The purely geometrical azimuthal orthographic projection can be entirely visualized as a physical model. The easiest case, a polar aspect, is presented here for the Northern hemisphere.
Suppose the Earth lying over a plane parallel to the
Equator. Light rays emanating from a point infinitely far away
on the north-south polar axis pierce a semitransparent northern
hemisphere and "paint" its features onto the plane. The southern
hemisphere is considered completely transparent.
Alternatively, imagine an observer infinitely far away on that axis. Parallel light rays emanating from Earth's surface hit an intervening plane perpendicular to the rays, where the image is developed. Since all rays are parallel, i.e., the perspective is "cylindrical", the plane may or may not be tangent to the sphere without affecting the result. Anyway, only one hemisphere can be seen at any time.
Other perspective azimuthal projections can be created just by changing the light source's position. For a practical presentation, a cartographer could conceivably paint coastlines and other geographical features onto a glass globe or bowl and, using either reflected sunlight or a strong flashlight at a convenient distance, literally project the globe shadows on a wall, thus creating a variety of azimuthal projections like the orthographic and stereographic.
Geometrically, the azimuthal orthographic projection can be imagined as converting polar coordinates to a point in 3-D cartesian space, then flattening it, i.e., ignoring one coordinate. The forward equations for a point are easily derived in polar coordinates:
Only points with (for the north polar case) or
(south polar) are visible.
Converting to Cartesian coordinates,
The conversion of the forward equations to inverse mapping is straightforward.
A more general aspect, either equatorial or oblique, can be obtained by first rotating Earth coordinates in 3D space, then applying the polar equations. Before digital computers became generally available, cartographers drafted general orthographic maps by first plotting the graticule of a polar map, then using it to place the parallels on an equatorial aspect; finally, both are used to create the oblique version by locating key graticule intersections marked with sets of parallel lines.