol)
plot(
c(-100, 100), c(-100, 100), type = "n",
xlab = "", ylab = "", axes = FALSE, asp = 1
)
rasterImage(image, -100, -100, 100, 100)
par(opar)

Here is how to apply the Möbius transformation for one value of the power $$t$$:

Mobius <- M_power_t(gamma = 0.7 - 0.3i, t = ...)
a <- Mobius["a"]
b <- Mobius["b"]
c <- Mobius["c"]
d <- Mobius["d"];
f <- Vectorize(function(x, y){
q0 <- x + 1i*y
q <- (a*q0 + b) / (c*q0 + d)
if(Mod(q) > 0.9999 || (Im(q) == 0 && Re(q) <= 0)){
return(bkgcol)
}
tau <- -1i * log(q) / pi
z <- eta(tau)
colormap(z)
})

x <- y <- seq(-1, 1, len = 2000)
image <- outer(x, y, f)

Then it suffices to run this code for $$t$$ varying from $$0$$ to $$2$$, and to save the image for each value of $$t$$. But this would be very slow. Actually I implemented the image generation with Rcpp. Here is the result:

My Rcpp code is available in the Github version of the jacobi package. The R code which generates an image for one value of $$t$$ is:

x <- seq(-1, 1, len = 2000L)
gamma <- 0.7 - 0.3i
t <- ...
image <- jacobi:::Image_eta(x, gamma, t)
opar <- par(mar = c(0,0,0,0), bg = bkgcol)
plot(
c(-100, 100), c(-100, 100), type = "n",
xlab = "", ylab = "", axes = FALSE, asp = 1
)
rasterImage(image, -100, -100, 100, 100)
par(opar)

You can also play with jacobi:::Image_E4 and jacobi:::Image_E6, which respectively generate a visualization of the Eisenstein series of weight $$4$$ and a visualization of the Eisenstein series of weight $$6$$.