The Stereographic projection, particularly on hemispheres as shown here:
was very common on antique maps of the world, and here is an example of its use in an old atlas:
Another common use for the Stereographic projection was to present the world as "Land" and "Water" hemispheres, which here I approximate in whole degrees, centering the land hemisphere on 2 degrees West longitude and 47 degrees North latitude:
It is sufficiently important that I feel it deserves to also be depicted in a more pictorial version, as produced by the program G-Projector:
This is a map projection that was known to the ancient Greeks. Like the Mercator projection, it is conformal. In addition to being used for maps, it is also sometimes used in crystallography.
It has the interesting and unusual property that all circles on the globe, small or great, appear as circles on the projection. This simplifies drawing it.
The Stereographic is constructed by projecting the globe onto a tangent plane from a "light source" located at the point on the globe antipodal to the plane's point of tangency - the spot where the plane on which one is drawing the map touches the globe.
The following diagram:
illustrates this principle.
Why is the Stereographic projection conformal?
This diagram reveals the reason.
The line going from the center of the globe to a point on its surface makes twice the angle that the line of projection does from the normal to the center of the projection. This is because an isosceles triangle is formed, having two equal arms, one from the center of the globe to the projection point, and one from the center of the globe to the location on the globe being placed on the map.
Because of that, the angle the surface of the globe makes to the line of projection, and the angle the surface of the map makes to the projection, are the same, although in opposite directions, and so the foreshortening of both from the viewpoint of the projection ray is the same.