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.

Two completely equivalent visualizations of the
azimuthal orthographic geometry

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.

The point P at 75°E 55°N mapped by the polar aspect of the azimuthal
projection: the Earth resting on the projection plane, and the
the point already projected.

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.

The resulting map.

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.