Stereographic projection

Stere­o­graphic projec­tion is the mapping of a sphere (with a pole punc­tured) onto a plane by the following law. For a point on the sphere, its image on the plane lies on the ray connecting the pole of the sphere — the centre of the projec­tion — with the point on the sphere. This mapping is mutu­ally unique and “works” in both direc­tions.

Stere­o­graphic projec­tion preserves angles between lines, and trans­lates any circles on the sphere into circles on the plane. More precisely: circles not passing through the projec­tion centre turn into circles in the plane and those passing through it (not just merid­ians!) turn into straight lines.

Another prop­erty: the curves symmetric on the sphere with respect to the equator turn inversely to each other on the plane with respect to the equator projec­tion.

One can realize a stere­o­graphic projec­tion by placing a small (point) light source at the pole of the sphere.

A beau­tiful model: stripes on the sphere which are the (pro)image of a square grid on the plane. It can be made by sticking the stripes on a trans­parent sphere or, even better, by printing the model on a 3D-printer.