The Stereographic Projection (all content)
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Contents
Aims
This TLP is designed to give you a good working knowledge of the stereographic projection and to enable you to identify and plot poles. This TLP will be looking solely at the stereographic projection of the cubic crystal system in order to keep it simple.
Before you start
This TLP assumes a good knowledge of the atomic structure of crystals, and the indexing of planes with Miller indices. It is also closely linked with the TLP on slip, which gives more information on the mechanisms of slip in single crystals.
Therefore, it would be useful to read the following TLPs: Atomic scale structure, Crystallography, Miller indices and Slip in Single Crystals
Introduction
In crystal geometry, the most important aspect of the lattice is the angular relationship between various planes and other symmetry elements, not the relative translational position of planes. The importance of this is demonstrated by the expression of the crystal structure through the outside planes of a macrocrystal, for example as in the fluorite crystal shown here:
(It should be noted that this is not common. To grow crystals so that their external structure represents their internal structure requires a very controlled set of conditions. This makes it only the more important to understand angular relationships, as they cannot be seen simply by looking at a crystal.)
We need to be able to describe these angular relationships in an easily understandable manner, and so we use the stereographic projection, which presents a 3D structure on the surface of a sphere. This can be extrapolated to a 2D structure, which allows direct measurement of angles between various rotational axes or normals to planes, and this is very useful. This has a long history, being first used by Neumann in 1823.
Basic concept
The basic idea is to represent planes as points on some representative surface, which maintains the angular relationship of the points to each other.
In the spherical projection, various structural features are expressed as points on a sphere. This sphere sits around the object being examined.
Because lattice planes always maintain the same angular relationship to each other, planes can be represented by a plane normal. It is the plane normal which is used to produce the point on the sphere. This point is referred to as the ‘pole’ of the plane.
[Click on the image and drag it, in order to observe the image from all angles. This can be done with several images in this TLP.]
The intersection of a plane normal with the sphere of projection results in a point on the sphere which is referred to as a ‘pole’.
It is possible to represent multiple planes on a single sphere, by extending the plane normals of all the planes. These can be joined to see angular relationships, as below.
The information can be stored as 3D spheres, but these are unwieldy as it would require you to carry around a sphere. It is a lot easier to use a 2D representation of the sphere.
The 2D representation is generated by projecting the sphere onto a ‘projection plane’, in this case to produce a sterographic projection. This is done by connecting the points on the sphere to some defined ‘projection point’. The projection point is typically defined as one of the poles of the sphere, as shown here:
An important aspect of these projections is the position of the projection plane. There are two positions commonly used, equatorial and above, shown here (click and drag image to rotate):
The stereographic projection is then built up by connecting points on the sphere to projection points, and then noting where the connecting lines intersect the plane of projection.
We can also project down the plane itself, as well as just its normal. The plane projects down as an arc on the projection plane, and is found at 90 degrees to its normal. Precisely what this means will be seen later.
At this stage of the process, it is useful to define a feature of the projection: the ‘primitive circle’. This is a circle on the projection plane, which is located where the sphere of projection intersects the projection plane.
This defines a boundary around the stereographic projection. Projected points may fall inside or outside of the primitive circle, depending on which pole is used as a projection point, as shown here:
Points may appear at a great distance outside the primitive circle, so typically, different projection points are used for features in different hemispheres. For example, in the above example, the South Pole would be used.
This is represented by using circles and dots depending on where a point is projected from. A pole projected from the North Pole is represented by a circle and a pole projected from the South Pole is represented by a dot. In this TLP, we will only be considering poles in the northern hemisphere, so the stereograms will only show dots.
Demonstration of projection
While we have seen the basis of projection for a very simple system, we may now more closely examine the production of a stereogram (another word for stereographic projection), by producing the stereogram for a cube.
Look at a cube:
The most obvious symmetry element is the fourfold rotational symmetry. This presents on the sphere as a projected rotational axis, which intersects with the sphere, and is then projected down onto the projection plane.
It can easily be seen that the projected points of the rotational axes maintain the same symmetry and angular relationship on the projection as they do in 3D.
This can be extended further by the addition of 3fold rotational axes which project from the vertices of the cube. (If these are not immediately obvious to you, please see the above image of a cube, and observe it down the [111] direction.)
To differentiate between the fourfold and threefold rotational axes, we introduce a new notation, such that various types of symmetry elements, when projected onto the plane, are illustrated in different ways. These typically are:
So adding in the diads that project through edges of the cube with the new notation
The projection at this point is evidently getting rather complex. But the 2D representation is still easily understood. Here are all rotation axes on a single diagram.
There are also a collection of planes of symmetry. These intersect with the sphere in an infinite number of places, and as such, present as curves on the plane of projection, as can be seen here.
The curves here take the form of ‘great circles’. This will be covered in more detail later, but for now, all you need to know is that planes which pass through the origin (i.e. the centre of the sphere) present as great circles, which intersect with the primitive circle at the opposite ends of a diameter of the projection. In the full stereogram, you will also see that axes of rotation which lie in planes of symmetry show on the great circle.
So, assembling the entire stereogram, we see:
As you can see, the stereogram holds a large amount of information in a method which can be easily interpreted when you understand the principles behind it.
Important properties of the stereographic projection
Preservation of angular truth: This is the main basis for use of the stereographic projection. The angle between poles of planes is the angle between those poles on the sphere. This is also the angle seen when the poles are projected down onto the projection plane. This has been seen in the case of the cube. However, the axis system of the stereographic projection is slightly more complicated, and will be investigated further when we look at the Wulff net.
The other important property is that any plane projects onto the projection plane as either a circle or a straight line. However, we do not necessarily see the entire circle. For example, planes which pass through the origin, if projected from a single point, present as a circle which falls both inside and outside of the primitive (click and drag image to rotate).
Typically, we would instead project some of the plane from both possible projection points. This leads to a ‘double’ arc.
The circle produced here is called a ‘great circle’. It will always pass through both ends of a diameter of the primitive circle. One special case of a great circle is the primitive circle, which we saw before.
The other possibility is a ‘small circle’. This appears when examining a plane which does not pass through the origin. This produces a circle on the projection plane which will not pass through opposite ends of a diagonal of the projection plane (click and drag image to rotate).
These will be looked at in more detail later.
The Wulff net
Having understood how the projection is constructed, we can now look at how it is examined using the Wulff Net. The net is a projection of a collection of great and small circles, which represent lines of latitude (small circles) and lines of longitude (great circles) on the sphere, as seen here:
If we consider rather more of each, we can project them downwards as before using the principle of projection to construct a stereographic projection of planes which are two degrees apart. This creates the Wulff Net.
This can be used to place planes and poles of planes on a stereogram with great accuracy. It allows the maintenance of angular truth, while still being easy to draw.
Use of the Wulff net in constructing a stereogram
There are a few important things to note about a stereogram. Any planes, whose poles lie upon a great circle, share a zone with any other plane whose pole is on that great circle. For example, in the cubic system, (100), (010), (
00) and (0 0) all lie on the primitive circle. The primitive circle is a special case of a great circle. Therefore, if you are trying to plot the pole of a plane on a stereogram and you know which zone it lies in, the use of a Wulff net will enable you to draw it relatively straightforwardlyWhen drawing stereographic projections, brackets are not used when defining poles representing the normals to planes. Hence, the normals to the planes (100), (010), (
00 ) and (0 0), etc. are written as 100, 010, 00 and 0 0 .For cubic crystals, the normal to (hkl) planes is parallel to the vector [hkl], so that the pole representing the normal to the (hkl) set of planes is also the pole representing the vector [hkl]. While this is also true for particular directions of the plane normal to the (hkl) set of planes and vectors [hkl] for crystals of lower symmetry, such as the normal to the (010) planes of an orthorhombic crystal being parallel to [010], it is not generally true.
The following animation goes through the basics of using a Wulff net for cubic crystals where the centre of the stereogram is 001.
Plotting poles on the stereogram through use of the Wulff net
It is possible to make use of the Wulff net to find the angular relationships of various poles of planes. However, to find these angular relationships, we need to plot the poles onto a Wulff net. There are various ways of doing this, described below.
Identifying poles on a stereogram through use of the Wulff net
To identify poles, find two great circles that intersect at the desired pole hkl. Find the two zone directions [u_{1}v_{1}w_{1}] and [u_{2}v_{2}w_{2}] of these two great circles by, in each case, identifying two poles lying in these zone directions, and then using the Weiss zone law condition to determine the two zone directions. The desired pole hkl is then the normal to the plane hkl which contains the directions [u_{1}v_{1}w_{1}] and [u_{2}v_{2}w_{2}]; this can also be determined from the Weiss zone law condition.
In general, directions such as [u_{1}v_{1}w_{1}] and [u_{2}v_{2}w_{2}] are referred to the real space lattice, while normals to planes (hkl) are referred to the reciprocal lattice.
However, since for cubic crystals, the normal to the plane hkl is parallel to the vector [hkl], the algebra required is equivalent to taking cross products to determine the zone directions [u_{1}v_{1}w_{1}] and [u_{2}v_{2}w_{2}], and then taking a cross product again to determine the desired pole hkl.
So, for cubic crystals and stereograms of cubic crystals, we can drop the distinction between the real lattice and the reciprocal lattice. Therefore, for example, we can identify poles on stereograms of cubic crystals using vector addition, e.g.:
Applications of the Stereographic Projection  Slip
Slip is an important deformation mechanism, and as such, it is important to understand it. Slip is typically defined in terms of systems, containing both a slip plane and a slip direction. The use of a Wulff net allows these to be found easily given the tensile axis, following the procedure shown here:
Interactive Wulff net
It is often useful to ‘play’ with a Wulff net in order to better understand the angular relationships between planes. Therefore, below is provided a Wulff net that will plot any plane with indices up to a maximum of 6.
Summary
The stereogram is a very useful tool in the studying of crystal structures, and the understanding of how those structures relate to planes within crystals. This allows interpretation of diffraction patterns and lattices, making them an important part of crystallography..
Questions
Deeper questions
The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.

Using a Wulff net, plot the {100} poles on a stereogram aligned so that the paper is in the xy plane. By measuring angles along great circles, with rotation of the Wulff net if necessary, plot 320, 323, 510 and 511. Find the angle between 323 and 511.

Plot the pole 141 by plotting two intersecting great circles on a Wulff net.

Plot the pole 211 by drawing small circles around 001, 111 and 110 on a Wulff net.

Identify all green poles on this diagram by use of vector addition and zonal relationships.
Going further
Books
A. Kelly and K.M. Knowles, Crystallography and Crystal Defects, 3rd Edition, Wiley, 2020.
D. McKie and C. McKie, Essentials of Crystallography, Blackwell Science Publications, 1986.
F.C. Phillips, An Introduction to Crystallography, 4th Edition, Oliver and Boyd, 1971.
Academic consultant: Kevin Knowles (University of Cambridge)
Content development: Rob Shaw
Web development: Lianne Sallows and David Brook
This DoITPoMS TLP was funded by the UK Centre for Materials Education and the Worshipful Company of Armourers and Brasiers'.