Stereographic map projection

The stereographic projection was likely known in its polar aspect to the ancient Egyptians, though its invention is often credited to Hipparchus, who was the first Greek to use it.

The earliest written description of it is Ptolemy's Planisphaerium, which calls it the "planisphere projection".

Its popularity in cartography increased after Rumold Mercator used its equatorial aspect for his 1595 atlas.

[1] It subsequently saw frequent use throughout the seventeenth century with its equatorial aspect being used for maps of the Eastern and Western hemispheres.

[2] In 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal.

[3] He used the recently established tools of calculus, invented by his friend Isaac Newton.

The sphere is normally chosen to model the Earth when the extent of the mapped region exceeds a few hundred kilometers in length in both dimensions.

For maps of smaller regions, an ellipsoidal model must be chosen if greater accuracy is required.

However, it is possible to show points arbitrarily close to the South Pole as long as the boundaries of the map are extended far enough.