Stereographic Projection of a Spherical Circle
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The diagram below contains a sphere with centre \(O\) on the \(xy\) plane. The equatorial circle is coloured black. The blue circle is the intersection of a plane with the sphere, and it is an example of a spherical circle. The orange circle in the diagram is the stereographic projection of the blue circle onto the \(xy\) plane. Again, the fact that this is a circle requires proof.
The point \(P'\) on the sphere is the centre of the blue circle. You can see that this is the centre by clicking and dragging to rotate the picture so that the point \(P'\) is lined up with the centre \(O\) of the sphere. By playing the animation, you can see the point \(X'\) moving around the blue circle, and the projection \(X\) moving around the orange circle in the \(xy\) plane.
The point \(P\) is the stereographic projection of \(P'\) onto the \(xy\) plane. We can see from the left-hand picture (where the orange circle is the projection of the blue circle onto the \(xy\) plane) that \(P\) does not correspond to the Euclidean centre of the orange circle. The exercise constructing a spherical circle will help you see how to construct the circle from the points \(P\) and \(R\).
The point \(P\) is the spherical centre of the orange circle. This is defined as the projection of \(P'\) from the sphere to the plane, but one can also define it as the point which is an equal distance from each point on the orange circle. This requires the notion of spherical distance, which is the stereographic projection of the great circle distance on the sphere. Another equivalent definition of the centre uses spherical lines (the stereographic projection of great circles). The spherical centre is the unique point \(P\) such that every spherical line through \(P\) intersects the circle orthogonally.
You can move the points \(P\) and \(R\) in the diagram below so that the orange circle is the unique circle passing through \(R\) with spherical centre \(P\). The circle on the sphere is the unique circle that projects to the orange circle using stereographic projection.
