**Stereographic projection**

**Definition. **Denote the unit sphere in $\mathbb{R}^{n+1}$ by $\mathbb{S}^{n}=\left\{{x \in \mathbb{R}^{n+1}|(x,x)=1}\right\}$ and its north pole by $\mathbf{N}=e_{n+1}$.

The *stereographic projection* is a map $\sigma\colon \mathbb{S}^{n}\to\mathbb{R}^{n} \cup \left\{{\infty}\right\}$ from $\mathbf{N}$ to the plane

$E=\left\{{x \in \mathbb{R}^{n+1}|x_{n+1}=0}\right\} \cong \mathbb{R}^{n}$ through the equator:

\[

\sigma(X)=

\left\{

\begin{array}{ll}

\infty & \textrm{, } X = \mathbf{N}\\

l_{NX}\cap E & \textrm{, } X \neq \mathbf{N}\\

\end{array}

\right.

\]

Analytically $\sigma$ is given by

\[

\sigma(X)= \sigma\ \left(\begin{pmatrix}

x_1\\\vdots\\x_{n+1}

\end{pmatrix} \right)

= \frac{1}{1-x_{n+1}}\begin{pmatrix}

x_1\\\vdots\\x_{n+1}

\end{pmatrix}

\]

since

\[N+\lambda(X-N)=e_{n+1}+\lambda \begin{pmatrix}

x_1\\\vdots\\x_{n+1}

\end{pmatrix}

= \begin{pmatrix}

y_1\\\vdots\\y_{n}\\0

\end{pmatrix}

\Rightarrow \lambda=-\frac{1}{x_{n+1}-1}=\frac{1}{1-x_{n+1}}, X\neq N.

\]

**Sphere inversion**

**Definition. **The *sphere inversion* $i_{M,r}\colon \mathbb{R}^{n} \cup \left\{{\infty}\right\} \to \mathbb{R}^{n} \cup \left\{{\infty}\right\}$ in the sphere

$\mathbf{S}_{M,r}=\left\{{x \in \mathbb{R}^{n}|\left\|x-M\right\|=r}\right\}$ is given by the following conditions:

- $X’$ lies on the ray $\overrightarrow{MX}$,
- $\left\|M-X\right\|\left\|M-X’\right\|=r^2$,
- $M\longleftrightarrow\infty$.

**Properties of sphere inversions**

- $i_{M,r}$ is an involution,
- $\mathbf{S}_{M,r}$ is fixed,
- affine subspaces containing M are mapped onto themselves.

**Lemma.**

**Proof.**

Since

\[

\left\|M-A\right\|\left\|M-A’\right\|=r^2 = \left\|M-B\right\|\left\|M-B’\right\| \iff

\frac{\left\|M-A\right\|}{\left\|M-B\right\|}=\frac{\left\|M-B’\right\|}{\left\|M-A’\right\|}

\]

we get that $\Delta AMB$ and $\Delta B’MA’$ are similar. In particular:

\[

\measuredangle MA’B’=\measuredangle ABM.

\measuredangle MAB=\measuredangle A’B’M.

\]

$\Box$

**Corollary.** Circles through M are mapped to lines. In particular, the line is parallel to the tangent to the circle at M.

**Remark. **All lines are thought to contain the point $\infty$ and are considered as circles of infinte radius.

**Corollary. **The stereographic projection is the restriction of the sphere inversion $i_{M,r}$ with $M=e_{n+1}$ and $r=\sqrt{2}$.

**Proof.**

- The equator of $\mathbb{S}^{n}$ is fixed.
- $\mathbb{S}^{n}$ is mapped to a plane (Corollary) through the equator (1).
- Both inversion and stereographic projection preserve the rays.
- The intersection points of a ray with $\mathbb{S}^{n}$ and $\mathbb{R}^{n}$ are unique, hence $i_{M,r}\Big|_{\mathbb{S}^{n}}=\sigma$.
- $N=M\longrightarrow\infty$ by $i_{M,r}$.

$\Box$

**Theorem.** An inversion in a sphere:

- maps spheres/hyperplanes to spheres/hyperplanes,
- is a conformal map, i.e. it preserves angles.

**Proof.**

(i) Consider the following sketch:

Hence the angles $\alpha$ and $\beta$ satisfy

\[

(\pi-\alpha)+\beta+\frac{\pi}{2}=\pi \Rightarrow \alpha-\beta=\frac{\pi}{2}.

\]

(ii) Conformality follows from the following picture

$\Box$

**Remark. **In general, the center of a sphere is *not* mapped to the center of its image by an inversion.

**Corollary.**

- The stereographic projection maps spheres through the north pole to hyperplanes and spheres not through the north pole to spheres.
- All spheres/hyperplanes are images of spheres on $\mathbb{S}^{n}$ under $\sigma$ (because $i_{0,1}$ is an involution).
- $\sigma$ is conformal.

**Formula for the sphere inversion.**

Consider the second figure showing the general idea of a sphere inversion. Since $X’$ lies on the ray $XM$ we have $X’=M+\lambda(X-M), \lambda > 0$ and from the second requirement for the sphere inversion we obtain

\begin{align*}

&&\underbrace{\left\|X’-M\right\|}_{X’-M=\lambda(X-M)} \left\|X-M\right\|&=r^2\\

\Rightarrow&& \lambda \left\|X-M\right\| \left\|X-M\right\|&=r^2\\

\Rightarrow&& \lambda = \frac{r^2}{\left\|X-M\right\|^2}.

\end{align*}

Hence the sphere inversion is given by

\[

i_{M,r}=M+\frac{r^2}{\left\|X-M\right\|^2}(X-M).

\]