85 3 Geodesy, Datums, Map Projec- tions, and Coordinate Systems Introduction Geographic information systems are distorted. This distortion may be difficult to different from other information systems detect on detailed maps that cover a small because they contain spatial data. These area, but the distortion is quite apparent on spatial data include coordinates that define large-area maps. Because measurements on the location, shape, and extent of geo- maps are affected by the distortion, we graphic objects. To effectively use GIS, we must use a map projection to reconcile the must develop a clear understanding of how portrayal of the Earth’s curved surface onto coordinate systems are established for the a flat surface. Earth, how these coordinates are measured The second main problem in defining a on the Earth’s curving surface, and how coordinate system results from the irregular these coordinates are converted for use in shape of the Earth. We learn early on that flat maps, either digital or paper. This chap- the Earth is shaped as a sphere. This is a ter introduces geodesy, the science of mea- valid approximation for many uses, how- suring the shape of the Earth, and map ever, it is only an approximation. Past and projections, the transformation of coordi- present natural forces yield an irregularly nate locations from the Earth’s curved sur- shaped Earth. These deformations affect face onto flat maps. how we best map the surface of the Earth, Defining coordinates for the Earth’s and how we define Cartesian coordinate surface is complicated by three main fac- systems for mapping and GIS. tors. First, most people best understand Thirdly, our measurements are rarely geography in a Cartesian coordinate system perfect, and this applies when measuring on a flat surface. Humans naturally per- both the shape of the Earth, and the exact ceive the Earth’s surface as flat, because at position of features on it. All locations human scales the Earth’s curvature is depend on measurements that contain some barely perceptible. Humans have been error, and on analyses that require assump- using flat maps for more than 40 centuries, tions. Our measurements improve through and although globes are quite useful for time, and so does the sophistication of our visualization at extremely small scales, analysis, so our positional estimates they are not practical for most purposes. improve; this evolution means our esti- A flat map must distort geometry in mates of positions change through time. some way because the Earth is curved. Because of these three factors, we When we plot latitude and longitude coor- often have several different sets of coordi- dinates on a Cartesian system, “straight” nates to define the same location on the sur- lines will appear bent, and polygons will be face of the Earth. Remember, coordinates 86 GIS Fundamentals Coordinates for a From Surveyor Data: Point Location Latitude (N) Longitude (W) NAD83(2007) 44 57 23.23074 093 05 58.28007 d Geo an de st ti NAD83(1986) 44 57 23.22405 093 05 58.27471 a c o S C u . r NAD83(1996) 44 57 23.23047 093 05 58.27944 v S . e y U X Y B k SPC MNS 317,778.887 871,048.844 MT ench mar SPC MNS 1,042,579.57 2,857,766.08 sFT UTM15 4,978,117.714 492,150.186 MT From Data Layers: X Y MN-Ramsey 573,475.592 160,414.122 sFT MN-Ramsey 174,195.315 48,893.966 MT SPC MNC 890,795.838 95,819.779 MT SPC MNC 2,922,552.206 314,365.207 sFT LCC 542,153.586 18,266.334 MT Figure 3-1: An example of different coordinate values for the same point. We may look up the coordinates for a well-surveyed point, and we may also obtain the coordinates for the same point from a number of dif- ferent data layers. We often find multiple latitude/longitude values (surveyor data, top), or x and y values for the same point (surveyor data, or from data layers, bottom). are sets of numbers that unambiguously this point. In this case, the three versions dif- define locations. They are usually x and y fer primarily due to differences in the mea- values, or perhaps x, y, and z values, or lati- surements used to establish the point’s tude and longitude values unique to a loca- location, and how measurement errors were tion. But these values are only “unique” to adjusted (the third factor, discussed above). the location for a specified set of measure- The GIS practitioner may well ask, which ments and time. The coordinates depend on latitude/longitude pair should I use? This how we translate points from a curved Earth chapter contains the information that should to a flat map surface (first factor, above), the allow you to choose wisely. estimate we use for the real shape of the Note that there are also several versions Earth (second factor), and what set of mea- of the x and y coordinates for the point in surements we reference our coordinates to Figure 3-1. The differences in the coordinate (the third factor). We may, and often do, values are too great to be due solely to mea- address these three factors in a number of surement errors. They are due primarily to different ways, and the coordinates for the how we choose to project from the curved same point will be different for these differ- Earth to a flat map (the first factor), and in ent choices. part to the Earth shape we adopt and the An example may help. Figure 3-1 shows measurement system we use (the second and the location of a U.S. benchmark, a precisely third factors). surveyed and monumented point. Coordi- We first must define a specific coordi- nates for this point are maintained by federal nate system, meaning we choose a specific and state government surveyors, and result- way to address the three main factors of pro- ing coordinates shown at the top right of the jection distortion, an irregularly shaped figure. Note that there are three different ver- Earth, and measurement imprecision. There- sions of the latitude/longitude location for after the coordinates for a given point are Chapter 3: Geodesy, Projections, and Coordinate Systems 87 fixed, as are the spatial relationships to other measured locations on the Earth’s surface measured points. But it is crucial to realize relative to the Sun or stars, reasoning they that different ways of addressing 1) the provided a stable reference frame. This Earth’s curvature, 2) the Earth’s deviation assumption underlies most geodetic observa- from our idealized shape, and 3) inevitable tions taken over the past 2000 years, and still inaccuracies in measurement, will result in applies today, with suitable refinements. different coordinate systems, and these dif- Eratosthenes, a Greek scholar in Egypt, ferences are the root of much confusion and performed one of the earliest well-founded many errors in spatial analysis. As a rule, measurements of the Earth’s circumference. you should understand the coordinate system He noticed that on the summer solstice the used for all of your data, and convert all data sun at noon shone to the bottom of a deep to the same coordinate system prior to analy- well in Syene, near Tropic of Cancer, so that sis. The remainder of this chapter describes the sun would be exactly overhead during how we define, measure, and convert among the summer solstice. He also observed that coordinate systems. 805 km north in Alexandria, at exactly the same date and time, a vertical post cast a Early Measurements shadow. The shadow/post combination o defined an angle that was about 7 12’, or In specifying a coordinate system, we about 1/50th of a circle (Figure 3-2). must first define the size and shape of the Earth. Humans have long speculated on this. Eratosthenes deduced that the Earth Babylonians believed the Earth was a flat must be 805 multiplied by 50, or about disk floating in an endless ocean, while the 40,250 kilometers in circumference. His cal- Greek Pythagoras, and later Aristotle, rea- culations were all in stadia, the unit of mea- soned that the Earth must be a sphere. He sure of the time, and have been converted observed that ships disappeared over the here to the metric equivalent, using our best horizon, the moon appeared to be a sphere, idea of a stadia length. Eratosthenes’s esti- and that the stars moved in circular patterns, mate differs from our modern measurements all observations consistent with a spherical of the Earth’s circumference by less than Earth. 4%. The Greeks next turned toward estimat- Posidonius, another Greek scholar, ing the size of the sphere. The early Greeks made an independent estimate of the size of the Earth by measuring angles from local Figure 3-2: Measurements made by Eratosthenes to determine the circumfer- ence of the Earth. 88 GIS Fundamentals Figure 3-3: Posidonius approximated the Earth’s radius by simultaneous measurement of zenith angles at two points. Two points are separated by an arc distance d measured on the Earth surface. These points also span an angle defined at the Earth center. The Earth radius is related to d and . Once the radius is calculated, the Earth circumference may be deter- mined. Note this is an approximation, not an exact estimate, but was appro- priate for the measurements avail- able at the time (adapted from Smith, 1997). vertical (plumb) lines to a star near the hori- was adopted by Ptolemy for his world maps.