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