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WOLFRAM|DEMONSTRATIONS PROJECT

The Riemann Sphere as a Stereographic Projection

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unwrap sphere
complex plane
axes/labels
surface
Riemann sphere
rainbow colors
mesh
zoom
translucent
show curve
none
curve thickness
projection lines on curve
sample points on sphere
rainbow colors on points
point size
point count (Δθ)
point count (Δh)
stereographic projection lines
rainbow colors on lines
line thickness
line count (Δθ)
line count (Δh)

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
z
plane, connect a line from
z
to a designated point that intersects both the sphere and the complex plane exactly once. In this Demonstration, the unit sphere is centered at
(0,0,1)
, and the stereographic projection is from the "north pole" of the sphere at
(0,0,2)
. 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.