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.