However, when plotting directional data in structural geology, they do represent the North and South geographic directions. • 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. W. Borchardt-Ott, Crystallography, 2nd Edition, Springer, New York, 1995 Stereographic Projection Let a sphere in three-dimensional Euclidean space be given. 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 The stereographic projection map, π : S2 −n−→ C, is described as follows: place a light source at the north pole n. For any point 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). In such a projection, great circles are mapped to circles, and loxodromes become logarithmic spirals. 1. The stereographic projection is a two-dimensional drawing of three-dimensional data. 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. Stereographic projection of a cantellated 24-cell. Stereographic projections have a very simple algebraic form that results immediately from similarity … 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. As defined in our projection, the N and S poles would plot directly above and below the center of the stereonet. The stereographic projection permits the mapping in two dimensions of crystallographic planes and directions in a convenient and straightforward manner. If Q is a point of Sn and E a hyperplane in En + 1, then the stereographic projection of a … 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. 3 = 0) is called stereographic projection from p~. 4.1. The geometry of all crystallographic planes and directions is reduced by one dimension. 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 Crystal Faces Page 3 of 6 9/7/2010 Geometrical Properties of Stereographic Projection (continued) 1.1. South Poles as defined in the projection above. Identify the complex plane C with the (x,y)-plane in R3. stereographic projection. 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