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.

• Forward equation: Lat,Long ===> x,y
• Inverse equation: x,y ====> Lat,Long

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:

• Area
• Distance
• Azimuths
• Shape
• Parallels evenly spaced
• Parallels-meridian right angle intersections

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:

• Conformal: shapes are correct; local angles about point are accurate; local scale about any point is constant. This is generally the most import characteristic; for military users, all NGA maps are conformal except those designed to show the entire world for general description.
• Equal area: areas correct everywhere
• Equidistant: distances constant
• Azimuthal: directions correct with respect to center of map

Special characters desired for some maps:

• Rhumb lines (constant direction) are straight
• Great circles (meridians): either straight, or circles
• Small circles (parallels): shown as circles

Geometric Projection types: distribute error differently depending on the geometry

• Cone
• Cylinder
• Plane 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.

Table of projection properties

Reference on projections:

• Snyder, J.P., 1987, Map projections -- a working manual: U.S. Geological Professional Paper 1395, 383 p.

Last revision 2/28/2022