# The Riemann Sphere as a Stereographic Projection

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.

⋃{∞}

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(0,0,1)

(0,0,2)