mobius
In lorentz: The Lorentz Transform in Relativistic Physics

knitr::opts_chunk$set(echo = TRUE)

Möbius addition

We define Möbius addition $\oplus$ on the unit disk $\mathbb{D} = \left\lbrace z\in\mathbb{C}\colon\left|z\right|<1\right\rbrace$ as follows:

\begin{equation} a\oplus z = \frac{a+z}{1+\overline{a}z} \end{equation}

(naturally we define $a\ominus z=a\oplus(-z)$ and it convenient to write $\ominus z=0\ominus z=-z$).

The $\oplus$ operator is neither commutative nor associative ($a\oplus b\neq b\oplus a$ and $a\oplus(b\oplus c)\neq (a\oplus b)\oplus c$ in general). However, it does posess the automorphic inverse property

\begin{equation} \ominus(a\oplus b) = \ominus a\ominus b \end{equation}

and the left cancellation law

\begin{equation} \ominus a\oplus (a\oplus z) = z \end{equation}

proof: [ \ominus(a\oplus b) = -(a\oplus b)=-\frac{a+b}{1+\overline{a}b}= =\frac{(-a)+(-b)}{1+\overline{a}b} =\frac{(-a)+(-b)}{1+\overline{(-a)}(-b)} =(-a)\oplus(-b) =\ominus a\ominus b ]

[ \ominus a\oplus (a\oplus z) = \ominus a\oplus\frac{a+z}{1+\overline{a}z} =\frac{-a + \frac{a+z}{1+\overline{a}z}}{1+\overline{(-a)}\frac{a+z}{1+\overline{a}z}} =\frac{-a\overline{a}z +z}{1-\overline{a}a} =\frac{z(1-a\overline{a})}{1-\overline{a}a} =z, ] the last equality following from the fact that $a\in\mathbb{D}$.

`%+%` <- function(x,y){(x+y)/(1+Conj(x)*y)}
`%-%` <- function(x,y){(x-y)/(1-Conj(x)*y)}

square <- function(a,b,len=100){ # a is lower left point, b is upper right
    a <- a+0i
    b <- b+0i
    c(
    seq(from=Re(a) + 1i*Im(a), to=Re(b) + 1i*Im(a),len=len),
    seq(from=Re(b) + 1i*Im(a), to=Re(b) + 1i*Im(b),len=len),
    seq(from=Re(b) + 1i*Im(b), to=Re(a) + 1i*Im(b),len=len),
    seq(from=Re(a) + 1i*Im(b), to=Re(a) + 1i*Im(a),len=len)
    )
}

gyrosetup <- function(...){
    par(pty="s")
    plot(NA,xlim=c(-1,1),ylim=c(-1,1),asp=1,axes=FALSE,xlab="",ylab="")
theta <- seq(from=0,to=2*pi,len=400)
points(sin(theta),cos(theta),type="l",col="gray",...)
segments(x0=-1, x1= +1,y0=0,col='gray')
segments(x0=0,y0=-1,y1=+1,col='gray')
text(0.9,0.06,"Re")
text(0.1,0.9,"Im")
}

polyz <- function(z,...){polygon(x=Re(z),y=Im(z),...)}

gyrosetup()

S <- square(-1-1i,1+1i)/6
points(S,type="l")

zs <- c(
    +0.2 + 0.38i,
    -0.2 + 0.58i,
    +0.2 - 0.38i,
    -0.3 - 0.38i
)*1.4

for(z in zs){
points(Re(z),Im(z))
polyz(S %+% z,col=rgb(1,0,0,alpha=0.3))
polyz(z %+% S,col=rgb(0,0,1,alpha=0.3))
}