In this demo you can explore the stereographic projection of the sphere onto the plane.
The unit sphere is projected from the North Pole onto the plane z=-1. z=-1 is used instead of the z=0 plane so the sphere remains completely visible.
When the mouse is over the sphere, dragging rotates the sphere.
Note that while the sphere is rotating there is a slight error in the texturing, going between the images of the two poles. When the sphere stops, this error is corrected.
Otherwise, dragging the mouse invokes the current tool on the world. Default tool is rotate tool (Tools are selectable from the tool bar.)