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Stereographic projection of the unit sphere from the north pole onto the plane z = 0, shown here in cross section. The unit sphere S 2 in three-dimensional space R 3 is the set of points (x, y, z) such that x 2 + y 2 + z 2 = 1.
The stereographic projection, also known as the planisphere projection or the azimuthal conformal projection, is a conformal map projection whose use dates back to antiquity. Like the orthographic projection and gnomonic projection , the stereographic projection is an azimuthal projection , and when on a sphere, also a perspective projection .
Stereographic projection of a pole. The upper sphere is projected on a plane using the stereographic projection. Consider the (x,y) plane of the reference basis; its trace on the sphere is the equator of the sphere. We draw a line joining the South pole with the pole of interest P.
Each torus is the stereographic projection of the inverse image of a circle of latitude of the 2-sphere. (Topologically, a torus is the product of two circles.) These tori are illustrated in the images at right.
Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner. (Notice that, since stereographic projection is conformal, round spheres are sent to round spheres or to planes.) A somewhat different way to think of the one-point compactification is via the exponential map. Returning to our picture ...
A stereographic projection of a Clifford torus performing a simple rotation Topologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together. In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles S 1 a and S 1
Stereographic projection as an inversion of a sphere. A stereographic projection usually projects a sphere from a point (north pole) of the sphere onto the tangent plane at the opposite point (south pole). This mapping can be performed by an inversion of the sphere onto its tangent plane.
Gall stereographic projection of the world. 15° graticule. Gall stereographic projection with 1,000 km indicatrices of distortion. The Gall stereographic projection, presented by James Gall in 1855, is a cylindrical projection. It is neither equal-area nor conformal but instead tries to balance the distortion inherent in any projection.