Pole figures, stereographic projection

Figure 1.16 Mappiitg of a minimal surface from real >ace to the complex plane. A point P on the surface, whose normal vector at P is n, is transformed to a point P by the Gauss map, given by the intersection of n (placed at the origin of the unit sphere O) with the sphere. P is mapped into a point P" on die complex plane (real and imaginary axes aand rresp.) by stereographic projection from the north pole of die sphere, N, onto the complex plane, which intersects the sphere in its equator.

FIGURE 2.13. The stereographic projection for representing, in two dimensions, the arrangement and directions of faces in a three-dimensional crystal, (a) The crystal is surrounded by a sphere. The points at which normals (perpendiculars) to the faces hit this sphere are noted by points labeled 100, 110, etc., the same cis the crystal faces they represent, (b) Each point (1 to 5, for example) representing a crystal face is joined to the opposite pole ( south if the point is in the northern hemisphere). 

Figure 8-7 shows the stereographic projection in a more complete form, with all poles of the type 100, 110, and 111 located and identified. Note that it was not necessary to index all the observed diffraction spots in order to determine the crystal orientation, which is specified completely, in fact, by the locations of any two 100 poles on the projection. The information given in Fig. 8-7 is therefore all that is commonly required. Occasionally, however, we may wish to know the Miller indices of a particular diffraction spot on the film, spot 11 for example. To find these indices, we note that pole 11 is located 35° from (001) on the great circle passing through (001) and (111). Reference to a standard projection and a table of interplanar angles shows that its indices are (112). 

These angles are shown stereographically in Fig. 9-10, projected, on a plane normal to the incident beam. The (111) pole figure in (a) consists simply of two arcs which are the paths traced out by 111 poles during rotation of a single crystal about [100]. In (b), this pole figure has been superposed on a projection of the reflection circle in order to find the locations of the reflecting plane normals. Radii drawn... 

A pole figure shows the distribution of a selected crystallographic direction relative to certain directions in the specimen. Texture data may also be presented in the form of an inverse pole figure, which shows the distribution of a selected direction in the specimen relative to the crystal axes. The projection plane for an inverse pole figure is therefore a standard projection of the crystal, of which only the unit stereographic triangle need be shown. Both wire and sheet textures may be represented. 

Figure 3. Stereographic projection. Pole P of the crystallographic plane projects to P on the projection plane (Ref. 14).

Figure 5 (a) Standard (001) stereographic projection of poles and zones circles for cubic crystals (after E A. Wood, Crystal Orientation Manual, Columbia Univ. Press, New York, 1963). 

For biaxial orientation the pole distribution t(, 4>) may be visualized as a density distribution defined on the surface of a sphere. The method of stereographic projection is then used to transcribe the density distribution from the spherical surface onto a sheet of paper. The contour map thus obtained is called a pole figure. 

The stereographic projection can be explained with the aid of Figure 3.20 as follows. The plane of projection is placed tangent to the sphere. The line drawn from the point of contact, A, to the center of the sphere, B, is extended, until it crosses the surface of the rear hemisphere at point C, which is then chosen as the point of projection. For a point P on the surface of the front hemisphere, the extension of the line from C to P gives its projection P on the projection plane. All the points on the surface of the front hemisphere are therefore mapped to points within the circle of projection of radius twice that of the sphere. The great half circle from the north pole N to the south pole S through A is mapped as the vertical diameter N AS, and the equatorial half circle EAW is mapped as the horizontal diameter E AW. Some of the more important properties of the stereographic projection can be described as follows ... 

Now, the angles measurement assume the poles alignment so that to be placed on the same big circle, operation performed by the rotation around an axis parallel to the equatorial plane of the stereographic projection, followed by the angular evaluation between the lines of latitude that correspond to them, see Figure 2.47. 

Being fixed the spherical grid, further on, one should consider its flat projection to be used in conjunction with the stereographic projection, through their poles. There are two types of flat projection of the spherical grid (always on the equatorial plane) as are shown in Figure 2.48. 

FIGURE 2.47 The spherical grid and the operation of poles aligning after Stereographic Projection (2003). 

In the Figure 2.49 the standard stereographic projection of the cubic crystal is considered oriented so that the direction [001] (thus along Oz) includes the North Pole of the sphere of projection. Only the information indicated in the northern hemisphere projection are required, those of the southern hemisphere being the duplicate with the opposite sign. 

Articles on x-ray diffraction measurements of orientation in this period were concerned with the orientation of cellulosic chains in fibers and in cell walls (29-33). A study of cotton, wood, flax, sisal, hemp, and other cellulosic structures established the occurrence of helically oriented cellulosic materials. Naturally occurring cellulose membranes were also investigated. Simon (32) proposed the use of pole figures, ie, stereographic projection of normals to crystallographic planes, to represent the orientation of the chains of crystalline cellulose (32). The development of uniaxial and uniplanar orientation was described based on the x-ray results. 

---