Stereographic Projection of a Great Circle
Back to Geometry homepage
The diagram below shows the unit sphere in \(\mathbb{R}^3\). The blue circle is a great circle, defined as the intersection of the sphere with a plane through the origin. The red circle is the projection of this great circle onto the \(xy\) plane, using the point \(N = (0,0,1)\) as a point of perspective (the fact that the projection is a circle requires proof, which we will do in the homework). This projection, defined using \(N = (0,0,1)\) as a point of perspective is called stereographic projection.
Note that the equator of the unit sphere is the intersection of this sphere with the \(xy\) plane, therefore it is mapped to itself by stereographic projection. Since a great circle (such as the blue circle in the diagram) always intersects the equator in two diametrically opposed points, then the stereographic projection of a great circle (such as the red circle in the diagram) must also intersect the equator in two diametrically opposed points.
In the diagram below, the green line through \(N\) passes through a point \(P\) on the great circle, and intersects the \(xy\) plane at the point \(P'\). You can move the points \(A\) and \(B\) in the diagram, and the applet will automatically draw the red circle (which passes through \(A\) and \(B\), and intersects the equatorial circle in diametrically opposed points). The great circle is then the circle on the sphere which projects to the red circle.
Equivalently, you can think of \(A\) and \(B\) as the stereographic projection of two points on the sphere, and the blue circle is the unique great circle passing through these points.
