In geometry, the stereographic projection, usually denoted by , 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.
Intuitively, then, the stereographic projection is a way of picturing the sphere as the 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.
In Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas
Let us show you an example making use of the projection. We assume verifies the following PDE
Then the transformed function , to be exact , given by
satisfies the following PDE
where denotes the Laplace-Beltrami operator with respect to the standard metric on .
Similarly, if function verifying the PDE
then a new function given by
will satisfy the following PDE
More general, PDE
In conclusion, using the stereographic projection we can transfer some geometric problems on sphere to ones in the whole space .