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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 .
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.
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 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.
On a stereographic projection map, a loxodrome is an equiangular spiral whose center is the north or south pole. All loxodromes spiral from one pole to the other. Near the poles, they are close to being logarithmic spirals (which they are exactly on a stereographic projection , see below), so they wind around each pole an infinite number of ...
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 ...
Twice the area of the purple triangle is the stereographic projection s = tan 1 / 2 ϕ = tanh 1 / 2 ψ. The blue point has coordinates (cosh ψ, sinh ψ). The red point has coordinates (cos ϕ, sin ϕ). The purple point has coordinates (0, s). Graph of the Gudermannian function. Graph of the inverse Gudermannian function.