Useful Map Properties: the Geodesic
What is the Shortest Path Between Two Points?
 |
| Equatorial Mollweide map with great
circle; graticule lines spaced 15° |
On any spherical surface, the shortest path between any two
points is part of a geodesic
line (also called geodetic, orthodrome
or great circle path) passing through the points and
centered on the sphere.
Except due to constraints like traffic patterns and weather
conditions, long-distance travellers like aircraft pilots
always seek for the shortest route. For that purpose, a
map showing great circles as straight lines is better
suited. Unfortunately, no map projection can show true
geodesics between any pair of points as such. The
following examples show in red a great circle connecting Campinas,
Brazil, and Tokyo, Japan.
The elliptical projection created
by Mollweide preserves area ratios but not directions.
Regions near the top and bottom suffer from greater distortion
the farther the distance from the map center. Notice how
map coordinates are translated prior to projection in order to
move the region of interest to a lesser-distortion area and to
emphasize the circular (on Earth) red path.
 |
| Equidistant cylindrical map with great
circle |
The equidistant
cylindrical projection does not preserve either areas or
directions. Only distances are preserved along the meridians
and the Equator. Sadly, most cylindrical projections are
probably chosen due to their neat rectangular lay-out rather
than any outstanding cartographic property.
The azimuthal family
of projections shows true directions from the center point
(azimuth) only. The azimuthal
stereographic projection is circle-preserving, as any
circle upon the sphere (every geodesic, parallel and meridian)
is still mapped to a circle; in particular, all geodesic lines
crossing the central point map to circles with infinite
diameter, i.e. straight lines. Unfortunately, it can show only
one hemisphere at a time. Another very important
azimuthal projection, the gnomonic, maps into
straight lines all great circles, even those not passing
through the central point, but can present even less than one
hemisphere. The azimuthal
equidistant can include the whole world and presents
true direction and distance to any point from the center
while suffering from lesser distortion near the map
periphery.
 |
 |
 |
| Stereographic map |
Gnomonic map |
Azimuthal equidistant map |
Projection distortion and unfamiliar shapes could make
difficult realizing the red lines in previous maps as actual
circles. The azimuthal
orthographic projection clearly shows the Earth's
sphericity as seen from a vantage point far away in
space. This closely mirrors the actual expedient of finding
a geodesic by applying a taut line against a globe.
 |
 |
 |
| The same great circle in
three different azimuthal orthographic views |
 |  |  |  |  | | www.progonos.com/furuti September 21, 2002 |