The Riemann Sphere as a Stereographic Projection

Wolfram Demonstrations Project (2015) · Complex Analysis, Geometry

The Riemann Sphere and the Extended Complex Plane

The Riemann sphere is a geometric representation of the extended complex plane $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$, the complex numbers with an added point at infinity. Compactifying $\mathbb{C}$ this way (a one-point compactification of a topological space into a compact space) gives meromorphic functions a natural home: a pole of $f$ at $z_0$ just means $f(z_0) = \infty$ on $\hat{\mathbb{C}}$.

To visualize this compactification, we perform a stereographic projection of the unit sphere onto the complex plane. For each point in the $z$ plane, draw a line from $z$ to a designated point on the sphere; the line intersects both the sphere and the plane exactly once.

The Projection

In this Demonstration, the construction lives in $\mathbb{R}^3$ with the plane at $z = 0$ representing $\mathbb{C}$. The unit sphere is centered at $(0, 0, 1)$, and the stereographic projection is from the "north pole" of the sphere at $N = (0, 0, 2)$.

Given a point $(x_0, y_0, z_0)$ on the unit sphere, parameterize the line from $N$ through that point:

$$L_1(t) = (0, 0, 2) + t\big((x_0, y_0, z_0) - (0, 0, 2)\big)$$

Setting $z = 0$ and solving $0 = 2 + t(z_0 - 2)$ gives the value of $t$ at which the line hits the plane. The image $L_1\!\left(\frac{-2}{z_0 - 2}\right)$ is the projected point.

An alternative construction centers the unit sphere at the origin and projects from $(0,0,1)$. Either way, the result is a conformal map with a designated "point at infinity" at the north pole.

Key Geometric Properties

Stereographic projection is conformal: it preserves angles between curves. A pair of curves meeting at some angle on the sphere will meet at the same angle in the plane.

It also maps circles on the sphere to circles or lines in the plane. A circle on $S^2$ that passes through the north pole projects to a straight line (a "circle of infinite radius"), while every other circle projects to a circle. The Demonstration lets you see this directly: set "show curve" to "circle" and enable "projection lines on curve" to watch the projection lines from $N$ through the circle on the sphere down to the corresponding circle in the plane.

The "Unwrap Sphere" Slider

To animate the projection, the Demonstration defines a one-parameter family of surfaces. Rescale $L_1(t)$ so that $L_2(0) = (x_0, y_0, z_0)$ and $L_2(1)$ is the projected point on the plane. Then parameterize the sphere using cylindrical coordinates $(x(u,v), y(u,v), z(u,v))$ and define:

$$F_s(u,v) = \big(x(u,v),\, y(u,v),\, z(u,v)\big) + s\left(L_1\!\left(\tfrac{-2}{z(u,v)-2}\right) - \big(x(u,v),\, y(u,v),\, z(u,v)\big)\right)$$

At $s = 0$ this is the unit sphere; at $s = 1$ it is the plane $z = 0$. Dragging the slider from 0 to 1 continuously deforms the sphere into the plane, visually illustrating the bijection.

Interactive Demonstration

The Wolfram Demonstration lets you interact with this projection in several ways: "unwrapping" the sphere, showing stereographic projection lines, viewing the image of a set of points on the sphere under the projection, and picking a curve to view its image under the projection. The rainbow coloring on the sphere is a convenient visual tool for comparing where points on the sphere map to on the plane. You can plot a parabola or hyperbola to get geometric intuition about the "point at infinity."

Snapshots

Here are some snapshots from the Demonstration showing different features of stereographic projection:

Riemann sphere sitting on the complex plane with Re and Im axes labeled
The Riemann sphere on the complex plane with rainbow coloring
Circle on the Riemann sphere with projection lines showing the corresponding circle in the plane
A circle on the sphere projects to a circle in the plane
Surface visualization showing the sphere partially unwrapped onto the plane
The surface visualization with the sphere and projected surface
Stereographic projection lines connecting points on the sphere to the plane
Projection lines from the north pole through the sphere to the plane

The hyperbola snapshot below shows how curves in the plane lift to the sphere. A hyperbola in $\mathbb{C}$ maps to a curve on $S^2$ that passes through the north pole (since the asymptotes "go to infinity"):

Hyperbola in the complex plane projected onto the Riemann sphere
A hyperbola in the plane and its image on the Riemann sphere

Where This Comes Up

The Riemann sphere appears throughout complex analysis and algebraic geometry. Möbius transformations $z \mapsto \frac{az + b}{cz + d}$ are automorphisms of $\hat{\mathbb{C}}$, and they correspond to rotations and other rigid motions of $S^2$. In algebraic geometry, $\hat{\mathbb{C}}$ is the complex projective line $\mathbb{P}^1(\mathbb{C})$, the simplest example of a projective variety. Rational functions on $\hat{\mathbb{C}}$ are the meromorphic functions, and every compact Riemann surface admits a nonconstant meromorphic function, making the Riemann sphere the base case in the classification of Riemann surfaces.

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