Mercator map: loxodrome or rhumb line in blue; part of a geodesic line or
great circle in red
A conformal (or
orthomorphic) map locally
preserves angles. Thus, any two lines in the map follow
the same angle as the corresponding original lines on the
Earth; in particular, projected graticule lines
always cross at right angles (a necessary but not sufficient
condition). Also, at any particular point scale is the same in all
directions. It does not follow that shapes are
always preserved across the map, as any conformal map includes
a scaling distortion somewhere (that is, scale is not
the same everywhere).
stereographic or Mercator maps are
A straight line drawn on a Mercator map connecting Campinas,
Brazil, to Seoul, South Korea is a loxodrome at a constant
angle of approximately 79°39' from any meridian. An
aircraft taking off from Campinas would easily land in
Seoul following this fixed bearing (disregarding factors
like traffic airlanes, wind deviation, weather, national
airspaces and fuel range; actual customary routes go westwards
but are in fact similar) along the whole trip.
However, that easy route would not be the most economical
choice in terms of distance, as the
geodesic line shows.
The same loxodrome and great circle in part of a polar
azimuthal equidistant map
The two paths almost coincide only in brief routes.
Although the rhumb line is much shorter on the
Mercator map, an azimuthal
equidistant map tells a different story, even though the
geodesic does not map to a straight line since it does not
intercept the projection center.
Since there is a trade-off:
following the geodesic would imply constant changes of
direction (those are changes from the current compass bearing
and are only apparent, of course: on the sphere, the trajectory
is as straight as it can be)
following the rhumb line would waste time and fuel,
use a protractor and read the bearings for each
navigate each segment separately following its corresponding
The same great circle (this time covering 360°) and loxodrome
in a "Lagrange" conformal map
Moving circles between globe and map
Suppose a set of concentric circles, with radiuses increasing in
1500km steps, centered on Campinas, Brazil. These are true
circles on Earth, which could demonstrate the theoretical range
of radio waves, airplanes, or missiles; they are represented here
in blue and are identical in all maps in this table. In each pair
of maps with blue shapes, the one on the left is a simple
aspect, while its right counterpart is usually
an oblique aspect centered on Campinas.
On the other hand, the orange lines were directly drawn as circles
on a map on the left column; their true shape on the globe is
presented on the right (they are the same curves only for each pair
of maps). There is only one projection and aspect
where both size and shape of circles are identical on globe and map.
Like all azimuthal projections, the
equidistant preserves the shape of any circle centered
on the map's center of projection, but not necessarily of others;
however, for those whose center does coincide
with the map's, the scale is also
preserved: on the second
map, notice how the radiuses are linearly spaced. Given
a proper aspect, this projection is the only correct tool
for graphically finding ranges on a flat map.
A Mercator map
is conformal, thus
preserving shapes locally but not globally. Scale changes quickly
towards the top and bottom of the map, especially vertically.
Also conformal, an
stereographic map preserves the shape of all
circles, even those not centered on the map. Nevertheless, their
scale is not preserved: they do not "grow" linearly, and those on
the first map, perhaps surprisingly, are not concentric.
A very simple design, the Plate Carrée is a particular case of
cylindrical projection. The scale is the same on the Equator
and all meridians (as measured on the equatorial aspect).
Therefore the width and height are identical for the "circles" on
the second map only.
exaggerate horizontal scales towards the top and bottom of maps,
if circles are naively drawn with a pair of compasses on a Plate
Carrée map, their correct shapes on Earth are pinched, as shown by
the azimuthal equidistant counterpart.
Transferring circles drawn on a Gall's orthographic to the real
globe yields shapes even farther from correct. Because scales
are stretched between 45°N and 45°S, and compressed
elsewhere, the true shapes are correspondingly deformed on the globe.
projection is pseudocylindrical
and equal-area; its poles are shown as lines, but with lesser
horizontal exaggeration than in cylindrical projections. On the other
hand, horizontal distortion depends on the longitude.
An interesting shape to study is the circumference, the
set of points at a fixed distance from a center, and
the circle, the set of points it encloses. Since both
scale and shape are often distorted, how faithfully are circles
drawn on a globe translated to a map? Or, conversely, should one
draw a circle on a map with a pair of compasses, which shape does
it actually represent on Earth? The circle could represent the
range of a radio station, or the autonomy of a vehicle. When
rangefinding, correct usage of map projections could mean the
difference between a successful trip and a mayday call.