## Section1.3Stereographic projection

### Subsection1.3.1Stereographic projection $S^1\to \extR$

Let $S^1$ denote the unit circle in the $x,y$-plane.

\begin{equation} S^1 = \{(x,y)\colon x^2+y^2=1\}\label{s1defn}\tag{1.3.1} \end{equation}

Let $N=(0,1)$ denote the "north pole" (that is, the point at the "top" of the unit circle). Given a point $P=(x,y)\neq N$ on the unit circle, let $s(P)$ denote the intersection of the line $\overline{NP}$ with the $x$-axis. See Figure 1.3.1. The map $s\colon S^1\setminus \{N\}\to \R$ given by this rule is called stereographic projection. Using similar triangles, it is easy to see that $s(x,y)=\frac{x}{1-y}\text{.}$

Draw the relevant similar triangles and verify the formula $s(x,y) = \frac{x}{1-y}\text{.}$

We extend stereographic projection to the entire unit circle as follows. We call the set

\begin{equation} \extR=\R\cup \{\infty\}\label{extendedrealsdefn}\tag{1.3.2} \end{equation}

the extended real numbers, where "$\infty$" is an element that is not a real number. Now we define stereographic projection $s\colon S^1 \to \extR$ by

\begin{equation} s(x,y) = \left\{ \begin{array}{cc} \frac{x}{1-y} \amp y\neq 1\\ \infty \amp y=1 \end{array} \right..\label{stereoproj1defn}\tag{1.3.3} \end{equation}

### Subsection1.3.2Stereographic projection $S^2\to \extC$

The definitions in the previous subsection extend naturally to higher dimensions. Here are the details for the main case of interest.

Let $S^2$ denote the unit sphere in $\R^3\text{.}$

\begin{equation} S^2 = \{(a,b,c)\in \R^3\colon a^2+b^2+c^2=1\}\label{s2defn}\tag{1.3.4} \end{equation}

Let $N=(0,0,1)$ denote the "north pole" (that is, the point at the "top" of the sphere, where the positive $z$-axis is "up"). Given a point $P=(a,b,c)\neq N$ on the unit sphere, let $s(P)$ denote the intersection of the line $\overline{NP}$ with the $x,y$-plane, which we identify with the complex plane $\C\text{.}$ See See Figure 1.3.3. The map $s\colon S^2\setminus \{N\}\to \C$ given by this rule is called stereographic projection. Using similar triangles, it is easy to see that $s(a,b,c)=\frac{a+ib}{1-c}\text{.}$

We extend stereographic projection to the entire unit sphere as follows. We call the set

\begin{equation} \extC=\C\cup \{\infty\}\label{extendedcomplexsdefn}\tag{1.3.5} \end{equation}

the extended complex numbers, where "$\infty$" is an element that is not a complex number. Now we define stereographic projection $s\colon S^2 \to \extC$ by

\begin{equation} s(a,b,c) = \left\{ \begin{array}{cc} \frac{a+ib}{1-c} \amp c\neq 1\\ \infty \amp c=1 \end{array} \right..\label{stereoprojdefn}\tag{1.3.6} \end{equation}

### Exercises1.3.3Exercises

###### Formulas for inverse stereographic projection.

It is intuitively clear that stereographic projection is a bijection. Make this rigorous by finding a formula for the inverse.

###### 1.

For $s\colon S^1\to \extR\text{,}$ find a formula for $s^{-1}\colon \extR\to S^1\text{.}$ Find $s^{-1}(3)\text{.}$

\begin{equation*} s^{-1}(r) = \begin{cases} \left(\frac{2r}{r^2+1},\frac{r^2-1}{r^2+1}\right) \amp \text{ if } r\neq \infty\\ (0,1)\amp \text{ if } r=\infty \end{cases} \end{equation*}
$s^{-1}(3) = (3/5,4/5)$
###### 2.

For $s\colon S^2\to \extC\text{,}$ find a formula for $s^{-1}\colon \extC\to S^2\text{.}$ Find $s^{-1}(3+i)\text{.}$

\begin{equation*} s^{-1}(z) = \left(\frac{2\re(z)}{|z|^2+1},\frac{2\im(z)}{|z|^2+1}, \frac{|z|^2-1}{|z|^2+1}\right) \end{equation*}
$s^{-1}(3+i) = (6/11,2/11,9/11)$
###### Conjugate transformations.
Let $\mu \colon X\to Y$ be a bijective map. We say that maps and $f\colon X\to X$ and $g\colon Y\to Y$ are conjugate transformations (with respect to the bijection $\mu$) if $f = \mu^{-1}\circ g\circ \mu\text{.}$
###### 3.

Show that the maps $S^1\to S^1$ given by $(x,y)\to (x,-y)$ and $\extR\to \extR$ given by $x\to 1/x$ are conjugate transformations with respect to stereographic projection.

###### 4.

Show that the map $R_{Z,\theta}\colon S^2\to S^2$ given by $(a,b,c)\to (a\cos\theta-b\sin\theta,a\sin\theta+b\cos\theta,c)$ (a rotation about the $z$-axis by angle $\theta$) and the map $T_{Z,\theta}\colon \extC\to \extC$ given by $z\to e^{i\theta}z$ are conjugate transformations with respect stereographic projection.

###### 5.

Show that the map $R_{X,\pi}\colon S^2\to S^2$ given by $(a,b,c)\to (a,-b,-c)$ (rotation about the $x$-axis by $\pi$ radians) and the map $T_{X,\pi}\colon \extC\to \extC$ given by $z\to 1/z$ are conjugate transformations with respect to stereographic projection.

###### 6.

Show that the map $R_{X,\pi/2}\colon S^2\to S^2$ given by $(a,b,c)\to (a,-c,b)$ (rotation about the $x$-axis by $\pi/2$ radians) and the map $T_{X,\pi/2}\colon \extC\to \extC$ given by $z\to \frac{z+i}{iz+1}$ are conjugate transformations with respect to stereographic projection.

###### 7.Projections of endpoints of diameters.

Show that $s(a,b,c)(s(-a,-b,-c))^\ast=-1$ for any point $(a,b,c)$ in $S^2\text{.}$ Conversely, suppose that $zw^\ast=-1\text{.}$ Show that $s^{-1}(z)=-s^{-1}(w)\text{.}$