Deducing the Azimuthal Orthographic Projection
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.
|
|
| First construction method |
Second method |
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 "draw" its features onto the plane. The southern
hemisphere is considered completely transparent.
Alternatively, one can imagine an observer infinitely far away
on that axis. Parallel light rays emanating from Earth's
surface hit an intervening plane (or disc) perpendicular to the
rays, where the image is developed. Since all rays are parallel
("cylindric perspective"), 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.
Geometrically, the orthographic projection can be imagined as
converting polar coordinates to 3-D cartesian, then flattening
it (ignoring one coordinate). The projection formulae are
easily derived in polar coordinates:
r = R cos 
, 0 <=
<= 
/2, 
= 
, 
=
r
 |
|
On the left, Earth set for a polar azimuthal orthographic projection,
resting on the projection plane; distance of projected point to center
of map depends only on latitude. On the right, Earth and projection
plane viewed from above; angle of point around center of map equals
longitude |
Only points with >= 0 (for the north polar case) or <= 0
(south polar) are visible.
Converting to
Cartesian coordinates,
| x = cos = R
cos cos |
| y = sin = R
cos sin |
Conversion of the equations to inverse mapping is straightforward.
|
| North polar aspect of azimuthal orthographic
map; parallel spacing manifests the distortion pattern, especially
near the rim |
A more general aspect, either equatorial or
oblique, can be obtained by first rotating Earth coordinates in
3D space, then applying the previous equations. Early
cartographers designed general orthographic maps by first
plotting the polar aspect, then locating key grid points by a
set of parallel lines drawn from the polar map.
Two other important azimuthal projections are 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 reflected sunlight or a strong flashlight
at a convenient distance, project the globe shadows on the wall,
thus creating a variety of azimuthal projections like the
orthographic and stereographic.
 |  |  |  |  |  | | www.progonos.com/furuti January 28, 2003 |