---
title: "Generalized circles and Möbius transformations"
author: "Stéphane Laurent"
date: '2019-03-12'
tags: R, maths, geometry
output:
  md_document:
    variant: markdown
    preserve_yaml: true
  html_document:
    highlight: kate
    keep_md: no
prettify: yes
linenums: yes
prettifycss: minimal
highlighter: pandoc-solarized
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(collapse = TRUE)
```

It is well known that a Möbius transformation maps a generalized circle to a 
generalized circle. 
In this blog post, not only we prove this fact, but we also provide the 
equation of the image generalized circle.


# Generalized circle

A *generalized circle* is a circle or a line. 
We shall see that a generalized circle $\mathcal{C}$ has equation
$$
Az\bar{z} + \bar{\gamma}z + \gamma\bar{z} = D 
$$
where $A$ and $D$ are real numbers, $\gamma \in \mathbb{C}$, and

- $A=0$, $D \geq 0$, $\gamma \neq 0$, when $\mathcal{C}$ is a line;

- $A \neq 0$ when $\mathcal{C}$ is a circle, 
and ${|\gamma|}^2 + AD > 0$.


## Circle equation

The circle with center $z_0$ and radius $R$ has equation $|z-z_0|=R$. 
Now, 
$$
|z-z_0|=R \ \ \iff \ \ {|z-z_0|}^2=R^2 
\ \ \iff (z-z_0)\overline{(z-z_0)} = R^2
$$
$$
\tag{1}\iff \ \ 
z \bar z - \bar{z_0}z - z_0\bar{z} = R^2 - {|z_0|}^2.
$$

Now, let the equation
$$
Az\bar{z} + \bar{\gamma}z + \gamma\bar{z} = D 
$$
with $A$, $D$ real numbers, $A \neq 0$, and $\gamma \in \mathbb{C}$.

It is equivalent to
$$
z\bar{z} + \frac{\bar{\gamma}}{A}z + \frac{\gamma}{A}\bar{z} 
 = \frac{D}{A}.
$$

Then it has the same form as $(1)$ with
$$
z_0 = - \frac{\gamma}{A}
$$
and 
$$
R^2 = {|z_0|}^2 + \frac{D}{A} > 0.
$$
Thus
$$
\dfrac{{|\gamma|}^2}{A^2} + \frac{D}{A} > 0, 
$$
that is ${|\gamma|}^2 + AD > 0$.


## Line equation

The general equation of a line is 
$$
\cos\theta \cdot x + \sin\theta \cdot y = d 
$$
where $\theta \in [0,2\pi)$ is the *direction* of the line, 
and $d \geq 0$ is its *offset*.

Letting $z = x+iy$, it is equivalent to 
$$
\bar{\gamma}z + \gamma\bar{z} = D 
$$
with $\cos\theta = \frac{\Re(\gamma)}{|\gamma|}$ and 
$\sin\theta = \frac{\Im(\gamma)}{|\gamma|}$, 
hence $\theta = \textrm{arg}(\gamma)$, and $d = \frac{D}{2|\gamma|}$. 
Indeed, 
$$
\bar{\gamma}z + \gamma\bar{z} = D  
\ \ \iff \ \ 
\frac{\bar{\gamma}}{|\gamma|} z + \frac{\gamma}{|\gamma|}\bar{z} 
 = \frac{D}{|\gamma|} 
$$
$$
\ \ \iff
\left( \frac{\bar{\gamma} + \gamma}{|\gamma|} \right) x 
 + \left( \frac{\bar{\gamma} - \gamma}{|\gamma|} \right) i y 
 = \frac{D}{|\gamma|} 
$$
$$
\ \ \iff
\left( \frac{\Re(\gamma)}{|\gamma|} \right) x 
 + \left( \frac{\Im(\gamma)}{|\gamma|} \right) y 
 = \frac{D}{2|\gamma|}. 
$$

Conversely, if the equation $\cos\theta \cdot x + \sin\theta \cdot y = d$ is 
given, then one can take $\gamma = \cos\theta + i \sin\theta$, and $D = 2d$.


# Möbius transformations

A *Möbius transformation* is a function 
$M \colon \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ of the form
$$
M(z) = \frac{az+b}{cz+d},
$$
where $a$, $b$, $c$, $d$ are complex numbers satisfying $ad - bc \neq 0$, 
and we agree with 

- $M(\infty) = \infty$ if $c = 0$ (hence $d \neq 0$);

- otherwise, $M(\infty) = a/c$ and $M(-d/c) = \infty$. 

We say that the matrix
$$
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
$$
is *associated* with $M$. 
Note that $A$ is invertible because of the condition $ad - bc \neq 0$. 

Also note that $M(z) \equiv z$ means that the identity matrix is associated 
with $M$.


## Composition of Möbius transformations

Let $M_1$ and $M_2$ be two Möbius transformations, $A_1$ a matrix associated 
with $M_1$ and $A_2$ a matrix associated with $M_2$. 
Then $M := M_1 \circ M_2$ is a Möbius transformation and $A_1A_2$ is a matrix 
associated with $M$.  

As a consequence, any Möbius transformation $M$ is invertible and its inverse 
is the Möbius transformation associated to the matrix 
$\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ if 
$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is a matrix associated with $M$. 

## Image of a generalized circle 

We shall see that a Möbius transformations maps a generalized circle to a 
generalized circle. 
We denote by $M$ the Möbius transformation associated to 
$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

### Image of a circle

Let $\mathcal{C}$ be the circle of center $z_0$ and radius $R$. 
We have seen that $\mathcal{C}$ has equation
$$
z \bar z - \bar{z_0}z - z_0\bar{z} = R^2 - {|z_0|}^2.
$$
We set $H = R^2 - {|z_0|}^2$. 
Inserting $z = M^{-1}(w) = \frac{dw - b}{a - cw}$, we get
$$
(dw - b)\overline{(dw - b)} 
 - \bar{z_0}(dw - b)\overline{(a - cw)}
 - z_0 \overline{(dw - b)}(a-cw) \\
 = H (a - cw)\overline{(a - cw)}, 
$$
that is, 
$$
\begin{align}
& \Bigl( d\bar{d} + d\bar{c}\bar{z_0} + \bar{d}c z_0 
  - H c\bar{c} \Bigr) w \bar{w} \\ 
\qquad &  - \Bigl( d\bar{b} + d\bar{a}\bar{z_0} + c\bar{b}z_0 
    - H c\bar{a} \Bigr) w \\ 
\qquad &  - \Bigl( b\bar{d} + b\bar{c}\bar{z_0} + \bar{d}a z_0 
      - H a\bar{c} \Bigr) \bar{w} \\
 = & \; H a\bar{a} - b\bar{b} - b\bar{a}\bar{z_0} - \bar{b}a z_0.
\end{align}
$$
Now, 
$$
\begin{align}
d\bar{d} + d\bar{c}\bar{z_0} + \bar{d} c z_0 - H c\bar{c} 
 & = (d + c z_0)\overline{(d + c z_0)} - R^2 c\bar{c} \\
 & = {|d+c z_0|}^2 - R^2{|c|}^2 \\ & =: A,
\end{align}
$$
$$
\begin{align}
H a\bar{c} - b\bar{d} - b\bar{c}\bar{z_0} - \bar{d}a z_0   
 & = R^2 a\bar{c} - (a z_0 + b)\overline{(c z_0 + d)} \\ & =: \gamma,
\end{align}
$$
$$
\begin{align}
H a\bar{a} - b\bar{b} - b\bar{a}\bar{z_0} - \bar{b}a z_0   
 & = R^2 a\bar{a} - (b + a z_0)\overline{(b + a z_0)} \\ 
 & = R^2{|a|}^2 - {|b + a z_0|}^2 \\ & =: D.
\end{align}
$$
The equation is 
$$
A w\bar{w} + \bar\gamma w + \gamma\bar{w} = D.
$$

If $\boxed{A \neq 0}$, that is $|d+c z_0| \neq R|c|$, this is the equation of the 
circle with center $-\gamma/A$ and radius 
$\sqrt{\frac{{|\gamma|}^2}{A^2} + \frac{D}{A}}$.

Let's check with R. 
To do so, we use the `circumcircle` function below, which returns the center 
and the radius of the circle passing by `z1`, `z2`, `z3`.

```{r}
circumcircle <- function(z1,z2,z3){
  x1 <- Re(z1); y1 <- Im(z1); p1 <- c(x1,y1)
  x2 <- Re(z2); y2 <- Im(z2); p2 <- c(x2,y2)
  x3 <- Re(z3); y3 <- Im(z3); p3 <- c(x3,y3)
  a <- det(cbind(rbind(p1,p2,p3),1))
  q1 <- c(crossprod(p1))
  q2 <- c(crossprod(p2))
  q3 <- c(crossprod(p3))
  q <- c(q1,q2,q3)
  x <- c(x1,x2,x3)
  y <- c(y1,y2,y3)
  Dx <- det(cbind(q,y,1))
  Dy <- -det(cbind(q,x,1))
  c <- det(cbind(q,x,y))
  center <- c(Dx,Dy)/a/2
  r <- sqrt(c(crossprod(center-p1)))
  list(center = center[1] + 1i*center[2], radius = r)
}
```

Let's take an example:

```{r}
Mobius <- function(a, b, c, d){
  function(z){
    (a*z+b) / (c*z+d)
  }
}
a <- 4; b <- 1i; c <- 2i; d <- 1+3i
M <- Mobius(a,b,c,d)
# define circle C
z0 <- 2+1i; R <- 2
# three points on M(C)
z1 <- M(z0+R); z2 <- M(z0-R); z3 <- M(z0 + 1i*R)
# circumcircle of z1, z2, z3
cc <- circumcircle(z1, z2, z3)
# calculation of A, gamma, D
A <- Mod(d+c*z0)^2 - R^2*Mod(c)^2
gamma <- R^2*a*Conj(c) - (a*z0 + b) * Conj(c*z0 + d)
D <- R^2*Mod(a)^2 - Mod(b + a*z0)^2
# center
-gamma/A
cc$center
# radius 
sqrt(Mod(gamma)^2/A^2 + D/A)
cc$radius
```

We have the following alternative expressions of the center and the radius:
$$
w_0 = M\left(z_0 - R^2\overline{\left(\frac{d}{c}+z_0\right)} \right), \\
R' = \bigl| w_0 - M(z_0+R) \bigr|.
$$

```{r}
z <- z0 - R^2/Conj(d/c+z0) 
( w0 <- M(z) ) # center
Mod(w0 - M(z0+R)) # radius
```

This code also works for $c = 0$:

```{r}
MC <- function(a, b, c, d, z0, R){
  M <- Mobius(a,b,c,d)
  z <- z0 - R^2/Conj(d/c+z0) 
  w0 <- M(z) 
  list(center = w0, radius = Mod(w0 - M(z0+R)))
}
c <- 0
M <- Mobius(a,b,c,d)
z1 <- M(z0+R); z2 <- M(z0-R); z3 <- M(z0 + 1i*R)
circumcircle(z1, z2, z3)
MC(a, b, c, d, z0, R)
```


Now, consider the case when $\boxed{A=0}$, that is $|d/c+z_0| = R$. 
The equation is 
$$
\bar\gamma w + \gamma\bar{w} = D.
$$
And this is the equation of a line, whose direction and offset are given by 
the formulas we have previously seen.


### Image of a line

Now, let $\mathcal{L}$ be the line having equation
$$
\bar{\gamma_0} z + \gamma_0\bar{z} = D_0.
$$
Inserting $z = M^{-1}(w) = \frac{dw - b}{a - cw}$, we get the following 
equation for $M(\mathcal{L})$:
$$
\bar{\gamma_0} (dw - b)\overline{(a-cw)} 
 + \gamma_0 \overline{(dw - b)}(a-cw) 
 = D_0(a-cw)\overline{(a-cw)}.
$$
It is equivalent to 
$$
\begin{align*}
& -\Bigl( \bar{\gamma_0}d\bar{c} + \gamma_0 \bar{d}c 
    + D_0 c\bar{c} \Bigr) w\bar{w}  \\
\qquad & 
 + \Bigl( \bar{\gamma_0}d\bar{a} + \gamma_0 \bar{b}c + D_0 c\bar{a} \Bigr) w \\ 
\qquad & 
 + \Bigl( \bar{\gamma_0}b\bar{c} + \gamma_0 \bar{d}a + D_0 \bar{c}a \Bigr) \bar{w} \\
 & \qquad = \; D_0 a \bar{a} + \bar{\gamma_0}b\bar{a} + \gamma_0\bar{b}a, 
\end{align*}
$$
that is 
$$
A w\bar{w} + \bar{\gamma}w + \gamma\bar{w} = D
$$
with 
$$
A = -2\;\Re(\gamma_0 c \bar{d}) - D_0 {|c|}^2, \\ 
\gamma = \bar{\gamma_0}b\bar{c} + \gamma_0 \bar{d}a + D_0 \bar{c}a, \\ 
D = D_0{|a|}^2 + 2\;\Re(\gamma_0 \bar{b}a).
$$

Again, there are two cases. We get a circle if $A \neq 0$, and a line 
otherwise. 

Let's check an example with $A=0$. Our current value of $c$ in R is $0$, so we 
have $A=0$. 

```{r}
# define the line L
theta <- 1
offset <- 2
# three points on M(L)
x <- c(0,1,2)
y <- (offset - cos(theta)*x) / sin(theta)
z1 <- M(x[1] + 1i*y[1]) 
z2 <- M(x[2] + 1i*y[2]) 
z3 <- M(x[3] + 1i*y[3])
# these points are aligned:
Im((z2-z1)/(z3-z1)) # aligned <=> 0
```

Now let's check an example with $A \neq 0$:

```{r}
# Mobius transformation
c <- 6 # don't take c = 0, otherwise A=0
M <- Mobius(a,b,c,d)
# three points on M(L)
z1 <- M(x[1] + 1i*y[1]) 
z2 <- M(x[2] + 1i*y[2]) 
z3 <- M(x[3] + 1i*y[3])
# calculation of A, gamma, D
gamma0 <- cos(theta) + 1i * sin(theta)
D0 <- 2 * offset
A <- -2 * Re(gamma0*c*Conj(d)) - D0*Mod(c)^2 
gamma <- Conj(gamma0)*b*Conj(c) + gamma0*Conj(d)*a + D0*Conj(c)*a
D <- D0 * Mod(a)^2 + 2*Re(gamma0*Conj(b)*a)
# circumcircle
cc <- circumcircle(z1, z2, z3)
# center
-gamma/A
cc$center
# radius 
sqrt(Mod(gamma)^2/A^2 + D/A)
cc$radius
```

# R code 

```{r}
Mobius <- function(a, b, c, d){
  function(z){
    (a*z+b) / (c*z+d)
  }
}

Mod2 <- function(z){
  Re(z)^2 + Im(z)^2
}

MobiusImageOfCircle <- function(a, b, c, d, z0, R){
  gamma <- R*R*a*Conj(c) - (a*z0 + b) * Conj(c*z0 + d)
  D <- R*R*Mod2(a) - Mod2(b + a*z0)
  x1 <- Mod2(d+c*z0)
  x2 <- R*R*Mod2(c)
  if(x1 != x2){
    A <- x1 - x2
    out <- list(center = -gamma/A, radius = sqrt(Mod2(gamma)/A/A + D/A))
    attr(out, "type") <- "circle"
  }else{
    out <- list(theta = Arg(gamma) %% (2*pi), offset = D/2/Mod(gamma))
    attr(out, "type") <- "line"
  }
  out
}

MobiusImageOfLine <- function(a, b, c, d, theta, offset){
  gamma0 <- cos(theta) + 1i*sin(theta)
  D0 <- 2 * offset
  A <- -2 * Re(gamma0*c*Conj(d)) - D0*Mod2(c) 
  gamma <- Conj(gamma0)*b*Conj(c) + gamma0*Conj(d)*a + D0*Conj(c)*a
  D <- D0*Mod2(a) + 2*Re(gamma0*Conj(b)*a)
  if(A != 0){
    out <- list(center = -gamma/A, radius = sqrt(Mod2(gamma)/A/A + D/A))
    attr(out, "type") <- "circle"
  }else{
    out <- list(theta = Arg(gamma) %% (2*pi), offset = D/2/Mod(gamma))
    attr(out, "type") <- "line"
  }
  out
}

MobiusImage <- function(a, b, c, d, gcircle){
  if(attr(gcircle, "type") == "circle"){
    MobiusImageOfCircle(a, b, c, d, gcircle$center, gcircle$radius)
  }else{
    MobiusImageOfLine(a, b, c, d, gcircle$theta, gcircle$offset)
  }
}

# plot ####
drawLine <- function(line, ...){
  theta <- line$theta; offset <- line$offset
  if(sin(theta) != 0){
    abline(a = offset/sin(theta), b = -1/tan(theta), ...)
  }else{
    abline(v = offset/cos(theta), ...)
  }
}

drawCircle <- function(circle, ...){
  plotrix::draw.circle(Re(circle$center), Im(circle$center), 
                       circle$radius, ...)
}

drawGCircle <- function(gcircle, color = "black", ...){
  if(attr(gcircle, "type") == "circle"){
    drawCircle(gcircle, border = color, ...)
  }else{
    drawLine(gcircle, col = color, ...)
  }
}
```


# References 

- D. A. Brannan, M. F. Esplen, J. J. Gray. 
*Geometry*, second edition. 
Cambridge University Press, 2012.

- D. Mumford, C. Series, D. Wright. 
*Indra's Pearls: The Vision of Felix Klein*. 
Cambridge University Press, 2002.