The stereographic projection is a
conformal azimuthal
map projection, that is,

There is zero angular distortion in the immediate neighborhood of each
point.

All angles with their vertex at a particular central point are shown the
same as they are on the surface of the Earth.

The projection does not preserve area; areas on the map farther from the
central point will be relatively larger than similar areas closer to the
central point.
Along with several other azimuthal projections, the stereographic projection
also has the amazing property of being a true
perspective projection, with
the
point of perspective being on the surface of the globe.
Imagine a (spherical) world globe made of glass with the continents
etched on its surface. Now imagine a vertical projection screen
placed in such a way that one point on the globe touches one point on the
screen (i.e. the plane of the screen is tangent to the globe). Call this
point T.
Now place a light source directly on the globe at the point antipodal
to T. Call this point L. When you
turn on the light, it will shine through the globe, but the continents
etched on the globe's surface will cast shadows on the projection screen.
/

/


/

/______

_ _

/
\ 
//
\ 
/

Lo     +T
\

\\
/ 
\_
_/ 
______

\

\


\

\

The shadow continents on the projection screen will appear to be shaped
correctly, but they will also appear larger and larger the further the
distance from T. This distortion makes it pointless
to show more than one hemisphere with any one stereographic projection.
Many of the world maps published in the 16th and 17th centuries employed
the stereographic projection, with two separate hemispherical views. Today,
however, the stereographic projection is most often used for very small
scale views of the polar regions.
We can calculate coordinates for a stereographic projection with very
simple formulae. Like other developed
map projections, we start by calculating polar coordinates from latitude
and longitude, then transforming to x and y. For any one point,
r = 2R tan (c/2)
theta = Azimuth
where R is the radius of the Earth (or whatever).
If the point of perspective L is either of the poles,
c is the point's latitude, and Azimuth is
the point's longitude.
If L is away from the poles, we need to use spherical
trigonometry to calculate c and Azimuth.
Those formulae are, however relatively simple.