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Stereographic projection is conformal, meaning that it preserves the angles at which curves cross each other (see figures). On the other hand, stereographic projection does not preserve area; in general, the area of a region of the sphere does not equal the area of its projection onto the plane. The area element is given in (X, Y) coordinates by
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.
Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres) In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point.
The stereographic projection provides a coordinate system for the sphere in which conformal flatness is explicit, as the metric is proportional to the flat one. In general relativity conformally flat manifolds can often be used, for example to describe Friedmann–Lemaître–Robertson–Walker metric. [5]
The stereographic projection is a homeomorphism between the unit sphere in with a single point removed and the set of all points in (a 2-dimensional plane). If G {\displaystyle G} is a topological group , its inversion map x ↦ x − 1 {\displaystyle x\mapsto x^{-1}} is a homeomorphism.
The stereographic projection maps the -sphere onto -space with a single adjoined point at infinity; under the metric thereby defined, {} is a model for the -sphere. In the more general setting of topology , any topological space that is homeomorphic to the unit n {\displaystyle n} -sphere is called an n ...
n-sphere S n: Hopf chart. Hyperspherical coordinates. Sphere S 2: Spherical coordinates. Stereographic chart Central projection chart Axial projection chart Mercator chart. 3-sphere S 3: Polar chart. Stereographic chart Mercator chart. Euclidean spaces: n-dimensional Euclidean space E n: Cartesian chart: Euclidean plane E 2: Bipolar coordinates
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