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.