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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
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:

a navigator could follow a hybrid procedure:
  1. trace the geodesic on an azimuthal equidistant or gnomonic map
  2. break the geodesic in segments
  3. plot each segment onto a Mercator map
  4. use a protractor and read the bearings for each segment
  5. 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
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 is identical on globe and map.
Azimuthal equidistant map Azimuthal equidistant map centered on Campinas
Like all azimuthal projections, the azimuthal equidistant preserves the shape of any circle centered on the map's center of projection, but not necessarily of others (above left); however, for those whose center does coincide with the map's, the scale is also preserved: on the map on the right, 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.
Mercator map Mercator map centered on Campinas
A Mercator map is conformal, thus preserving shapes locally but not globally. Scales change quickly towards the top and bottom of the map, especially vertically.
Stereographic map Stereographic map centered on Campinas
Also conformal, an azimuthal stereographic projection preserves the shape of all circles, even those not centered on the map (left). Nevertheless, their scale is not preserved: they do not "grow" linearly, and those on the left, perhaps surprisingly, are not concentric.
Equidistant cylindrical map Equidistant cylindrical map centered on Campinas
A very simple design, the Plate Carrée is a particular case of the equidistant 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 right only.
Because all cylindrical projections 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 map on the right. Equidistant cylindrical map Azimuthal equidistant map centered on Campinas
Equal-area cylindrical (Gall-Peters) map Equal-area cylindrical map (Gall-Peters) centered on Campinas
A particular case of Lambert's equal-area cylindrical projection, Gall's orthographic (also known as Gall-Peters or "Peters") projection preserves areas but strongly distorts shapes, with vertical scale changing very nonuniformly.
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. Equal-area cylindrical map Azimuthal equidistant map centered on Campinas
Eckert VI  map Eckert VI map centered on Campinas
Eckert's sixth 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. Eckert VI map Azimuthal equidistant map centered on Campinas
Goode homolosine map Fuller Dymaxion(TM) map
Most interrupted maps are split at specific lines depending on the mapmaker's priorities; therefore changing the aspect usually either robs the map's purpose or demands a whole new set of interruptions. Here are normal aspects of Goode's interrupted homolosine (left) and Fuller's Dymaxion™ maps.

Circles on Earth and on Maps

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.


Work in Progress



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Copyright © 1996, 1997 Carlos A. Furuti