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.

Ste`re*o*graph"ic (?), Ste`re*o*graph"ic*al (?), a. [Cf. F. st'er'eographique.]

Made or done according to the rules of stereography; delineated on a plane; as, a stereographic chart of the earth.

Stereographic projection Geom., a method of representing the sphere in which the center of projection is taken in the surface of the sphere, and the plane upon which the projection is made is at right andles to the diameter passing through the center of projection.

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