Stereographic projection Representation of relationship of planes and directions in 3D on a 2D plane. Useful for the orientation problems. A line (direction)  a point.

(100)

A plane (Great Circle)  trace http://courses.eas.ualberta.ca/eas233/0809winter/EAS233Lab03notes.pdf A plane (Great Circle)  trace

Pole and trace http://en.wikipedia.org/wiki/Pole_figure

Great circle Equal angle with respect to N or S pole

Construction of latitude (Parallels) and longitude (Meridians) of Wulff net! http://www.quadibloc.com/maps/maz0202.htm

Meridians: great circle Parallels except the equator are small circles

Using a Wulff net: How to address the shorted distance between two locations? Connecting two points with the great circle!

Measure the angle between two points: Bring these two points on the same great circle; counting the latitude angle.

Angle between the planes of two zone circles is the angle between the poles of the corresponding

 Finding the trace of a pole:

Rotation of a projection about an axis in the projection plane

Rotation about a direction (pole) that is inclined to the projection plane To rotate about the pole B1 by 40°

Movement of pole when rotated along A axis for 35.3o. The (112) pole is brought to the center.

 Determining Miller indices for poles: [001] [010] [100]

Stereographic projection of different Bravais systems Cubic (001)

How about a standard (011) stereographic projection of a cubic crystal? 𝑥 𝑦 𝑧 Start with what you know! What does (011) look like?

𝑥 [01 1 ] [ 1 00] (011) [100] 𝑧 𝑦 [0 1 1] [11 1 ] [1 1 1] 109.47o (011) 70.53o

[011] [001] [0 1 1] 𝑥 [01 1 ] [ 1 00] (011) [100] 𝑧 𝑦 [001] [0 1 1] 45o [011]

[011] [111] [100] [100] [111] 𝑥 [01 1 ] [ 1 00] (011) 𝑧 𝑦 [0 1 1] 1 10 [100] [111] 𝑥 [01 1 ] 35.26o 70.53o [ 1 00] (011) 𝑧 𝑦 111 [0 1 1] [011] [011] [001] [0 1 1]

Trigonal Hexagonal 3a [ 2 111] [0001] c tan −1 3𝑎 𝑐 3a [ 2 110]

Orthorhombic Monoclinic

 Stereographic projections of non-cubic crystals: two stereographic projections is required (one for the surface normal (poles) and the other the directions).

Two convections used in stereographic projection (1) plot directions as poles and planes as great circles (2) plot planes as poles and directions as great circles (plot the pole of the plane and the great circles of the direction)

Example: [001] stereographic projection; cubic (2) Zone axis B.D. Cullity

 Applications of the Stereographic projections: (1) Representation of point group symmetry

(2) Representation of preferred orientation (texture or fabric): e.g. A rolled sheet of polycrystalline cubic Metal. A {100} pole figure RD: rolling direction TD: transverse direction Successive levels of shading correspond to the contours of the orientations of plane normals and directions. {111} pole figure Showing the orientation of {111} planes {100} plane normals are spreading out toward the transverse direction

Worthwhile reading: http://www.doitpoms.ac.uk/tlplib/stereographic/index.php http://folk.uib.no/nglhe/e-modules/Stereo%20module/1%20Stereo%20new.swf