Constructing a spherical line
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Using stereographic projection we can project all the points on the unit sphere except for the north pole to the plane, and the north pole projects to the point at infinity. Using this projection, we can transfer the geometry of the sphere to the plane. The advantage of this is that we can now visualise the sphere on a 2-dimensional plane where we can use a ruler and compass to construct lines and circles, rather than inside a 3-dimensional space, however the disadvantage is that the geometry of the sphere is different to the usual Euclidean geometry of the plane.
Recall that stereographic projection maps the points on the equator of the unit sphere to the unit circle in the plane. This equatorial circle will be the basis for understanding spherical geometry in the plane.
Recall that a line in spherical geometry is a great circle on the sphere. The stereographic projection of a great circle is a circle in the plane which intersects the equatorial circle in two diametrically opposed points (because a great circle will intersect the equator in two diametrically opposed points and the equator maps to itself under stereographic projection). From now on we define a spherical line to be a circle in the plane which intersects the equatorial circle in two diametrically opposed points.
The diagram below shows the equatorial circle coloured in black, with centre \(O\). Your task is to construct a spherical line through the points \(A\) and \(B\) using the ruler and compass tools on the diagram. You will find the homework exercises very useful, so I encourage you to try these first. The diagram already contains an example of a spherical line through the points \(P\) and \(Q\) (the green circle). Notice that the green circle intersects the equatorial circle in two diametrically opposed points. This is illustrated in the diagram, which shows that the line through these two points passes through the centre of the equatorial circle.
Once you have done the construction, you can check your answer by drawing a line through the two points of intersection with the equatorial circle, and checking that this line passes through the centre \(O\).
