1. 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.

3. (100)

4. A plane (Great Circle)  trace
A plane (Great Circle)  trace

5. Pole and trace

7. Great circle
Equal angle
with respect
to N or S pole

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

9. Meridians: great circle
Parallels except the
equator are small circles

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

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

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

13.  Finding the trace of a pole:

14. Rotation of a projection about an axis in the projection plane

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

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

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

18. Stereographic projection of different Bravais systems
Cubic (001)

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

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

21. [011] [001] [0 1 1] 𝑥 [01 1 ] [ 1 00] (011) [100] 𝑧 𝑦 [001] [0 1 1]
45o
[011]

22. [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]

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

24. Orthorhombic
Monoclinic

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

26. 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)

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

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

29. (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

30. Worthwhile reading: