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ScottMcMichael edited this page Nov 14, 2013 · 1 revision

Map Projections

Map projections are necessary to represent a planetary surface (a sphere) with a map (a plane). All map projections distort the surface in some way. Depending on the projection, shape, area, direction, bearing, or distance (or some combination) may be preserved, but no projection can preserve all features at the same time.

Equirectangular

Alternate names:

  • plate carrée
  • equidistant cylindrical projection
  • geographic projection
  • simple cylindrical
  • longitude-latitude projection

The equirectangular projection is the simplest projection of all. Longitude and latitude are directly mapped onto x and y: x = longitude, y = latitude (hence the alias of "longitude-latitude projection"). If you are given a DEM or map that doesn't list its projection, you can probably assume that it is equirectangular.

DEMs from USGS in the equirectangular projection will give you:

  • Minimum / Maximum Latitude
  • Easternmost / Westernmost Longitude

These measurements define the transformation between pixel coordinates and lat-long coordinates:

  • latitude = [(Maximum Latitude - Minimum Latitude) / (Image Height)] * PixelY + Minimum Latitude
  • longitude = [(Westernmost Longitude - Easternmost Longitude) / (Image Width)] * PixelX + Easternmost Longitude

Also note that the "Map Resolution" given is:

  • Map Resolution [pixel/degree] = (Image Height) / (Maximum Latitude - Minimum Latitude)

More info: http://en.wikipedia.org/wiki/Equirectangular and http://www.progonos.com/furuti/MapProj/Normal/ProjCyl/projCyl.html#Rectangular

Mercator

The Mercator projection is a cylindrical projection that is useful because it represents lines of constant course as straight segments.

Google Maps currently uses a Mercator projection for its map images. Despite its obvious scale distortions at small scales, the projection is well-suited as an interactive world map that can be zoomed seamlessly to large-scale (local) maps, where there is relatively little distortion due to the projection's conformal nature. (See http://en.wikipedia.org/wiki/Mercator_projection#Uses)

More info: http://en.wikipedia.org/wiki/Mercator_projection and http://www.progonos.com/furuti/MapProj/Normal/ProjNav/projNav.html#Mercator

Other resources

'''READ THIS:''' Map projections and ISIS (includes interactive demos!): http://isis.astrogeology.usgs.gov/IsisWorkshop/index.php/Learning_About_Map_Projections

A very complete book by USGS on map projections (includes detailed mathematics) (pdf): http://pubs.er.usgs.gov/usgspubs/pp/pp1395

Map projection list by USGS (with pictures and comparisons): http://egsc.usgs.gov/isb/pubs/MapProjections/projections.html

Map projection list (with pictures): http://www.progonos.com/furuti/MapProj/Normal/ProjTbl/projTbl.html

Catalog of projections & parameters for GeoTIFFs (includes Proj.4 parameters): http://www.remotesensing.org/geotiff/proj_list

Coordinates

Latitude

The equator is used as a the point of reference for latitude. For a spheroid that is slightly flattend by its rotation, there are a variety of definitions of latitude. The most common definitions are geographic and geocentric latitude, called "planetographic" and "planetocentric" latitude for planets other than earth. Most maps of Mars since 2002 use planetocentric coordinates.

For a great diagram of planetographic vs planetocentric latitude, see: http://en.wikipedia.org/wiki/Latitude#Geocentric_latitude

Longitude

Since there is no intrinsic reference point for longitude, one must be chosen and agreed upon (called the prime meridian).

  • On the earth in WGS84 (the latest revision of the World Geodetic System), the prime meridian is the IERS Reference Meridian: http://en.wikipedia.org/wiki/Prime_Meridian#IERS_Reference_Meridian
  • The prime meridian of the Moon lies directly in the middle of the face of the moon visible from Earth and passes near the crater Bruce.
  • The prime meridian of Mars is defined by the crater Airy-0.

ECEF (Cartesian coordinates)

From http://en.wikipedia.org/wiki/ECEF (also has a great diagram):

ECEF stands for Earth-Centered, Earth-Fixed, and is a Cartesian coordinate system, and is sometimes known as a "conventional terrestrial" system. It represents positions as an X, Y, and Z coordinate. The point (0,0,0) denotes the mass center of the earth, hence the name Earth-Centered. The z-axis is defined as being parallel to the earth rotational axes, pointing towards north. The x-axis intersects the sphere of the earth at the 0° latitude, 0° longitude. This means the ECEF rotates with the earth around its z-axis. Therefore, coordinates of a point fixed on the surface of the earth do not change, hence the name earth-fixed.