The Riemann Sphere as a Stereographic Projection
The Riemann sphere is a geometric representation of the extended complex plane (the complex numbers with the added point at infinity, ). To visualize this compactification of the complex numbers (transformation of a topological space into a compact space), one can perform a stereographic projection of the unit sphere onto the complex plane as follows: for each point in the plane, connect a line from to a designated point that intersects both the sphere and the complex plane exactly once. In this Demonstration, the unit sphere is centered at , and the stereographic projection is from the "north pole" of the sphere at . You can interact with this projection in a variety of 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 the 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 under the projection.
Snapshot 1: a stereographic projection of a circle between the complex plane and the Riemann sphere that also shows the stereographic projection lines
Snapshot 2: a partially "unwrapped" Riemann sphere under the stereographic projection
Snapshot 3: a visual representation of the bijection between points on the sphere and plane at all points where the stereographic projection is defined, with a selection of stereographic projection lines shown
Snapshot 4: a mapping of a hyperbola on the complex plane onto the sphere under the stereographic projection, which may help to give a bit of intuition about the notion of a "point at infinity"
While this Demonstration is constructed in , the plane at is meant to represent the complex plane.
The stereographic projection is constructed as follows: Suppose that is a point on the unit sphere. Then the projection begins by parameterizing a line connecting and : . Next is to find at which value intersects the plane at by solving the equation and substituting into . That is, is the point at which intersects . Then rescale to be so that and by defining . Parameterize the sphere using cylindrical coordinates, and define a parameterized family of surfaces . This parameterized family of surfaces is such that is the unit sphere, is the plane , and is an "in between" surface that allows the Demonstration to visually illustrate the "unwrapping" process with a slider.
An alternative stereographic projection centers the unit sphere at the origin. However, whichever construction is chosen, we obtain a conformal map with a designated "point at infinity." You can get an intuitive or geometric understanding of the "point at infinity" by plotting a parabola or hyperbola.