Stereographic projection

Representation of relationship of planes and directions in 3D ona 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

http://en.wikipedia.org/wiki/Pole_figure

Pole and trace

Great circle

Equal anglewith respectto N or S pole

http://www.quadibloc.com/maps/maz0202.htm

Construction of latitude (Parallels) and longitude (Meridians)of Wulff net!

Meridians: great circle

Parallels except theequator 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 anglebetween two points:

Bring these two points onthe same great circle; counting the latitude angle.

Angle between the planes of two zone circles is the anglebetween 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 thepole 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:

[100]

[010]

[001]

cos)/(

;cos)/(

;cos)/(

lc

d

kb

d

ha

d

cos:cos:cos:: cbalkh

Stereographic projection of different Bravais systems

Cubic (001)

How about a standard (011) stereographicprojection of a cubic crystal?

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Start with what you know!

What does (011) look like?

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(011)

[100]

[01]

[]

[]

(011)

[][11]109.47o

70.53o

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[100]

[01]

[] (011)

[011][][001]45o

[011] [001] [01]

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[01]

[] (011)

[011] []

35.26o

[011] [111] [100]

[100][111]

1 10

111

[011] [001] [01]

70.53o

HexagonalTrigonal 3a

3a

c[0001]

[110]

[111]

tan−1( 3𝑎𝑐 )

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

B.D. Cullity

Zoneaxis

(2)

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

(2) Representation of preferred orientation (texture or fabric): e.g.

A rolled sheet ofpolycrystalline cubicMetal.

A {100} pole figureRD: rolling directionTD: transverse directionSuccessive levels ofshading correspond tothe contours of theorientations of planenormals and directions.

{100} plane normals are spreading out toward thetransverse direction

{111}pole

figure

Showingthe

orientationof {111}planes

http://www.doitpoms.ac.uk/tlplib/stereographic/index.php

Worthwhile reading:

http://folk.uib.no/nglhe/e-modules/Stereo%20module/1%20Stereo%20new.swf