In Cartesian coordinates ( x, y, z) on the sphere and ( X, Y) on the plane, the projection and its inverse are given by the formulas The plane z = 0 runs through the center of the sphere the "equator" is the intersection of the sphere with this plane.įor any point P on M, there is a unique line through N and P, and this line intersects the plane z = 0 in exactly one point P′, known as the stereographic projection of P onto the plane. Let N = (0, 0, 1) be the "north pole", and let M be the rest of the sphere. The unit sphere S 2 in three-dimensional space R 3 is the set of points ( x, y, z) such that x 2 + y 2 + z 2 = 1. Stereographic projection of the unit sphere from the north pole onto the plane z = 0, shown here in cross section He used the recently established tools of calculus, invented by his friend Isaac Newton.ĭefinition First formulation In 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal. įrançois d'Aguilon 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 star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy. It is believed that already the map created in 1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Roze (1542), Rumold Mercator (1595), and many others. In the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. The term planisphere is still used to refer to such charts. One of its most important uses was the representation of celestial charts. Planisphaerium by Ptolemy is the oldest surviving document that describes it. It was originally known as the planisphere projection. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians. It demonstrates the principle of a general perspective projection, of which the stereographic projection is a special case. Illustration by Rubens for "Opticorum libri sex philosophis juxta ac mathematicis utiles", by François d'Aguilon.
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