GEOG 486
Cartography and Visualization

Datum and Projection

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A geodetic datum is a spatial reference system that describes the shape and size of the earth and establishes an origin for coordinate systems. Two main types of datums include horizontal datums and vertical datums. Horizontal datums are used to describe what we typically think of as x and y coordinates. Vertical datums describe position in the vertical direction and are often based on height above sea level. For the remainder of this concept gallery item, we will focus on horizontal datums.

Before we discuss how datums are developed and chosen, it is useful to consider how we can create a model of the earth. In general, we would like to be able to use the simplest model of the earth that we can create, as this will make mathematical calculations of distance, area and direction much easier to perform. We can think of this modeling process as a gradual process of simplification, beginning with the complex surface of the earth, and progressing to an ellipsoid or a spheroid shape (see figure below). The first simplification that we might make is to remove all of the earth’s surface topography (e.g., mountain ranges) and consider the shape of the earth at mean sea level. This shape is called a geoid. Because the earth’s mass is not evenly distributed (due to differing masses of the materials the earth is made of), the geoid is not regularly shaped, but has bulges and depressions. Once we have this basic shape, we might choose to create the ellipsoid (the shape created when you rotate an ellipse around its vertical axis) that most closely matches the shape of the geoid. This step simplifies calculations significantly. We can then go even one step further by creating a spheroid that has the same volume as the ellipse. Although the shape of the spheroid will not match the shape of the geoid as well as the ellipsoid does, it has the advantage that distance, direction and area calculations are even simpler.

A diagram to show the progression in the generalization of the earth's shape.
Figure 7.cg.1 Increasing generalization of the earth’s shape, progressing from the relatively irregular earth’s surface to the regular shape of a sphere.
Credit: After Geoscience Australia, 2004

People have created hundreds of datums that are in use around the world today. The main reason that people have developed different datums for different places is so that they can choose an ellipsoid that best matches the shape of the earth at an area of local interest (usually a country). There are two main types of datums: local datums and geocentric datums. In local datums, a point of the ellipsoid is matched to a point on the earth’s surface (e.g., the North American Datum of 1927 intersects the surface of the earth at Meades Ranch in Kansas, while the Australian Geodetic Datum intersects with the Johnston Geodetic Station in the Northern Territory). Geocentric datums, on the other hand, are based on the earth’s center of mass. Our knowledge of where that center of mass is located has improved with modern satellite data. Many countries are now shifting to geocentric datums because GPS measurements are based on a geocentric datum. This switch avoids the need for transforming GPS-collected data from one coordinate system to another. It is important to understand which datum was used when your data were created, because the position of features may be different depending on which datum was used. In some cases, there may be a positional discrepancy of up to one kilometer (see Figure 7.cg.2, below)! These shifts are especially important in large-scale mapping applications, as these discrepancies will be much larger than any projection-induced error.

Diagram to illustrate the fact that position shifts vary depending on datums.
Figure 7.cg.2 This figure shows the position of the Texas Capitol Dome Horizontal Benchmark in several different coordinate systems. You can see that position shifts (from the baseline position of WGS 84) for commonly used North American datums (e.g., NAD27) are relatively small, shifts from other datums can be quite substantial.
Credit: After Dana, 1994

A map projection is a mathematical transformation that we use to convert data that are stored in spherical coordinates (e.g., latitude and longitude) into a plane coordinate system (see Figure 7.cg.3, below). The most important reasons for using plane coordinate systems are that many numerical properties (e.g. area and distance) are much easier to calculate on a plane than a sphere, and that most maps are still produced in two-dimensional media (e.g., on a computer screen or on paper). The process of moving from a shere to a flat surface will, by necessity, involve introducing distortion of some type. One way of proving this to yourself is to sketch the outline of the graticule onto an orange and try to first pull off the peel without tearing it (other than one cut from the north pole to the south pole along a meridian), and then flatten it. Can you flatten it completely without it tearing along the polar edges? The amount of distortion a map projection introduces increases as the map scale decreases (i.e., it is most problematic in small-scale mapping). You can again use an orange to prove this to your self. Compare the amount of tearing you need to completely flatten the whole peel with the amount of tearing you would need to flatten a very small piece of the whole peel (say 1/1000th of the total peel). You will certainly find a much smaller amount of tearing is needed for the smaller piece of peel!

Graphic illustration to show that the transformation from geographic coordinates to plane coordinates is systematic.
Figure 7.cg.3 Map projections are systematic transformations from geographic coordinates to plane coordinates.
Credit: Illustration by David DiBiase

There are three types of deformation that may be present in a map projection: stretching, shearing and tearing (see Figures 7.cg.4, 7.cg.5 and 7.cg.6, below). These deformations can lead to distortions of one or more map projection properties: equivalence (area), conformality (angle), equidistance (distance) and azimuthality (direction). Some map projections minimize particular types of deformation at the expense of increasing other types of deformation. For example, interrupted projections (i.e., those that divide hemispheres of the earth into lobes), often severely deform oceans by creating tears through them, but because this tearing concentrates distortion near the tears, it also minimizes distortion over land areas. To refresh your memory on map projection properties, review Lesson 2, Geometric Properties Preserved and Distorted of Geog 482.

Mercator projection
Figure 7.cg.4In the Mercator projection, latitude-longitude quadrangles are stretched along the x-axis and y-axis as you move away from the equator.
Boone projection
Figure 7.cg.5 In the Bonne or Werner projection, latitude-longitude quadrangles are sheared (e.g., squares are distorted into parallelograms) as you move away from the center line (0° longitude).
Goode homolosine projection
Figure 7.cg.6 In an extreme case of distortion (of the oceans), the Goode’s Homolosine Equal Area projection tears oceans to minimize distortion over land areas. Since the Earth has no edges, actually any map projection that shows an edge of the surface of the Earth (which includes all 2D projections) suffers from tearing.