Riemann sphere

The Riemann sphere is the extended complex plane $\hat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}$, visualized as a sphere via stereographic projection.

Image: Stereographic projection from the north pole onto a plane, CC BY-SA 4.0

Stereographic projection

Place a unit sphere on the complex plane, with the south pole at the origin. Draw a line from the north pole through any point on the sphere — except the north pole itself — and extend it to the complex plane. The intersection with the plane is the image of that sphere point.

This gives a bijection between the sphere minus the north pole and $\mathbb{C}$. The north pole itself corresponds to the point at infinity, denoted $\infty$.

Why this is useful

• Removes the “point at infinity” oddity. On $\hat{\mathbb{C}}$, $\infty$ is just another point — geometrically equivalent to any finite point. Functions like $f(z)=1/z$ swap $0$ and $\infty$ but are otherwise unremarkable on the sphere.
• Unifies “circles and lines.” Lines in $\mathbb{C}$ pass through $\infty$ on the sphere; from the sphere viewpoint, every “line” is actually a circle that happens to pass through the north pole. So the family “circles and lines in $\mathbb{C}$” = “circles on the sphere.”
• Compact. The Riemann sphere is compact (closed and bounded), unlike $\mathbb{C}$. Many theorems become cleaner on a compact space.

Möbius transformations as rotations of the sphere

Every Möbius transformation corresponds to a smooth, orientation-preserving conformal map of the Riemann sphere. The rotations of the sphere — the rigid orientation-preserving motions that fix the center — form a subgroup of the Möbius transformations. They’re parameterized by matrices

$\left(\begin{array}{cc}a& -\bar{b}\\ b& \bar{a}\end{array}\right),|a{|}^2+|b{|}^2=1,$

acting via $f(z)=(az-\bar{b})/(bz+\bar{a})$. These matrices form the group $\mathrm{SU}(2)$.

The correspondence is two-to-one: the matrices $M$ and $-M$ give the same Möbius transformation (overall sign cancels in the formula), but they’re different elements of $\mathrm{SU}(2)$. So the rotation group of the sphere is

$\mathrm{SO}(3)=\mathrm{SU}(2)/\{\pm I\},$

with $\mathrm{SU}(2)$ a double cover of $\mathrm{SO}(3)$. This double cover is the same one that shows up in quantum mechanics as the spin-$1/2$ structure of electrons — the math is identical because the sphere acts as the configuration space of spin orientations.

In particular, the map $1/z$ swaps $0$ and $\infty$. On the Riemann sphere this is a $180°$ rotation of the sphere, exchanging south and north poles — perfectly symmetric, with no special role for either point.

Singularities at $\infty$

A function $f(z)$ has a singularity at $\infty$ if and only if $g(w)=f(1/w)$ has a singularity at $w=0$. The classification (removable, pole of order $m$, essential) transfers:

• $f(z)=z^n$ ($n>0$): pole of order $n$ at $\infty$ (since $1/w^n$ has a pole of order $n$ at $w=0$).
• $f(z)=1/z^n$: zero of order $n$ at $\infty$ (since $w^n\to 0$).
• $f(z)=e^z$: essential singularity at $\infty$ (since $e^{1/w}$ has an essential singularity at $0$).

A function is rational iff its only singularities on the Riemann sphere (including at $\infty$) are poles. Equivalently, rational functions are exactly the meromorphic functions on the Riemann sphere.

In context

The Riemann sphere is the natural setting for:

• Möbius transformations: act as conformal automorphisms of the sphere.
• Rational functions: their natural domain — they’re “everywhere meromorphic” on the sphere.
• Algebraic geometry: $\hat{\mathbb{C}}={\mathbb{CP}}^1$ is the simplest complex projective space.
• Conformal mapping: Riemann mapping theorem and its variants live most naturally on the sphere.

In Vector Calculus and Complex Analysis, the Riemann sphere mostly serves as a clean way to handle the “point at infinity” in Möbius transformations and the Smith chart — providing a uniform language for the pole of a transformation and its behavior at infinity.