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