Stereographic projection

Stereographic projection
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians. It was originally known as the planisphere projection. Planisphaerium by Ptolemy is the oldest surviving document that describes it. One of its most important uses was the representation of celestial charts. The term planisphere is still used to refer to such charts.

It is believed that the earliest existing world map created by Gualterious Lud of St Dié, Lorraine, in 1507 is based upon the stereographic projection, mapping each hemisphere as a circular disk.
The equatorial aspect of the stereographic projection, commonly used for maps of the Eastern and Western Hemispheres in the 17th and 18th centuries (and 16th century - Jean Roze 1542; Rumold Mercator 1595), was utilized by the ancient astronomers like Ptolemy.

François d'Aiguillon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles (Six Books of Optics, useful for philosophers and mathematicians alike).

In 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal.
He used the recently established tools of calculus, invented by his friend Isaac Newton.

Sources:
http://mathworld.wolfram.com/StereographicProjection.html
http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Stereographic_projection.html
http://en.wikipedia.org/wiki/Stereographic_projection
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