Map Projections: Properties: Preserving Shapes

Map Projections

Useful Map Properties: Shapes

Are Shapes Preserved?

Loxodrome (blue) and geodesic (red) in Mercator map

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).
Any azimuthal stereographic or Mercator maps are conformal.

Loxodromes and geodesics

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.

azimuthal equidistant map

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,

a navigator could follow a hybrid procedure:

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

Corresponding map in "Lagrange" projection

The same great circle (this time covering 360°) and loxodrome in a "Lagrange" conformal map