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