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stereographic projection
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(Definition)
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The -dimensional Euclidean unit sphere is defined as a subset of
:
The stereographic projection maps all points of to the -dimensional Euclidean space
except one. Let
be this point (it is usually called the north pole). Then the stereographic projection is defined by
Here, is an arbitrary real number. If , the projection degenerates; in all other cases, however, is a smooth bijective mapping.
The image of a point under can be geometrically constructed as follows. Embed
into
as a hyperplane at . Unless , the straight line defined by and intersects with
in precisely one point, . The most common values for are and , see figures and .
Figure 1: Stereographic projection of the one dimensional unit sphere for
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Figure 2: Stereographic projection of the one dimensional unit sphere for
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Let
be the map
, then
(a suitably restricted composition) maps all points of except the south pole
smoothly and bijectively to
. Together, and
form an atlas of , so is an -dimensional smooth manifold.
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"stereographic projection" is owned by GrafZahl.
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(view preamble)
Cross-references: smooth manifold, atlas, composition, intersects, line, straight, hyperplane, image, mapping, bijective, smooth, projection, real number, Euclidean space, points, maps, subset, Euclidean
There are 6 references to this entry.
This is version 2 of stereographic projection, born on 2005-05-24, modified 2005-05-25.
Object id is 7110, canonical name is StereographicProjection.
Accessed 7178 times total.
Classification:
| AMS MSC: | 51M15 (Geometry :: Real and complex geometry :: Geometric constructions) | | | 54C05 (General topology :: Maps and general types of spaces defined by maps :: Continuous maps) | | | 54C25 (General topology :: Maps and general types of spaces defined by maps :: Embedding) | | | 54E40 (General topology :: Spaces with richer structures :: Special maps on metric spaces) |
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Pending Errata and Addenda
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