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).
Any azimuthal
stereographic or Mercator maps are
conformal.

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 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,

- trace the geodesic on an azimuthal equidistant or gnomonic map
- break the geodesic in segments
- plot each segment's endpoints onto a Mercator map
- use a protractor and read the bearings for each segment
- navigate each segment separately following its corresponding constant bearing.

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.

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.

Copyright © 1996, 1997 Carlos A. Furuti