Stereographic Projection

This page is a sub-page of our page on Calculus of One Complex Variable.

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The sub-pages of this page are:

The Mercator projection

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Related sources of information:

Generalizations of Stereographic Projection
Hyperbolic Stereographic Projection

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Stereographic Projection:

Stereographic projection of a circle on the sphere onto the plane
(with the pre-image of the center of the image circle)
:

The interactive simulation that created this movie.

Stereographic projection takes a circle on the sphere to a circle in the plane, but it does NOT map the center point of the original circle to the center point of the image circle. As shown in the movie, the point that is mapped to the center point of the image circle is the vertex point of the cone that is tangent to the original circle.

Stereographic projection of a circle on the sphere onto the plane (moving camera):

The interactive simulation that created this movie.

Stereographic projection (corresponding circular arcs):

The interactive simulation that created this movie.

Read the text in the window above the simulation and switch the mentioned colors off and on by clicking (and re-choosing) the box to the left of each formula.

Proof that stereographic projection is conformal (= preserves angles)

Stereographic-projection-conformal-proof

Interactive simulation of the proof that sterographic projection is a conformal map. The proof is in the text above the simulation window. It describes the elements in the 3D scene. The text window can be enlarged by pulling down at the bottom of it.

Stereographic projection as inversion:

The interactive simulation that created this movie.

Stereographic backprojection of a rectangular grid:

The interactive simulation that created this movie.

Stereographic backprojection of a polar grid:

The interactive simulation that created this movie.

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