![]() ![]() This construction plays a role in algebraic geometry and conformal geometry. The stereographic projection presents the quadric hypersurface as a rational hypersurface. As before, the stereographic projection is conformal and invertible outside of a "small" set. is the unique point of intersection of QP with E. In Cartesian coordinates ( x, y, z) on the sphere and ( X, Y) on the plane, the projection and its inverse are given by the formulas The plane z = 0 runs through the center of the sphere the "equator" is the intersection of the sphere with this plane.įor any point P on M, there is a unique line through N and P, and this line intersects the plane z = 0 in exactly one point P ′, known as the stereographic projection of P onto the plane. Let N = (0, 0, 1) be the "north pole", and let M be the rest of the sphere. 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. He used the recently established tools of calculus, invented by his friend Isaac Newton.ĭefinition First formulation Stereographic projection of the unit sphere from the north pole onto the plane z = 0, shown here in cross section In 1695, Edmond Halley, motivated by his interest in star charts, was the first to publish a proof. In the late 16th century, Thomas Harriot proved that the stereographic projection is conformal however, this proof was never published and sat among his papers in a box for more than three centuries. įrançois d'Aguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles (Six Books of Optics, useful for philosophers and mathematicians alike). In star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy. It is believed that already the map created in 1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Roze (1542), Rumold Mercator (1595), and many others. In the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. The term planisphere is still used to refer to such charts. One of its most important uses was the representation of celestial charts. Planisphaerium by Ptolemy is the oldest surviving document that describes it. It was originally known as the planisphere projection. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians. It demonstrates the principle of a general perspective projection, of which the stereographic projection is a special case. 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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