3.2: Earth

Latitude and Longitude

Any feature can be referenced by its latitude and longitude, which are angles measured in degrees from the Earth’s center to a point on the Earth’s surface (see Figure 3.1). Across the spherical Earth, latitude lines stretch horizontally from east to west (left image in Figure 3.2), and they are parallel to each other, hence their alternative name, parallels. Longitude lines, also called meridians, stand vertically and stretch from the North Pole to the South Pole (center image in Figure 3.2). Together these “north to south” and “east to west” lines meet at perpendicular angles to form a graticule, a grid that encompasses the Earth (right image in Figure 3.2).

Figure 3.1: Latitude and longitude are angles measured in degrees from the Earth’s center to a point on the Earth’s surface.

Figure 3.2: Latitude, longitude, and the Earth’s graticule.

Midway between the poles, the equator stretches around the Earth, and it defines the line of zero degrees latitude (left image in Figure 3.2). Relative to the equator, latitude is measured from 90 degrees at the North Pole to -90 degrees at the South Pole. The Prime Meridian is the line of zero degrees longitude (center image in Figure 3.2), and in most coordinate systems, it passes through Greenwich, England. Longitude runs from -180 degrees west of the Prime Meridian to 180 degrees east of the same meridian. Because the globe is 360 degrees in circumference, -180 and 180 degrees is the same location.

Earth’s Shape

If the geographic extent of your project area was small, like a neighborhood or a portion of a city, you could assume that the Earth is flat and use no projection. This is referred to as a planar surface or even a planar “projection,” but with the understanding that it does not use a projection. Planar representation does not significantly affect a map’s accuracy when scales are larger than 1:10,000. In other words, small areas do not need a projection because the statistical differences between locations on a flat plane and a 3-dimensional surface are not significant.

For small-scale maps (those that encompass a large area, see Figure 2.3), you must consider the Earth’s shape. Our assumption that the Earth is round or spherical does not accurately represent it. The Earth’s constant spinning causes it to bulge slightly along the equator, ruining its perfect spherical shape. The slightly oval nature of the Earth’s geometric surface makes the terms ellipsoid and spheroid more accurate in describing its shape, but they are not perfect terms either since differences in material weights (for instance iron is denser than sedimentary deposits) and the movement of tectonic plates makes the Earth dynamic and constantly changing. The Earth is a geoid with a slight pear shape; it is a little larger in the southern hemisphere and includes other bulges. The difference, however, between the ellipsoid and the geoid is minor enough that it does not affect most mapping. Until recently, projections based on geoids were rare because of the complexity and cost of collecting the necessary data to create the projection, but satellite imagery has helped with measurement and geoid projections are now more common.

Map Projections

Globes do not need projections, and even though they are the best way to depict the Earth’s shape and to understand latitude and longitude, they are not practical for most applications that require maps. We need flat maps. This requires a reshaping of the Earth’s 3-dimensions into a 2-dimensional surface. This reshaping cannot be done without introducing some error. To illustrate this point, imagine taking a cardboard globe, cutting it in half at the equator, and then cutting both the northern and southern hemispheres into four equal parts apiece. Resting on a table, the pieces are not flat; they arch in the center. Try flattening one of the pieces. If you succeed, part of the cardboard will be scrunched together and other parts will tear apart. By flattening it, you modify its geography.

Map projections enable the reshaping of the Earth by mathematically transforming spherical coordinates (x, y, and z) to 2-dimensional (x and y) space. They are the foundations we use to represent the Earth’s surface or portions of it.

Projections are abstractions, and they introduce distortions to either the Earth’s shape, area, distance, or direction (and sometimes to all of these properties). Different map projections cause different map distortions.

One way to classify map projections is to describe them by the characteristic they do not distort. Usually only one property is preserved in a projection. This chapter confines its focus to just two properties—area and shape—because the projections that preserve these properties—equal-area and conformal—are the most common.

Equal area (or equivalent) projections preserve the area (or the amount of space) within features. On a small-scale political map of the world, the areas within each country are preserved. In reality, the area of Mexico and Greenland is similar, and in the right-hand map in Figure 3.3, which is drawn in an equal area projection called Mollweide, the two territories are approximately the same size. Equal area projections, however, distort all the other properties. Shape, distance, and direction are not preserved.

Figure 3.3: Projections transform the 3-D Earth into a 2-D plane. Some projections, like Mercator, attempt to preserve area while other projections like Mollweide preserve the area contained within the landmasses.

Conformal (also known as orthomorphic) maps preserve shape by preserving the angles of feature boundaries like countries and continents. Maintaining angles, however, distorts the area within the features (see left map in Figure 3.3). In the Mercator projection, Greenland looks like Greenland, but it is far larger than Mexico, its spatial equivalent. In addition, no conformal projection preserves the shapes of features that extend close to the poles (notice Antarctica).

Another way to classify map projections is by their projection surface. Imagine a translucent globe with country boundaries and latitude and longitude lines drawn upon it in black. In addition, imagine a light bulb positioned at the globe’s center. If you placed large pieces of paper on or around the translucent globe and turned on the interior light, you would see the country boundaries and the latitude and longitude lines projected onto the paper. If those projected lines were imprinted onto the paper, the paper could be removed from the globe, cut, and flattened to produce a 2-dimensional map.

What you just imagined is the way many projections were first conceived. Even today, in the age of computer modeling, most of the map projections we use are variations on three basic projection surfaces: planar, conic, and cylindrical (see Figure 3.4).

Figure 3.4: Projection surfaces.

Planar projections, the least common, can be conceptualized by placing a flat sheet in contact (at one point) with the translucent globe, usually at the North or South Pole, and the lines on the globe are projected onto the sheet. The projected map creates a circular graticule (see top row of Figure 3.4). Direction, one of the properties not described, is usually preserved from the center of the map outward. Some planar projections preserve area or distance. Consider using a planar projection if your research area is at one of the poles.

1. Conic projections, a common projection surface, are conceptualized by placing a paper cone on the globe, and the lines on the globe are projected to the cone. After unraveling the cone, the graticule appears fan shaped (middle row of Figure 3.4). Conic projections preserve different properties including area and shape, but never both in a single projection. Distortion across the map varies. No distortion exists along the parallel (latitude) where the cone touches the globe, but distortion increases in both directions away from this line of tangency. Consider using a projection with this surface if your study area is in the mid-latitudes including the U.S.
2. Cylindrical projections are developed by wrapping paper around the globe in the shape of a cylinder. The lines on the globe are projected to the cylinder (bottom row of Figure 3.4), and the resultant graticule is rectangular. There is no distortion along the equator (its point of tangency), but distortion increases toward the Earth’s poles. This projection surface preserves different properties including area and shape (but again, both are not preserved in a single projection). Consider using this projection surface if your study area is worldwide or in the tropics.

Many variations can be made using these three projection surfaces. Instead of having the paper come to a simple point or line of tangency with the globe, you could cut the globe’s surface (called secant), so that conic and cylindrical projections intersect the globe at two lines (latitude) and plane projections create a single circle. No distortion occurs anywhere the projection surface (the paper) intersects with the globe. Where the projection surface is outside the globe, features appear larger than they are in reality. Where the projection surface is inside the globe, features appear smaller.

Additional variations result when you move the position of the globe’s interior light or combine multiple projection surfaces. In addition, with computers, mathematical projections not based on these projection surfaces exist, and some of these projections are very popular.

There are thousands of different projections, but only a few dozen projections are noteworthy and used. Examples include Albers Equal Area Conic, Lambert Conformal Conic, Mercator, Miller Cylindrical, and Robinson. Many of these projection names include words like equal area, conformal, conic, and cylindrical; they provide clues to the projection’s characteristics and projection surfaces.

As mentioned in Chapter 2, it is important to choose an appropriate projection for your GIS project to achieve accurate results. Are you interested in calculating the area of features? If so, you must use a projection that preserves area, or your calculations will be inaccurate. Improper projections distort attribute accuracy, positional accuracy, and thus the information in your final maps and reports. As Chapter 6 describes, choosing an unsuitable map projection is one way to lie with maps.

How accurate do your locations need to be? If you are making a world map with the locations of the largest ports, precise locations are probably not necessary. If you are drilling a train tunnel, however, positional accuracy is required. Some projections use spheres to model the Earth’s shape. Remember (from the first part of this chapter) that a sphere is the most generalized shape of the world and the least accurate. Many projections are based on spheres, and these projections are suitable for world maps and large world regions that do not require a high degree of positional accuracy. Most projections today, however, are based on ellipsoids (and spheroids) that distort the uniformity of the sphere to bulge a bit at the equator. Statistically speaking, there is no significant difference between most ellipsoids and the true shape of the Earth for most mapping purposes. Still, for those projects that require even more precision, there are projections based on geoids. These projections were rare until recently because of the time it took to calculate the projections and the difficulty of the math and measurement. With increasing satellite imagery, however, they are more common.

Coordinate Systems

Projections and coordinate systems are two separate things. As described above, projections convert the Earth from 3-dimensional space to a 2-dimensional map. Coordinate systems are referencing systems used to describe specific locations and measure distances on maps. They provide x, y locations (sometimes designated as Eastings and Northings) for features, and, within GIS, they are used to spatially register layers of features that occupy the same area.

While coordinate systems are not projections, they usually use them. Latitude and longitude, the best known coordinate system, however, does not use a projection, but in most cases, coordinate systems incorporate a map projection, reference spheroid, datum, one or more standard parallels, a central meridian, and possible shifts in x or y (easting and northing) directions.

Like projections, there are many coordinate systems. Some of these coordinate systems focus locally and some globally. The most common at the world level is latitude and longitude, but because it is not a “projected” coordinate system, plotted points usually have a high degree of distance and shape distortion when plotted on a 2-dimensional flat map (and so it should not be used for making two-dimensional maps). Latitude and longitude uses the Prime Meridian and the Equator as reference planes, and it is best used when conceptualizing the Earth as a globe.

There are two ways of providing latitude and longitude coordinates. One method uses degrees, minutes and seconds. For instance, the CSUS Geography department is located at 38*N 33’ 32” latitude and 121*W 25’ 31” longitude. Another method is decimal degrees, and the same coordinates are represented as 38.55889 latitude and -121.42527 longitude.

The following is a description of a few of the most used coordinate systems in the United States: Universal Transverse Mercator, State Plane Coordinate System, and United States National Grid. Latitude and longitude, also widely used, was described earlier in this chapter.

Developed in the 1940s by the U.S. Army Corps of Engineers, Universal Transverse Mercator (UTM) is a coordinate system that largely covers the globe. The system reaches from 84 degrees north to 84 degrees south latitude, and it divides the Earth into 60 north-south oriented zones that are 6 degrees of longitude wide (see Figure 3.5). Each individual zone uses a defined transverse Mercator projection. The contiguous U.S. consists of 10 zones. In the Northern hemisphere, the equator is the zero baseline for Northings (Southern hemisphere uses a 10,000 km false Northing). Each zone has an arbitrary central meridian of 500 km west of each zone’s central meridian (called a false Easting) to insure positive Easting values and a central bisecting meridian. In UTM, the CSUS Geography Department is located at 4,269,000 meters north; 637,200 meters east; zone 10, northern hemisphere.

Figure 3.5: Universal Transverse Mercator zones.

The State Plane Coordinate System (SPCS) is a projected coordinate system that divides the U.S. and its possessions into over 120 zones (see Figure 3.6). Some smaller states use a single zone while larger states are divided into several zones. California has six zones (see map in lower-left corner of Figure 3.6). Each zone provides a local reference system that has its own parameters. Zones oriented east to west use the Lambert Conformal Conic projection while zones stretching more north to south use Transverse Mercator (not to be confused with UTM). Used principally by cities, many counties, and some states, they are popular projected coordinate systems. In SPCS, the CSUS Geography Department is located at 599200.796 feet; 2050091.975 feet; CA zone 2.

Figure 3.6: State Plane Coordinate System.

Many GIS projects cover more than one SPCS or UTM zone. In response to calls for a single coordinate system that covers the entire U.S., the United States National Grid (USNG), was created in 2001. After the attacks of September 11, 2001 and the increasing use of GPS-enabled and location tracking devices, a consistent grid takes on additional importance, and in 2005, the Department of Homeland Security (DHS) recommended that any DHS grant should reference their data to USNG. USNG incorporates the U.S. Military Grid Reference System’s hierarchy (not described here), but the basic zones are identical to UTM.

Datums

A datum is a starting point for locating features on the Earth’s surface; it is the origin point of a coordinate system. It defines the position of the ellipsoid (or spheroid) relative to the Earth’s center. There are many different datums and hence many different starting positions. Like both projections and coordinate systems, international organizations and individual nations have established datums for their specific needs. The World Geodetic System 1984 (WGS84) is the most widely used datum internationally. In the U.S., the two most used datums are North American Datum 1927 (NAD27) and North American Datum 1983 (NAD83). NAD83 updates NAD27 by using a more accurate ellipsoid for North America, derived from better satellite imagery, and changing reference units from feet to meters.