The stereographic projection permits the mapping in two dimensions of crystallographic planes and directions in a convenient and straightforward manner.
The stereographic projection is a two-dimensional drawing of three-dimensional data.
The geometry of all crystallographic planes and directions is reduced by one dimension.
Planes are plotted as great circle lines.
Directions are plotted as points.
Also, the normal to a plane completely describes the orientation of a plane.
THE STEREOGRAPHIC PROJECTION
representing angles and planes
representing angles and planes
representing angles and planes
great circles: diameter equal to that of sphere
• Great circles project as the arcs of circles
• Vertical great circles project as straight lines
representing angles and planes
010
100
001
representing angles and planes
Planes that mutually intersect along a common direction form the planes of a zone, and the line of intersection is called the zone axis.
Example:
[111] direction as a zone axis.
There are three {110} planes that pass through the [111] direction. There are also three {112} planes and six {123} planes, as well as a number of higher indice planes that have the same zone axis.
[111] zone axis
{112} and {123} planes
all of the poles of a same zone axis fall on the great circle representing the stereographic projection of the (111) plane
[111] zone axis
If the axis of a zone is given by the indices [u v w], and if a plane belongs to that zone denoted by the indices (h k l), then:
Weiss Zone Law: hu + kv + lw = 0
The Weiss rule is independent of the crystal system.
If two planes of (h1k1l1) and (h2k2l2) belong to one zone axis of [u v w], the following relationships are obtained:
h1u + k1v + l1w = 0 and h2u + k2v + l2w = 0
(ph1 + qh2)u + (pk1 + qk2)v + (pl1 + ql2)w = 0
where p and q are arbitrary integers.
In other words, if a zone axis [u v w] contains two planes (h1k1l1) and (h2k2l2), planes represented by p( h1k1l1 ) + q(h2k2l2 ) also belong the same zone.
Weiss Zone Law
Wulff net
Stereographic projection of latitude and longitude lines in which the north–south axis is parallel to the plane of the paper.
The latitude and longitude lines of the Wulff make possible graphical measurements (angles).
Rotation About an Axis in the Line of Sight
Measuring from the centre
• To measure the angle between two poles rotate the Wulff net until both lie on a common great circle. The angle required is measured along the great circle using the scale on the net.
Measuring angle between two poles
Locating plane normals
Angle between two planes
Rotation about the North–South Axis of the Wulff Net
Rotation about the North–South Axis of the Wulff Net
STANDARD PROJECTIONS
STANDARD PROJECTIONS
STANDARD PROJECTIONS
THE STANDARD STEREOGRAPHIC TRIANGLE FOR CUBIC CRYSTALS
Cubic symmetry
THE STANDARD STEREOGRAPHIC TRIANGLE FOR CUBIC CRYSTALS
cubic P orthorhombic P
Lower symmetry systems
Lower symmetry systems
z
x
y001
(100)
(010) 010
100
hexagonal
Lower symmetry systems
stereogram of hexagonal system
stereogram of monoclinic system
Study:
Exercices 1.13 and 1.14 – Reza Abbaschian, Lara Abbaschian, Robert E. Reed-Hill, pg 28.
Applications 1, 3, 7 and 9 Barrett & Massalski, pg 47