Stereographic projection onto the sphere

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The following picture shows the sphere and the \(xy\)-plane in \(\mathbb{R}^2\). A straight line through the north pole passes through another point on the sphere (the red point) and intersects the \(xy\)-plane in the black point in the picture. This defines a projection of the sphere minus the north pole onto the \(xy\)-plane, called the stereographic projection of the sphere.

In the diagram below you can move the sliders for \(u\) and \(v\) to move the black point around the \(xy\)-plane. You can then see how the values of \(u\) and \(v\) then parametrise the sphere minus the north pole.