Map Projection

The globe is not the ideal map: it is too small to see any amount of detail, and you can't get a straight edge or protractor on it to measure distances or angles. The central problem in cartography thus becomes flattening a round earth onto a flat piece of paper or a computer screen.

The earth's shape is really extremely close to spherical, but precise maps require a shape model or datum.

A projection is a mathematical equation or series of equations to take a three dimensional location on the earth (even when given in the seeming two dimensional form of latitude and longitude, earth coordinates have an implicit third dimension, the earth radius) and provide two dimensional coordinates to plot on a paper or computer screen.

Without the use of a chart table and tools, accurate plotting with lat/long coordinates is not easy.  The x,y cartesian coordinates make plotting much easier, even if the xy coordinates do not appear on the final map.  In other cases, such as the UTM/MGRS coordinates used by the military's ground forces, the cartesian coordinates appear much more prominently on the map than the lat/long graticule.


Example for the Mercator Projection

The use of a single R term indicates this is a spherical projection.
The inclusion of the terms a and e indicate this is an ellipsoidal projection. 

Note the much greater inclusion of trigonometric and log/exponents, which are expensive (slow) to compute.


A Perfect projection would preserve:

You can't get it all (except very large scale maps of small areas when the earth's curvature can be ignored, and they only generally get one thing perfect, and the others are "close" enough that you cannot see the distortion.)

Projection Terms:

Special characters desired for some maps:

Geometric Projection types: distribute error differently depending on the geometry

Geometric Projection types: distribute error differently depending on the geometry

  • Plane--a plane is a cone whose apex and base coincide
  • Cone
  • Cylinder--a cylinder is a cone whose apex is located infinitely far from the base

Take the geometric shape, project from the earth onto the shape, and then open up and flatten the shape. Different projections have different properties based on this geometric starting point and how the projecting works.






Diagrams from Wikipedia.



Grids: ease of plotting, with rectangular coordinates; may or may not coincide with lat-long. UTM best known example.

Album of projections.

Table of projection properties

Reference on projections:

Last revision 2/28/2022