Illustration by
Rubens for "Opticorum libri sex philosophis juxta ac mathematicis utiles", by François d'Aiguillon. It demonstrates how the projection is computed.
In geometry, a stereographic projection is a function that maps the points of a sphere onto a plane. The projection is defined on the entire sphere, except for one point, called the projection point.
Intuitively, the stereographic projection is a way of picturing a sphere as a plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a stereonet or Wulff net.
A simple example of such a projection, encountered in everyday life is the sun casting a shadow of a globe onto the ground.
Images for kids

An extension of the Wulff net, showing how meridians and parallels extend beyond the hemisphere

Stereographic projection of the world north of 30°S. 15° graticule.



Stereographic projection of the spherical panorama of the Last Supper sculpture by Michele Vedani in Esino Lario, Lombardy, Italy during Wikimania 2016

A Cartesian grid on the plane appears distorted on the sphere. The grid lines are still perpendicular, but the areas of the grid squares shrink as they approach the north pole.

A polar grid on the plane appears distorted on the sphere. The grid curves are still perpendicular, but the areas of the grid sectors shrink as they approach the north pole.

The sphere, with various loxodromes shown in distinct colors

Illustration of steps 1–4 for plotting a point on a Wulff net

Animation of Kikuchi lines of four of the eight zones in an fcc crystal. Planes edgeon (banded lines) intersect at fixed angles.

The rational points on a circle correspond, under stereographic projection, to the rational points of the line.

Stereographic projection is used to map the Earth, especially near the poles, but also near other points of interest.

A crystallographic pole figure for the diamond lattice in [111] direction

Use of lower hemisphere stereographic projection to plot planar and linear data in structural geology, using the example of a fault plane with a slickenside lineation