# Introduction to Groups and Geometries

## Section1.3Stereographic projection

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

Let $$S^1$$ denote the unit circle in the $$x,y$$-plane:
$$S^1 = \{(x,y)\in \R^2 \colon x^2+y^2=1\}\,.\tag{1.3.1}$$
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 $$s(x,y)=\frac{x}{1-y}\text{.}$$

#### Checkpoint1.3.2.

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
$$\extR=\R\cup \{\infty\}\tag{1.3.2}$$
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
$$s(x,y) = \begin{cases} \frac{x}{1-y} \amp y\neq 1\\ \infty \amp y=1. \end{cases}\tag{1.3.3}$$

### 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 $$\R^3\text{:}$$
$$S^2 = \{(a,b,c)\in \R^3\colon a^2+b^2+c^2=1\}\,.\tag{1.3.4}$$
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 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
$$\extC=\C\cup \{\infty\}\tag{1.3.5}$$
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
$$s(a,b,c) = \begin{cases} \frac{a+ib}{1-c} \amp c\neq 1\\ \infty \amp c=1. \end{cases}\tag{1.3.6}$$

### Subsection1.3.3Conjugate 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 $$f = \mu^{-1}\circ g\circ \mu\text{.}$$ Exercise Group 1.3.4.3–6 demonstrate examples of this definition for which $$\mu$$ is stereographic projection.

#### Checkpoint1.3.4.

Show that $$f=\mu^{-1}\circ g \circ \mu$$ if and only if $$\mu\circ f = g\circ \mu\text{.}$$

### Exercises1.3.4Exercises

#### 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) = \begin{cases} \left(\frac{2\re(z)}{|z|^2+1},\frac{2\im(z)}{|z|^2+1}, \frac{|z|^2-1}{|z|^2+1}\right)\amp \text{ if } z\neq \infty\\ (0,0,1) \amp \text{ if } z=\infty \end{cases} \end{equation*}
$$s^{-1}(3+i) = (6/11,2/11,9/11)$$

#### Transformations that are conjugate via stereographic projection.

Suggestion: Use Checkpoint 1.3.4 for the exercises that follow.
##### 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 to 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 $$S^2$$ with $$|c|\neq 1\text{.}$$ Conversely, suppose that $$zw^\ast=-1$$ for some $$z,w\in \C\text{.}$$ Show that $$s^{-1}(z)=-s^{-1}(w)\text{.}$$