Map Projections for GIS

Map projections for GIS

J. M. Stroh, 2002

Read the document for background. I have placed one of the reviews at the beginning, in bold.

You need to know that most GIS data, especially for Washington State, comes in two datums:

North American Datum 1927 (NAD27). All older maps, many current maps and data sets.

North American Datum 1983 (NAD83). Many current maps and data sets.

Most GIS data come in three different grids:

Universal Transverse Mercator (UTM) in both datums.

State plane coordinate system (SPCS) in NAD27.

State plane coordinate system (SPCS) in NAD83.

Note the SPCS has a different definition in the two datums.

Evergreen campus GIS data comes in a local TESC grid as well as the SPSC grid (both datums).

What do you need to remember about the basic projections for GIS applications?

All maps in an application must have the same projection, and have the same ellipsoid (datum). In the 1980s the North American Datum of 1983 replaced the North American Datum of 1927.

Only a few projections are used for most mapping in the USA. These are:

Albers equal-area (conical, equal area, small-scale state maps, National Atlas)
Lambert conformal conic (conical, conformal, small-scale state maps, detailed state plane coordinate system for states long in E-W direction, topographic maps). See table 1 below for Washington State parameters.
Transverse Mercator (cylindrical, conformal, detailed state plane coordinate system for states long in N-S direction, topographic maps).
Universal Transverse Mercator (cylindrical, conformal, world wide, except poles, many applications for base maps, base for many digital GIS products like Digital Elevation Models).
Polyconic (conical, not conformal, equal-area, or equidistant, obsolete but used for older large-scale topographic maps).
Latitude-longitude still the basic coordinate system.

What to do if the source of GIS information comes in different projections, but with the same ellipsoid? Most GIS software can convert from one projection to another, ArcView without extensions has limited capability though. Be aware that the shape of a map will change with the new projection, so maps after conversion might not fit together.

Try to find information all in the same projection to begin with.

If you have an application that covers a large area, join all maps in the old projection, then convert to the new projection.

Remember a GIS will either not function if information uses more than one projection, or it will give bogus results.

Over very small areas, a few km by a few km, differences in the projection used will produce error smaller than errors in the source if paper maps are used. This does not apply to high accuracy surveys.

Table 1. Washington State Plane Coordinate System parameters.

Washington / Std Parallel / Std Parallel / Origin* / Origin*
first / second / Long. / Lat.
North (1) / 47 30 / 48 44 / 120 50 / 47 00
South (2) / 45 50 / 47 20 / 120 30 / 45 20

*At origin x = 2,000,000 ft.; y = 0 ft for NAD27

*At origin x = 500,000 m.; y = 0 m for NAD83

Map projections and coordinate systems play a critical role in GIS. How do we locate information in space? The earth has approximately a spherical shape, and more accurately the shape of an oblate spheroid. Over the last two centuries geodesists have developed ever more accurate models for the shape of the earth. Note: Europeans generally use the term spheroid for the oblate spheroid, while in the USA the term ellipsoid is used.

Several ellipsoids are in use world wide. In the USA the Clarke 1866 ellipsoid was used as the base for all mapping until recently. Currently the GRS 80 (global reference system) and the WGS (world geodetic system) 84 ellipsoids are used for all USA mapping, and globally. GRS 80 and WGS 84 are functionally equivalent. See table 1 for a comparison. This produces no end of grief for GIS users. New data comes in a different ellipsoid from older data so you need to understand this and how to convert from one to the other. More on this later.

Table 1. Ellipsoids in use by USA. Source. Defense Mapping Agency Technical Manual 8358.1: Datums, Ellipsoids, Grids, and Grid Reference Systems, 1998.

Name / Date / Equatorial
Radius,
a, m / Polar
Radius,
b, m / Flattening
f / Use
Clarke 1866 / 1866 / 6,378,206.4
exact / 6,356,583.8 / 1/294.9786982 / N. America
Philippines
GRS 80 / 1980 / 6,378,137
exact / 6,356,752.31 / 1/298.257222101 / Adopted USA global
WGS 84 / 1984 / 6,378,137
exact / 6,356,752.31 / 1/298.257223563 / NASA, US Military global

Projecting coordinates from the sphere or ellipsoid to a flat plane requires a developable surface. There are three developable shapes, the plane, the cone and the cylinder. For the cone and cylinder if you cut one side you can lay the surface out flat. The plane is already flat. We have planar, conic (or conical), and cylindrical (or cylindric) projections. Each has advantages and disadvantages.

Going from the sphere or ellipsoid to a flat surface introduces distortion in the new coordinates, no exceptions. In all cases for GIS the transformation goes from angular coordinates, latitude and longitude, to X-Y, Cartesian, coordinates. Different projections minimize distortion in different ways. Note that the distortion varies for every location on the map.

The conformal projection conserves shape, angular relationships, while area and distance get distorted.

The equal-area projection conserves area at the expense of shape and distance.

The equidistant projection conserves distance at the expense of shape and area.

In all cases the scale distortion can be calculated for any point on the map. Many projections attempt to minimize two or more of these distortions. They are not conformal, equal-area, nor equidistant.

The amount of distortion varies with scale. Projecting coordinates for a very small area introduces very limited distortion, while projecting coordinate for large parts of the earth generates very severe distortion. Remember that distortion varies in type and severity with the projection. For example the standard Mercator projection is a conformal, cylindrical, projection with the cylinder wrapped around the equator. It has no distortion on the equator, and minimal distortion in area a distance close to the equator. Far north of the equator area and distance get severely distorted. Greenland looks larger than most of south America for example. Using secant projections helps minimize distortion over a larger area.

What projection works best for what application?

The conformal obviously works best for applications requiring angular relationships, like navigation. It is the projection of choice for most surveying applications.

The equal-area projection works best in applications where area is important such as land resource assessment. The US National Atlas uses an equal-area projection.

The equidistant has fewer applications.

What do you need to remember about the basic projections for GIS applications?

All maps in an application must have the same projection, and have the same ellipsoid (datum). In the 1980s the North American Datum of 1983 replaced the North American Datum of 1927.

Only a few projections are used for most mapping in the USA. These are:

Albers equal-area (conical, equal area, small-scale state maps, National Atlas)
Lambert conformal conic (conical, conformal, small-scale state maps, detailed state plane coordinate system for states long in E-W direction, topographic maps).
Transverse Mercator (cylindrical, conformal, detailed state plane coordinate system for states long in N-S direction, topographic maps).
Universal Transverse Mercator (cylindrical, conformal, world wide, except poles, many applications for base maps, base for many digital GIS products like Digital Elevation Models).
Polyconic (conical, not conformal, equal-area, or equidistant, obsolete but used for older large-scale topographic maps).
Latitude-longitude still the basic coordinate system.

Try to find information all in the same projection to begin with.

If you have an application that covers a large area, join all maps in the old projection, then convert to the new projection.

Remember a GIS will either not function if information uses more than one projection, or it will give bogus results.

Review.

Earth is approximately spherical, and very close to an ellipsoid.

Several ellipsoids are used in different “datums.”

A projection “flattens the earth” by going from a curved surface to a plane.

A projection requires a developable surface, plane, cone, or cylinder.

A projection always introduces distortion.

Conformal projections minimize shape (angle) distortion.

Equal area projections minimize area distortion.

Equidistant projections minimize distance distortion.

GIS applications must use the same ellipsoid and projection, unless the area covered is very small.

Ellipsoids and projections in common use in the USA are:

Clarke 1866, GRS 80, and WGS 84 ellipsoids.
Latitude and longitude
Lambert conic conformal projection.
Albers equal-area projection.
Transverse Mercator projection.
Universal Transverse Mercator projection.

References used for this document.

Maling, D.H. 1992. Coordinate Systems and Map Projections, 2nd Ed. Pergamon Press. Oxford.

Robinson, Arthur H., Randall D. Sale, Joel L. Morrison, and Phillip C. Muehrcke. 1984. Elements of Cartography, 5th Ed. John Wiley & Sons. New York.

Snyder, John P. 1993. Flattening the Earth: Two Thousand Years of Map Projections. University of Chicago Press. Chicago, IL.

------. 1987. Map Projections--A Working Manual. U.S. Geological Survey Professional Paper 1395. U.S. Government Printing Office. Washington, D.C.

Web Sources:

1. Yahoo list of Cartography Web Sites

2. National Geographic Society

3. Hunter College's Geography Department (CUNY)

4. The Geographer's Craft Project , Department of Geography, University of Texas at Austin

Some illustrations.

The Lambert conformal conic. Secant, used for the contiguous state maps, which all fit together.

The Mercator projection.

Conformal

Normal aspect – wrapped on Equator

Any straight line is a rhumb line (a bearing for direct navigation).

Easy to construct, used for many years for navigation

Does not show equal angles.

Transverse aspect – cylinder is wrapped on any meridian, North-South (at 90 degrees to equator, ie. Transverse to equator). This is the base for many state plane systems in West

Other Mercators:

Oblique Mercator – wrapped around pole.

Peters projection

Equal area

“Evergreen politically-correct projection of the world”