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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.
Wulff net, step of 10° Wulff net, pole and trace of a plane Pole figure of a diamond lattice in 111 direction. A Wulff net is used to read a pole figure. The stereographic projection of a trace is an arc. The Wulff net is arcs corresponding to planes that share a common axis in the (x,y) plane.
The Hopf fibration can be visualized using a stereographic projection of S 3 to R 3 and then compressing R 3 to a ball. This image shows points on S 2 and their corresponding fibers with the same color. For unit radius another choice of hyperspherical coordinates, (η, ξ 1, ξ 2), makes use of the embedding of S 3 in C 2.
As the name indicates, the UPS system uses a stereographic projection. Specifically, the projection used in the system is a secant version based on an elliptical model of the earth. The scale factor at each pole is adjusted to 0.994 so that the latitude of true scale is 81.11451786859362545° (about 81° 06' 52.3") North and South.
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
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 .
The inverses of these two stereographic projections are maps from the complex plane to the sphere. The first inverse covers the sphere except the point ( 0 , 0 , 1 ) {\displaystyle (0,0,1)} , and the second covers the sphere except the point ( 0 , 0 , − 1 ) {\displaystyle (0,0,-1)} .
The stereographic projection from the north pole (1, 0, 0, 1) of this sphere onto the plane x 3 = 0 takes a point with coordinates (1, x 1, x 2, x 3) with + + = to the point (,,,). Introducing the complex coordinate ζ = x 1 + i x 2 1 − x 3 , {\displaystyle \zeta ={\frac {x_{1}+ix_{2}}{1-x_{3}}},} the inverse stereographic projection gives ...