This method works for cubic crystals, but can be adapted to work with other crystal systems. In this example the pole of the plane (123) in a cubic crystal is plotted.
1. Draw a 4” diameter circle on a piece of graph paper. Mark both the N-S line and the equatorial line. The N-S line represents the Z-Axis of cartesian space - this axis is parallel to the [001] direction in the cubic crystal. The Equatorial line represents the Y-Axis of cartesian space and is parallel to the [010] axis of the cubic crystal.
2. Determine the angle betwen the normal to the plane (hkl) and the Z-Axis. In a cubic system the normal to the plane is the direction [hkl] so for the plane (123) the normal is [123]. The angle between two vectors is...
so
3. Next, draw the vector from the centre of the circle to the circumference using the calculated angle. Then from the point where the normal intersects the circumference draw a line to the South pole of the cirlce. Where this second line intersects the equator is the radial displacement, r’, of the pole of (123) from the centre of the projection.
4. Next, draw a new 4” diameter circle, which represents the equatorial plane of the stereographic projection, and transfer onto this plane a cirlce of radiaus r’.
5. Now set l=0 and draw the plane (120) or {hk0}. The intercpet on the Y-Axis is b/2 and on the X-axis is a/1. Since a=b then the points are 1/2 on Y and 1 on X.
Then construct the normal to the plane.
The point where the normal intersects the small (blue) circle is the pole (123).
The remaining poles of the planes {123} can be obtained either by plotting each out or by the applying the rotational and mirror symmetry of the cubic crystal. The example below shows just 8 of the possible 48 variants of {123} obtained by reflection and/or a 90° rotation about the [001] axis.
All 48 poles of the group {123} are plotted above. You should note that those with negative l lie at the same point as those with positive l, so just 24 poles are actually visible.
Now, we shall explore how to plot all the poles of a given set of planes using trigonometry and from there how to plot poles of planes in non-cubic crystal systems...
Plotting the Zone [uvw] on a stereographic projection