History Illustration by Rubens for "Opticorum libri sex philosophis juxta ac mathematicis utiles", by François d'Aguilon. Sometimes stereographic computations are done graphically using a special kind of graph paper called a stereographic net, shortened to stereonet, or Wulff net. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection it finds use in diverse fields including complex analysis, cartography, geology, and photography. Intuitively, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. This is the spherical analog of the Poincaré disk model of the hyperbolic plane. A two-dimensional coordinate system on the stereographic plane is an alternative setting for spherical analytic geometry instead of spherical polar coordinates or three-dimensional cartesian coordinates. The metric induced by the inverse stereographic projection from the plane to the sphere defines a geodesic distance between points in the plane equal to the spherical distance between the spherical points they represent. The stereographic projection gives a way to represent a sphere by a plane. It is neither isometric (distance preserving) nor equiareal (area preserving). It maps circles on the sphere to circles or lines on the plane, and is conformal, meaning that it preserves angles at which curves meet and thus locally approximately preserves shapes. It is a smooth, bijective function from the entire sphere except the center of projection to the entire plane. In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the projection plane) perpendicular to the diameter through the point.
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