Details

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.