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Lat/Lon, Mercator's, Lambert's (Oh My!)

A grunt's introduction to map projections

by Tom Russo

This article originally appeared in Volume 9, Issue 8 of Lost... and Found.

Lat/Lon, Mercator's, Lambert's (Oh My!)

A grunt's introduction to map projections

[Note: While the text of this article is my own, most of the projection images in this article are copyright 1994 by Professor Peter H. Dana of the department of geography at the University of Texas at Austin, and are used with his permission from Map Projections lecture notes on the Geographer's Craft Project web site from the University of Colorado at Boulder. Please visit those web sites for more detailed information and references on this material.]

[For Further Reading: additional references and hyperlinks are available on Cibola's UTM Converter page.]

The purpose of this article is to introduce some basic concepts of map production, specifically the concept of projection. It is my hope that by the end of the article you'll have a slightly deeper understanding of these pieces of paper we play with every so often, and a clearer picture of the meanings of the various coordinate systems we use in the field.

The Problem

Roughly speaking, the problem addressed by projection is to deform pieces of a nearly spherical surface, say, like this one:
Photo by T. Russo
into a planar surface that you could fold up, stuff in your pack, and use for navigation. There are many ways to do that, and each method has its advantages for specific applications.

Latitude/Longitude: the Geographic coordinate system

Before diving into the mapping of the surface of the Earth onto a plane, let's recap the Latitude/Longitude system. Refer to the figure below (Figure 2): Copyright Peter H. Dana

The Latitude/Longitude (or "lat/lon") system is based on angles from two specific reference planes cutting the Earth. The circles on the surface of the Earth cut by these two planes are called "Great Circles" because they have the same diameter as the Earth itself. The "Prime Meridian" is the great circle that passes through the north and south poles, and through Greenwich, England. This meridian is given the arbitrary coordinate of 0 degrees Longitude. The circle we call the "Equator" is given the coordinate of 0 degrees Latitude.

Ignoring the fact that the Earth is not actually spherical (and thereby sweeping the difference between "geodetic" and "geocentric" latitude under the carpet), one can define a "polar" coordinate system based on these reference planes. Referring to Professor Dana's graphic below, the longitude is the angle marked as "theta" (the Greek letter that looks like an "O" with a horizontal bar), and the latitude is the angle "phi" (circle with vertical line through it). The Equatorial plane is the "X-Y" plane in this figure, and the plane in which the prime meridian lies is the "X-Z" plane.

Copyright Peter H. Dana

To avoid negative coordinates, longitudes always have value between 0 and 180 degrees and are given an additional designation of "West" or "East" depending on their position relative to Greenwich. Similarly, latitudes are always between 0 and 90 degrees, and are given an "North" or "South" designation based on their position relative to the Equator.

Unprojected maps

The simplest way to make a planar map from the (nearly) spherical Earth is to use the latitude and longitude coordinates as simple Cartesian coordinates, and plot the coordinates of map features on graph paper. This is called an "unprojected coordinate system," "Equidistant Cylindrical projection" or "Plate Carre" and it can be useful sometimes --- one popular APRS program does just this, transforming maps in projected coordinate systems into lat/lon and then plotting them along with untransformed lat/lon coordinates of APRS stations. Here's what most of North America looks like in an unprojected lat/lon map (notice the New Mexico State Police districts map in red, and wilderness areas in green).

Unfortunately, the unprojected lat/lon maps suffer from serious deficiencies: scale, area, and shape are all distorted. The distortion is worst near the poles, but is present everywhere. Since we often need maps that reflect accurately some quality of the real world, we need to find a different way of preparing them for those uses.

Map Projection

The mathematical process of mapping a curved surface onto a plane is called "projection." It is simplest (though not accurate) to imagine the process as if one were to place a light bulb at the center of a globe, and shine the light through the globe ("project" it) onto a map surface. One can imagine several ways to do this, but here are two of the more common projection types:

For each type of projection (conic, cylindrical, transverse cylindrical), there are many ways of making the mathematical mapping of details onto the projection surface, but some distortion always results from the process. If the projection is such that the distance scale at any point is the same in any direction, the projection is called "conformal." If the projection is such that all mapped areas have the same ratio to their area in the real world, the projection is called "equal area." A mapping can not be both conformal and equal area.

Mercator's Projection

Prior to the 1970s, one of the most common world maps you could find was a Mercator Projection world map (there was usually one in every primary school classroom). Mercator's projection is a cylindrical conformal projection. The Mercator projection suffers from extreme distortion the farther you get from the Equator, and this is why such a map shows Canada as if it occupies a huge fraction of the planet. Below is an example of a Mercator projection of continent coastlines (taken from the USGS publication "Cartographic Projection Procedures for the UNIX Environment"):

Note how the lines of latitude become farther and farther apart as one gets away from the Equator. The mapping being conformal, in any region of the map the scale is constant in all directions, but you can see how the scale must be different at the top of the map than it is at the Equator.

The primary feature that makes Mercator's projection so useful is that a straight line drawn between any two points is a loxodrome or rhumb line --- fancy words meaning "lines of constant true bearing." This is not in general true of any other map projection, and makes it phenomenally convenient for low-tech maritime navigation: draw a straight line between origin and destination, measure the angle that line makes with the meridians of longitude, and follow that true course --- you'll eventually get there. None of the other projections discussed here have that property, even though we tend to assume so when working on large scale maps in UTM.

Transverse Mercator Projection

The Transverse Mercator projection is just like the Mercator projection, only the circle of tangency is a meridian of longitude (the "central meridian" of the projection) instead of the Equator. A transverse Mercator projection is subject to all the same distortion of a Mercator projection, turned on its side. What makes it convenient is that the distortion is minimized near the central meridian, and maps can be prepared for each area using a central meridian close enough that the distortion is not so important.

For illustration, here is a figure of the Western hemisphere, in Transverse Mercator projection with central meridian of 90 degrees West:

Note how the distortion increases dramatically as one gets farther from the central meridian. In this figure, lines of longitude are drawn every fifteen degrees. Note the shape of the region inside the first pair of longitude lines. We'll return to that later.

Straight lines in Transverse Mercator projection are NOT rhumb lines.

The Universal Transverse Mercator Projection

The Universal Transverse Mercator (UTM) projection is just a specific use of the Transverse Mercator projection. UTM "zones" of six degree width are defined, and maps of regions inside those zones are prepared in Transverse Mercator projection using the meridian at the center of the zone. The width of the zone is small enough that distortion is within reasonable limits --- in fact one could use a given UTM projection for a zone extending as far as 4 degrees from the central meridian (a fact sometimes used for UTM projected maps that straddle zone boundaries). If you refer back to the image of the Transverse Mercator projection of the Western hemisphere, the 30 degree swath nearest the central meridian wasn't that badly distorted --- so a six (or even eight) degree swath must be even less distorted. The distortion is not significant for our purposes over the span of a seven and a half minute (i.e. one-eighth of a degree) USGS quadrangle map within that zone.

The first UTM zone (zone 1) extends from 180 degrees West longitude to 174 degrees West, with a central meridian of 177 degrees West. UTM Zone 13 (in which Albuquerque lies) has a central meridian of 105 degrees West longitude, and extends from 108 degrees West to 102 degrees West. Below is a figure showing the UTM zones and the alphabetic designators used within them.

Copyright Peter H. Dana

Note, by the way, that this image of UTM zones is an unprojected map of the world. Observe how it compares to the map of the world in Mercator projection and the map of the Western Hemisphere in Transverse Mercator projection.

While you're looking at the UTM zone chart, notice that there are in fact some exceptions to the rule that zones are 6 degrees wide. Look at UTM zone 32V, and zones 31X-37X, for example. This choice is made so that certain countries be mapped using one specific projection instead of straddling a zone boundary. That close to the North pole, widening a zone doesn't really lead to severe distortion of features, so this makes sense.

UTM is most appropriate for mapping regions with more North-South extent than East-West extent.

Lambert Conformal Conic projection

I include this projection here only because it is commonly used for aeronautical charts. We don't use them a lot in ground SAR, but those of you who are ICS section chiefs may well encounter them on air missions. The Lambert Conformal Conic projection is a secant conic projection, and you will find the projection parameters ("standard parallels") marked on the title page:

One might choose Lambert Conformal Conic for a map's projection when the area represented by the map has more East-West extent than North-South extent.

Polyconic Projection

This one I mention only in passing, primarily because very old USGS maps use it. For example, the Cubero, NM quad you can download in GEOTIFF format from sar.lanl.gov was created in 1957 and photorevised in 1971 --- it shows in the bottom left corner the following information:

(Note: the GEOTIFF file itself has been reprojected into the UTM projection by USGS --- the distortion created by transforming the digital image is apparent even in the tiny image above --- it explains why all the text has been rotated.) Even though the map projection used in these old maps has no bearing on how we use them (there is still both a UTM and lat/lon grid drawn on the map that we can use), it is important to realize that the map was prepared differently than others in the state. If you are working on a search in an area that straddles the boundary of one whose USGS map hasn't been updated recently, you might find not only that you're going to have incompatibility between map datum (NAD27 vs. NAD83), but also map projection. In this case, butting the two maps together and trying to tile them will not serve you well!

The polyconic projection is a compromise projection intended to minimize all types of distortion in the map, but it is neither conformal nor equal-area: no type of distortion is truly eliminated. Maps made in polyconic projection cannot be tiled well as these distortions begin to show up even when three maps are tiled side-by-side. The USGS used the polyconic projection in maps produced from around 1879 to about 1957. Even after the USGS stopped using polyconic projection, they continued to label some maps as being in that projection. (Reference: Snyder, J.P.: Map Projections used for large-scale quadrangles by the U.S. Geological Survey).

Conclusion

While we're familiar with the term "UTM" and use it as the name of a coordinate system, it is important understand that the coordinate systems we use is based on the projection used to prepare our maps. It is also important to understand the nature of the distortion created by map projection and how this distortion limits the utility to particular applications.

I have mentioned map coordinate systems only briefly. In a future article I will present a more technical discussion of the UTM coordinate system and its relation to latitude/longitude, and also introduce some other odd-ball coordinate grids that are present (and usually ignored by SAR grunts) on every USGS topo map.

References:

If the subject of map projections intrigues you, please consider further reading of definitive sources. I recommend: Map Projections: A Working Manual, Map Projections used for large-scale quadrangles by the U.S. Geological Survey, and An Album of Map Projections, all of them USGS publications authored by John Snyder. You might also wish to read a fun little book entitled "How to Lie with Maps" by Mark Monmonier.

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