Chapter 3
sponds to a geoid, the name for the shape with the most accurate correspon- dence to the actual oblate and irregular shape of the earth at a given time.
Projections have great importance for geographic information because most geographic information records the location of things and events on a two-dimensional coordinate system, called a “Cartesian coordinate system”
when the x and y axes intersect at right angles (see the right side of Figure 3.1). Polar coordinates, which record location in terms of one distance and an angle from a central point, are also used. Any projection of location from the round surface to a flat plane causes some form of distortion. This has important consequences for the accuracy of geographic information or maps and what you can do with a particular projection.
Traditionally, most books on cartography start by discussing projec- tions. Projections are one of cartography’s most important contributions to science and civilization. Projections are, and have been, the foundations for almost all representations of the earth or any part of the earth. Almost all geographic information also uses projections. The ancient Greek geogra- pher and astronomer Ptolemy invented several projections that were used by the Romans and by others for centuries afterward. Some centuries later, when European exploration and colonization commenced, because they were so important to the accurate determination of a ship’s location and showing geographical relationships between mother countries and colonies, projections quickly became an important mathematical activity. You can even think of the 400 years between Mercator’s publication of his global pro- jection in 1568 and 1968 as the “golden” years of projections. Although the choice of these years is somewhat arbitrary, it roughly coincides with the period of significant European colonization and ends soon after computers made the calculations for projections a much easier task. Before moving on to the concepts of projections, you should also know that while it is possible to record the location of things and events in three-dimensional coordinate systems, they are still rather uncommon in most of geography and cartogra- phy. They are very uncommon because of their relative complexity, the wide- Most projections of locations transform three-dimensional locations into two-dimen- sional locations.
spread use of two-dimensional coordinate systems, and the cost of trans- forming two dimensional coordinate systems. Chapter 4 will take a look at some of these systems, including their applications.
Key Concepts of Projections
Projections convert measured locations of things and events in three dimen- sions to two dimensions. Projections are important but also complicated because it is impossible using geometric or more complex mathematical methods to simultaneously preserve both the shape and the two-dimensional area of any three-dimensional object found either on the spherical surface of the earth, in the earth, or near the earth, when we depict it in a two-dimen- sional coordinate system. Each projection is an abstraction of the earth’s sur- face and introduces distortions that affect the accuracy of the geographic information or map. A projection starts with one of three representations of the earth’s irregular surface (geoid, ellipsoid, or spheroid) and converts it directly or through intermediary transformations to a f lat, or planar, trans- formation.
Choosing the right projection is important for controlling these distor- tions. Thankfully, choosing the right projection for a particular area is a task that has often been done by institutions and governments and made part of
World map from 1801 using a Mercator projection.
Fromwww.davidrumsey.com. Reprinted by permission of David Rumsey.
conventions or even laws that state what projection must be used for certain areas and activities (see Chapter 5). This is usually a good thing, but many institutions and governments require multiple projections.
Whatever you do with geographic information or maps, you need to know some projection concepts in order to understand projection distor- tions and their consequences. Some geographic information is stored in lati- tude and longitude coordinates and can be displayed or mapped on a flat screen or piece of paper, but these “unprojected geographic coordinates,” as they are usually called, have tremendous amounts of distortion when shown on a flat plane.
Four fundamental concepts are crucial to know when you use geo- graphic information and maps:
1. The earth is almost round, and always changing shape. Three models of the earth are used in making projections: sphere, ellipsoid, and geoid. A per- fectly round object, or sphere, is defined by the mathematical relationship between the center of the object and its surface, the radius. The surface of a sphere is a constant distance from the object’s center. This is the simplest model used in projections and is sufficient for geographic information and
Illustration from 1862 showing 15 projections.
Fromwww.davidrumsey.com. Reprinted by permission of David Rumsey.
maps of very large areas. However, because the spinning of the earth creates a centrifugal force that causes the earth to bulge at the equator and flatten at the poles, the distance from the center of the earth to any point on the equa- tor is greater than the distance between the center of the earth and the north or south poles. This more precise shape is known as an ellipsoid (but often called a spheroid) and comes much closer to describing the actual shape of the earth. It is accurate enough for most geographic information and maps of smaller areas. Because of different weights of material in the earth’s core, differences in magnetic fields, and movements of the earth’s tectonic plates, very detailed measurements of locations use a geoid for projections. A geoid
Abstractions of the earth used in making projections.
U.S. continental State–Plane Zones (NAD83). These zones are commonly used in the United States for state geographic information activities and are often defined by statute.
is the most accurate representation of the earth’s surface. It accurately describes the location of objects to a common reference at a certain point.
The difference between the sphere, ellipsoid, and geoid at any place can be as much as several hundred meters (yards). The ellipsoid and geoid models of the earth are defined and updated at irregular intervals. Should you become involved with very detailed and accurate measurements of location, you should also be aware that the geoid of the earth is constantly changing and locations recorded with an older geoid may not match a newer geoid.
(See Plate 1 for geoid undulations.)
2. A projection makes compromises. Every projection either preserves one projection property or makes some compromises between projection prop- erties. In either case, some projection properties are compromised by every projection. Because there are theoretically an unlimited number of projec- tions, it is important to organize projections by projection properties. Which projection is used in making geographic information or a map has much to do with how geographic characteristics and relationships are preserved. The four projection properties, along with the cartographic terms in parentheses for each, are:
Angles Preservation of the angles (including shapes) of small areas (conformal)
Areas Preservation of the relative size of regions (equivalent or equal area)
Distance Partial preservation of distance relationships (equidis- tant)
Direction Certain lines of direction are preserved (azimuthal) Most projections preserve area, although a large number are compromise projections, which means that they sometimes preserve area, but sometimes preserve shape. Usually compromise projections are used for showing the globe, but they can be used for smaller areas. All things considered, the pro- jections that preserve area are more common because people usually need maps of smaller areas where geographic relationships and area comparisons are very important. However, the projections showing the globe are signifi- cant because they are the only way for almost all people to see and under- stand the world. Global projections make very significant trade-offs between projection properties. One of the most common projections used for show- ing the entire world, the Mercator projection, is a classic case of how a pro- jection always trades off among projection properties. In the case of the Mercator projection, it preserves the shape and distance relationships of small areas, but only locally; it preserves lines of constant bearing; it fails to preserve area (the sizes of Greenland and Africa are greatly distorted); it par- tially preserves continuity, breaking Eurasia into two halves. These trade-offs mean that the Mercator projection is a good choice for representing small areas and large areas, but only for navigation.
3. Distortions will occur. Every projection, in making trade-offs between
the various projection properties, creates distortions. These distortions can be minimized by choosing a projection that corresponds as well as possible to characteristics of the area to be mapped and the known purposes and uses of the geographic information or map. Inappropriate and erroneous choice of projections can lead to significant errors and misrepresentations.
Since there are no rules for choosing optimal projections, you simply have to assess each projection individually and learn through practice and dis- cussion with other people what projection is best for a particular area, pur- pose, and use. In many places the projections of most geographic informa- tion and maps have already been determined. However, different people, institutions, and countries may use very different projections for the same area, requiring you to know the distortions that different projections cre- ate.
4. Geographic information from different projection should not be combined.
Geographic information is particularly prone to errors resulting from the combination of data from different projections. This also applies to maps, but since it is very time-consuming to trace two maps and overlay the trac- ings, in practice you should be most concerned with the consequences of combining geographic information from different projections, which is per- haps one of the easiest mistakes to make with GIS. Sometimes, although you may know the geographic information is for the same place, the combined
Example projections with their projection properties.
data is separated by a huge distance, possibly even many times the size of the earth. Sometimes—and this is why knowing the projection of geographic information is so important—the distances between geographic information objects can be minute, just a few inches or feet. However, because of differ- ences in projections, what may be minute differences in one place may be vast differences elsewhere.
Assessing projection distortions and determining the best projection for an activity and area remains a complex activity that is required for working with very accurate geographic information. (See Chapter 4.)
Projected or Unprojected Geographic Information
Geographic information or maps for large areas—for example, a continent or the world—are often projected, but they can also be unprojected. If they are unprojected, the distortion is very significant because the latitude and longi-
Positional uncertainty when coordinate system datum is unknown in North America (NAD27 or NAD83).
From Wieczorek, Guo, and Hijmans (2004). Reprinted by permission of Taylor & Francis Ltd.
tude values are converted to a two-dimensional orthogonal network of x, y values. The advantage is that unprojected geographic information can readily be transformed to other projections as needed. Smaller areas are usu- ally projected because the projected representations better correspond to conventional maps that people have used for many years. In areas with legally established coordinate systems or with clear conventions, the choice of projection can be easy. In other areas, a few choices may be preferable depending on the orientation, size, and accepted practices for the area in question.
Projections in Practice
You should look at the distortions of the Mercator projection and the recently popularized Peters projection in Figure 3.9. The widespread use of the Mercator projection to show things and events at a global scale (which, you should note, Mercator never did) leads to very sizeable distortions, espe- cially in areas near the poles, but also in the latitudes where most of Europe and North America are located (see Figure 3.10). These distortions led Arno Peters to promote his adaptation of older projections, the Peters’s projec- tion, which has been widely adopted even though it introduces other distor- tions. While the Peters’s projection does not solve all projection problems, it has made people more aware of the distortions inherent in projections.
Geographic Information and Maps Are Abstractions
Finally, we should note that projection is one type of abstraction, which can be misused and even lied with. Sometimes this is obvious, but careful editing can gloss over rough spots. Geographic information and maps involve many other abstractions, which is why one of Mark Monmonnier’s books on carto- graphic principles, uses, and abuses carries the title “How to Lie with Maps.”
Based on what you now know about projections, the claim that maps lie is easy enough to refute. All maps must have distortions; therefore, some would argue, what is called a “lie” is only a “distortion.”