The Riemann sphere

The unit sphere sits in the complex plane, meeting it along the equator, which is therefore the plane's own unit circle. Joining a point z of the plane to one pole of the sphere by a straight line marks a second point P where that line crosses the sphere: this correspondence is stereographic projection. The pole one projects from answers to z = ∞, and the opposite pole to z = 0, so that the sphere is the plane with a single point at infinity added — the Riemann sphere.

Drag the amber point z across the plane — it carries the plane's own hue, as its image P carries the sphere's; the projecting line follows both.
Drag empty space to orbit the sphere, and scroll to zoom. The sphere is translucent, so P stays visible on the far side.
Push z far out and watch P approach the pole marked ∞.
Tick “a figure at z” to carry the pennant flag of the other lessons along, and watch what the projection does to it. Its image is largest near z = 0 and shrinks as z moves out, by the factor λ in the panel.

Project from

Show a figure at z (and its image) Show the graticule Show mini-lesson

What to look for

Three collinear points. Pick a point z on the plane, join it to the south pole marked ∞; the line ray ∞z meets the sphere again at P, called the stereographic projection of z. The three points are collinear by construction. Notice that when z lies inside the unit circle, its stereographic projection P is on the upper hemisphere; when z lies outside the unit circle, its stereographic projection P is on the lower hemisphere .

Where the poles go. Drag z toward the origin and P climbs to the pole marked 0; push z far away in any direction and P settles toward the pole marked ∞, from whichever side it approached. An unbounded sequence in the plane approaches no point of the plane at all; its images on the sphere, however, converge to the pole marked ∞, and they do so whatever direction the sequence escaped along. One point of the sphere therefore serves for every such sequence, and that single added point is what makes the sphere compact — and what makes it the natural home of the plane.

The equator is not drawn. Look where the sphere meets the plane. No equator has been drawn anywhere: the curve you see is simply the plane's own unit circle, which is where the two surfaces cross. And the unit circle is exactly the set the projection fixes pointwise — put z on it and P lands on z itself, for either choice of pole.

Angles survive; areas do not. The projection is conformal: it preserves angles everywhere. Switch on the figure and drag it about — its corners keep their angles wherever it goes, though its size does not keep its own: the local magnification is λ = 2/(1 + |z|²), reported in the panel. It is largest at the origin and falls to zero as |z| grows, so an arbitrarily small neighbourhood of the pole marked ∞ is spread over the unbounded far reach of the plane.

Which pole to project from. That last fact settles a cartographic question. Whichever pole we project from is the one whose neighbourhood gets stretched beyond recognition, so we project from the pole we care least about. For a map of the Earth that is the south pole, and this lesson projects from the south by default; the Antarctic is then the region carried out beyond the edge of any map one is willing to print. There is a second reason, harder to see and just as real: projected from the south, a figure and its image on the sphere have the same handedness — switch the pole to the north and watch the flag's pennant change sides.