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Proof: Pick a circle on S not containing N and let A be the vertex of the cone tangent to S at this circle (Fig. • It is the stereographic projection of the grid of a conventional globe oriented so that the N´-S´ direction lies in the plane of projection. Stereographic Projection of Crystal Faces Page 3 of 6 9/7/2010 However, when plotting directional data in structural geology, they do represent the North and South geographic directions. The stereographic projection is a two-dimensional drawing of three-dimensional data. 1. Stereographic projection can be defined as a graphical technique for representing the angular relationships between planes and directions in crystal on a 2D piece of paper. The geometry of all crystallographic planes and directions is reduced by one dimension. More generally, stereographic projection may be applied to the n-sphere Sn in (n + 1)dimensional Euclidean space En + 1. W. Borchardt-Ott, Crystallography, 2nd Edition, Springer, New York, 1995 3 = 0) is called stereographic projection from p~. Theorem 2: Stereographic projection is circle preserving. Stereographic Projection Let a sphere in three-dimensional Euclidean space be given. 7). If Q is a point of Sn and E a hyperplane in En + 1, then the stereographic projection of a … A geometric construction known as stereographic projection gives rise to a one-to-one correspondence between the complement of a chosen point A on the sphere and the points of the plane Z 4.1. The stereographic projection map, π : S2 −n−→ C, is described as follows: place a light source at the north pole n. For any point Identify the complex plane C with the (x,y)-plane in R3. One can deﬁne a parametrization around the north pole similarly, by sending (u,v) to (u,−v,0) and then inverting stereographic projection from the south pole. A map projection obtained by projecting points P on the surface of sphere from the sphere's north pole N to point P^' in a plane tangent to the south pole S (Coxeter 1969, p. 93). Geometrical Properties of Stereographic Projection (continued) 1.1. The stereographic projection permits the mapping in two dimensions of crystallographic planes and directions in a convenient and straightforward manner. Stereographic projection of a cantellated 24-cell. South Poles as defined in the projection above. Stereographic projections have a very simple algebraic form that results immediately from similarity … It is deﬁned everywhere on S except at p~ itself. As defined in our projection, the N and S poles would plot directly above and below the center of the stereonet. STEREOGRAPHIC PROJECTION OF THE SPHERE 3 The metric of the sphere in terms of the projected coordinates is (again using 11) ds2 = 1 1+ ˆ2 4L2 2 dˆ 2 +r2d˚2 (15) = 1 1+ ˆ2 4L2 2 dˆ2 +ˆ2d˚2 (16) Note that this is the metric of the surface of the original sphere, and not of the projection. In such a projection, great circles are mapped to circles, and loxodromes become logarithmic spirals. STEREOGRAPHIC PROJECTION IS CONFORMAL Let S2 = {(x,y,z) ∈ R3: x2 +y2 +z2 = 1} be the unit sphere, and let n denote the north pole (0,0,1). stereographic projection. Of all crystallographic planes and directions is reduced by one dimension 2nd Edition, Springer, New York 1995. The stereonet data in structural geology, they do represent the North stereographic projection pdf. The center of the stereonet and directions is reduced by one dimension is everywhere! S except at p~ itself more generally, stereographic projection ( continued ) 1.1 Sn in ( n 1. 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