{"id":91,"date":"2014-01-24T18:09:15","date_gmt":"2014-01-24T18:09:15","guid":{"rendered":"http:\/\/opentextbc.ca\/natureofgeographicinformation\/?post_type=chapter&p=91"},"modified":"2015-05-27T15:41:20","modified_gmt":"2015-05-27T15:41:20","slug":"overview","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/natureofgeographicinformation\/chapter\/overview\/","title":{"raw":"Scales and Transformations","rendered":"Scales and Transformations"},"content":{"raw":"

2.1. Overview<\/h2>\r\nChapter 1 outlined several of the distinguishing properties of geographic data. One is that geographic data are necessarily generalized, and that generalization tends to vary with scale. A second distinguishing property is that the Earth's complex, nearly-spherical shape complicates efforts to specify exact positions on Earth's surface. This chapter explores implications of these properties by illuminating concepts of scale, Earth geometry, coordinate systems, the \"horizontal datums\" that define the relationship between coordinate systems and the Earth's shape, and the various methods for transforming coordinate data between 3D and 2D grids, and from one datum to another.\r\n\r\nCompared to Chapter 1, Chapter 2 may seem long, technical, and abstract, particularly to those for whom these concepts are new. Registered students will notice that we've allotted more time to work through the chapter and associated quizzes. Seven practice quizzes are available in ANGEL to help registered students get a grip on these concepts. Chapter 2 also includes a graded quiz in the same open-book format as the practice quizzes. If you do reasonably well on the practice quizzes, you should do well enough on the graded quiz too.\r\n\r\nObjectives<\/strong>\r\n\r\nStudents who successfully complete Chapter 2 should be able to:\r\n
    \r\n\t
  1. Demonstrate your ability to specify geospatial locations using geographic coordinates;<\/li>\r\n\t
  2. Convert geographic coordinates between two different formats;<\/li>\r\n\t
  3. Explain the concept of a horizontal datum;<\/li>\r\n\t
  4. Calculate the change in a coordinate location due to a change from one horizontal datum to another;<\/li>\r\n\t
  5. Estimate the magnitude of \"datum shift\" associated with the adjustment from NAD 27 to NAD 83;<\/li>\r\n\t
  6. Recognize the kind of transformation that is appropriate to georegister two or more data sets;<\/li>\r\n\t
  7. Describe the characteristics of the UTM coordinate system, including its basis in the Transverse Mercator map projection;<\/li>\r\n\t
  8. Plot UTM coordinates on a map;<\/li>\r\n\t
  9. Describe the characteristics of the SPC system, including map projection on which it is based;<\/li>\r\n\t
  10. Convert geographic coordinates to SPC coordinates;<\/li>\r\n\t
  11. Interpret distortion diagrams to identify geometric properties of the sphere that are preserved by a particular projection; and<\/li>\r\n\t
  12. Classify projected graticules by projection family.<\/li>\r\n<\/ol>\r\n \r\n

    Comments and Questions<\/h3>\r\nRegistered students are welcome to post comments, questions, and replies to questions about the text. Particularly welcome are anecdotes that relate the chapter text to your personal or professional experience. In addition, there are discussion forums available in the ANGEL course management system for comments and questions about topics that you may not wish to share with the whole world.\r\n\r\nTo post a comment, scroll down to the text box under \"Post new comment\" and begin typing in the text box, or you can choose to reply to an existing thread. When you are finished typing, click on either the \"Preview\" or \"Save\" button (Save will actually submit your comment). Once your comment is posted, you will be able to edit or delete it as needed. In addition, you will be able to reply to other posts at any time.\r\n\r\nNote: the first few words of each comment become its \"title\" in the thread.\r\n

    2.2. Checklist<\/h2>\r\n \r\n\r\nThe following checklist is for Penn State students who are registered for classes in which this text, and associated quizzes and projects in the ANGEL course management system, have been assigned. You may find it useful to print this page out first so that you can follow along with the directions.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Chapter 2 Checklist (for registered students only)<\/caption>\r\n
    Step<\/th>\r\nActivity<\/th>\r\nAccess\/Directions<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    1<\/th>\r\nRead<\/strong>\u00a0Chapter 2<\/td>\r\nThis is the second page of the Chapter. Click on the links at the bottom of the page to continue or to return to the previous page, or to go to the top of the chapter. You can also navigate the text via the links in the GEOG 482 menu on the left.<\/td>\r\n<\/tr>\r\n
    2<\/th>\r\nSubmit\u00a0eight practice quizzes<\/strong>including:\r\n
      \r\n\t
    • Map Scale<\/li>\r\n\t
    • Geographic Coordinate System<\/li>\r\n\t
    • Horizontal Datums<\/li>\r\n\t
    • Coordinate Transformations<\/li>\r\n\t
    • UTM Coordinate System<\/li>\r\n\t
    • SPC Coordinate System<\/li>\r\n\t
    • Map Projection Properties<\/li>\r\n\t
    • Classifying Map Projections<\/li>\r\n<\/ul>\r\nPractice quizzes are not graded and may be submitted more than once.<\/td>\r\n
    		Go to ANGEL > [your course section] > Lessons tab > Chapter 2 folder > [quiz]<\/td>\r\n<\/tr>\r\n
    3<\/th>\r\nPerform\u00a0\u201cTry this\u201d activities<\/strong>\u00a0Including:\r\n
      \r\n\t
    • Geographic coordinates practice application<\/li>\r\n\t
    • Calculate horizontal datum shift<\/li>\r\n\t
    • UTM coordinates practice application<\/li>\r\n\t
    • Explore SPC zone characteristics<\/li>\r\n\t
    • Interactive Album of Map Projections<\/li>\r\n<\/ul>\r\n\u201cTry this\u201d activities are not graded.<\/td>\r\n
    		Instructions are provided for each activity.<\/td>\r\n<\/tr>\r\n
    4<\/th>\r\nSubmit the\u00a0Chapter 2 Graded Quiz<\/strong><\/td>\r\nANGEL > [your course section] > Lessons tab > Chapter 2 folder > Chapter 2 Graded Quiz<\/td>\r\n<\/tr>\r\n
    5<\/th>\r\n\u00a0Read\u00a0comments and questions<\/strong>posted by fellow students. Add comments and questions of your own, if any.<\/td>\r\n\u00a0Comments and questions may be posted on any page of the text, or in a Chapter-specific discussion forum in ANGEL.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

    2.3. Scale<\/h2>\r\nYou hear the word \u201cscale\u201d often when you work around people who produce or use geographic information. If you listen closely, you\u2019ll notice that the term has several different meanings, depending on the context in which it is used. You\u2019ll hear talk about the scales of geographic phenomena and about the scales at which phenomena are represented on maps and aerial imagery. You may even hear the word used as a verb, as in \u201cscaling a map\u201d or \u201cdownscaling.\u201d The goal of this section is to help you learn to tell these different meanings apart, and to be able to use concepts of scale to help make sense of geographic data.\r\n\r\nSpecifically, in this part of Chapter 2\u00a0 you will learn to:\r\n
      \r\n\t
    1. Calculate map scale using representative fractions.<\/li>\r\n\t
    2. Describe the general relationship between map scale, detail, and accuracy.<\/li>\r\n<\/ol>\r\n

      2.4. Scale as Scope<\/h2>\r\nOften \u201cscale\u201d is used as a synonym for \u201cscope,\u201d or \u201cextent.\u201d For example, the title of an international research project called The Large Scale Biosphere-Atmosphere Experiment in Amazonia (1999) uses the term \u201clarge scale\u201d to describe a comprehensive study of environmental systems operating across a large region. This usage is common not only among environmental scientists and activists, but also among economists, politicians, and the press. Those of us who specialize in geographic information usually use the word \u201cscale\u201d differently, however.\r\n

      2.5. Map and Photo Scale<\/h2>\r\nWhen people who work with maps and aerial images use the word \u201cscale,\u201d they usually are talking about the sizes of things that appear on a map or air photo, relative to the actual sizes of those things on the ground.\r\n\r\nMap scale is the proportion between a distance on a map and a corresponding distance on the ground:<\/strong>\r\n(Dm \/ Dg).\r\n\r\nBy convention, the proportion is expressed as a \u201crepresentative fraction\u201d in which map distance (Dm) is reduced to 1. The proportion, or ratio, is also typically expressed in the form 1 : Dg rather than 1 \/ Dg.\r\n\r\nThe representative fraction 1:100,000, for example, means that a section of road that measures 1 unit in length on a map stands for a section of road on the ground that is 100,000 units long.\r\n\r\nIf we were to change the scale of the map such that the length of the section of road on the map was reduced to, say, 0.1 units in length, we would have created a\u00a0smaller-scale<\/strong>\u00a0map whose representative fraction is 0.1:100,000, or 1:1,000,000. When we talk about large- and small-scale maps and geographic data, then, we are talking about the relative sizes and levels of detail of the features represented in the data. In general, the larger the map scale, the more detail is shown. This tendency is illustrated below.\r\n\r\n \r\n\r\n \r\n\r\nGeographic data are generalized according to scale. Click on the buttons beneath the map to zoom in and out on the town of Gorham. (Adapted from Thompson, 1988)\r\n\r\nOne of the defining characteristics of topographic maps is that scale is consistent across each map and within each map series. This isn\u2019t true for aerial imagery, however, except for images that have been\u00a0orthorectified<\/strong>. As discussed in Chapter 6, large scale maps are typically derived from aerial imagery. One of the challenges associated with using air photos as sources of map data is that the scale of an aerial image varies from place to place as a function of the elevation of the terrain shown in the scene. Assuming that the aircraft carrying the camera maintains a constant flying height (which pilots of such aircraft try very hard to do), the distance between the camera and the ground varies along each flight path. This causes air photo scale to be larger where the terrain is higher and smaller where the terrain lower. An \u201corthorectified\u201d image is one in which variations in scale caused by variations in terrain elevation (among other effects) have been removed.\r\n\r\nYou can calculate the average scale of an\u00a0unrectified<\/strong>\u00a0air photo by solving the equation\u00a0Sp = f \/ (H-havg)<\/strong>, where f is the focal length of the camera, H is the flying height of the aircraft above mean sea level, and havg is the average elevation of the terrain. You can also calculate air photo scale at a particular point by solving the equation\u00a0Sp = f \/ (H-h)<\/strong>, where f is the focal length of the camera, H is the flying height of the aircraft above mean sea level, and h is the elevation of the terrain at a given point. You\u2019ll have a chance to practice calculating both map scale and air photo scale in a forthcoming practice quiz.\r\n

      2.6. Graphic Map Scales<\/h2>\r\nAnother way to express map scale is with a graphic (or \u201cbar\u201d) scale. Unlike representative fractions, graphic scales remain true when maps are shrunk or magnified.\r\n\r\n\"Example\r\n\r\nGraphic scales.\r\n\r\nIf they include a scale at all, most maps include a bar scale like the one shown above left. Some also express map scale as a representative fraction. Either way, the implication is that scale is uniform across the map. In fact, except for maps that show only very small areas, scale varies across every map. As you probably know, this follows from the fact that positions on the nearly-spherical Earth must be transformed to positions on two-dimensional sheets of paper. Systematic transformations of this kind are called\u00a0map projections<\/strong>. As we will discuss in greater depth later in this chapter, all map projections are accompanied by deformation of features in some or all areas of the map. This deformation causes map scale to vary across the map. Representative fractions may therefore specify map scale along a line at which deformation is minimal (nominal scale<\/strong>). Bar scales denote only the nominal or average map scale. Variable scales, like the one illustrated above right, show how scale varies, in this case by latitude, due to deformation caused by map projection.\r\n

      2.7. Map Scale and Accuracy<\/h2>\r\nOne of the special characteristics of geographic data is that phenomena shown on maps tend to be represented differently at different scales. Typically, as scale decreases, so too does the number of different features, and the detail with which they are represented. Not only printed maps, but also digital geographic data sets that cover extensive areas tend to be moregeneralized<\/strong>\u00a0than data sets that cover limited areas.\r\n\r\nAccuracy also tends to vary in proportion with map scale. The United States Geological Survey, for example, guarantees that the mapped positions of 90 percent of well-defined points shown on its topographic map series at scales smaller than 1:20,000 will be within 0.02 inches of their actual positions on the map (see the\u00a0National Geospacial Program Standards and Specifications<\/a>). Notice that this \u201cNational Map Accuracy Standard\u201d is scale-dependent. The allowable error of well-defined points (such as control points, road intersections, and such) on 1:250,000 scale topographic maps is thus 1 \/ 250,000 = 0.02 inches \/ Dg or Dg = 0.02 inches x 250,000 = 5,000 inches or 416.67 feet. Neither small-scale maps nor the digital data derived from them are reliable sources of detailed geographic information.\r\n\r\n\"Stage\r\n\r\nAreas (in gray) disqualified as potential sites for a low level radioactive waste storage facility depicted on a small scale map (original 1:1,500,000) mask small suitable areas large enough to contain the 500-acre facility (Chem-Nuclear Systems, Inc., 1994).\r\n\r\nSometimes the detail lost on small-scale maps causes serious problems. For example, a contractor hired to use GIS to find a suitable site for a low level radioactive waste storage facility in Pennsylvania presented a series of 1:1,500,000 scale maps at public hearings around the state in the early 1990s. The scale was chosen so that disqualified areas of the entire state could be printed on a single 11 x 17 inch page. A report accompanying the map included the disclaimer that \u201cit is possible that small areas of sufficient size for the LLRW disposal facility site may exist within regions that appear disqualified on the [map]. The detailed information for these small areas is retained within the GIS even though they are not visually illustrated\u2026\u201d\u00a0(Chem-Nuclear Systems, Inc. 1993, p. 20)<\/cite>. Unfortunately for the contractor, alert citizens recognized the shortcomings of the small-scale map, and newspapers published reports accusing the out-of-state company of providing inaccurate documents. Subsequent maps were produced at a scale large enough to discern 500-acre suitable areas.\r\n

      2.8. Scale as a Verb<\/h2>\r\nThe term \u201cscale\u201d is sometimes used as a verb. To scale a map is to reproduce it at a different size. For instance, if you photographically reduce a 1:100,000-scale map to 50 percent of its original width and height, the result would be one-quarter the area of the original. Obviously the map scale of the reduction would be smaller too: 1\/2 x 1\/100,000 = 1\/200,000.\r\n\r\nBecause of the inaccuracies inherent in all geographic data, particularly in small scale maps, scrupulous geographic information specialists\u00a0avoid enlarging source maps<\/strong>. To do so is to exaggerate generalizations and errors. The original map used to illustrate areas in Pennsylvania disqualified from consideration for low-level radioactive waste storage shown on an earlier page, for instance, was printed with the statement \u201cBecause of map scale and printing considerations, it is not appropriate to enlarge or otherwise enhance the features on this map.\u201d\r\n

      PRACTICE QUIZ<\/strong><\/h3>\r\nRegistered Penn State students should return now to the Chapter 2 folder in ANGEL (via the Resources menu to the left) to take a self-assessment quiz about Map Scale.\r\n\r\nYou may take practice quizzes as many times as you wish. They are not scored and do not affect your grade in any way.\r\n

      2.9. Geospatial Measurement Scales<\/h2>\r\nThe word \u201cscale\u201d can also be used as a synonym for a ruler\u2013a measurement scale. Because data consist of symbols that represent measurements of phenomena, it\u2019s important to understand the reference systems used to take the measurements in the first place. In this section we\u2019ll consider a measurement scale known as the\u00a0geographic coordinate system<\/strong>\u00a0that is used to specify positions on the Earth\u2019s roughly spherical surface. In other sections we\u2019ll encounter two-dimensional (plane) coordinate systems, as well as the measurement scales used to specify attribute data.\r\n\r\nIn this section of Chapter 2 you will:\r\n
        \r\n\t
      1. Demonstrate your ability to specify geospatial locations using geographic coordinates.<\/li>\r\n\t
      2. Convert geographic coordinates between two different formats.<\/li>\r\n<\/ol>\r\n

        2.10. Coordinate Systems<\/h2>\r\n\"A\r\n\r\nA Cartesian coordinate system.\r\n\r\nAs you probably know, locations on the Earth\u2019s surface are measured and represented in terms of\u00a0coordinates.<\/strong>\u00a0A coordinate is a set of two or more numbers that specifies the position of a point, line, or other geometric figure in relation to some reference system. The simplest system of this kind is a Cartesian coordinate system (named for the 17th century mathematician and philosopher Ren\u00e9 Descartes). A Cartesian coordinate system is simply a grid formed by juxtaposing two measurement scales, one horizontal (x) and one vertical (y). The point at which both x and y equal zero is called the\u00a0origin<\/strong>\u00a0of the coordinate system. In the illustration above, the origin (0,0) is located at the center of the grid. All other positions are specified relative to the origin. The coordinate of upper right-hand corner of the grid is (6,3). The lower left-hand corner is (-6,-3). If this is not clear, please ask for clarification!\r\n\r\nCartesian and other two-dimensional (plane) coordinate systems are handy due to their simplicity. For obvious reasons they are not perfectly suited to specifying geospatial positions, however. The\u00a0geographic coordinate system<\/strong>\u00a0is designed specifically to define positions on the Earth\u2019s roughly-spherical surface. Instead of the two linear measurement scales x and y, the geographic coordinate systems juxtaposes two curved measurement scales. The east-west scale, called\u00a0longitude<\/strong>\u00a0(conventionally designated by the Greek symbol\u00a0lambda<\/em>), ranges from +180\u00b0 to -180\u00b0. Because the Earth is round, +180\u00b0 (or 180\u00b0 E) and -180\u00b0 (or 180\u00b0 W) are the same grid line. That grid line is roughly the International Date Line, which has diversions that pass around some territories and island groups. Opposite the International Date Line is the\u00a0prime meridian<\/strong>, the line of longitude defined by international treaty as 0\u00b0. The north-south scale, called\u00a0latitude<\/strong>(designated by the Greek symbol\u00a0phi<\/em>), ranges from +90\u00b0 (or 90\u00b0 N) at the North pole to -90\u00b0 (or 90\u00b0 S) at the South pole. We\u2019ll take a closer look at the geographic coordinate system next.\r\n\r\n\"Geodetic\r\n\r\nThe geographic (or \u201cgeodetic\u201d) coordinate system.\r\n

        2.11. Geographic Coordinate System<\/h2>\r\n\"Geographic\r\n\r\nThe geographic coordinate system.\r\n\r\nLongitude<\/strong>\u00a0specifies positions east and west as the angle between theprime meridian<\/strong>\u00a0and a second\u00a0meridian<\/strong>\u00a0that intersects the point of interest. Longitude ranges from +180 (or 180\u00b0 E) to -180\u00b0 (or 180\u00b0 W). 180\u00b0 East and West longitude together form the International Date Line.\r\n\r\nLatitude<\/strong>\u00a0specifies positions north and south in terms of the angle subtended at the center of the Earth between two imaginary lines, one that intersects the\u00a0equator\u00a0<\/strong>and another that intersects the point of interest. Latitude ranges from +90\u00b0 (or 90\u00b0 N) at the North pole to -90\u00b0 (or 90\u00b0 S) at the South pole. A line of latitude is also known as a\u00a0parallel<\/strong>.\r\n\r\nAt higher latitudes, the length of parallels decreases to zero at 90\u00b0 North and South. Lines of longitude are not parallel, but converge toward the poles. Thus while a degree of longitude at the equator is equal to a distance of about 111 kilometers, that distance decreases to zero at the poles.\r\n

        TRY THIS!<\/strong><\/h3>\r\n

        GEOGRAPHIC COORDINATE SYSTEM PRACTICE APPLICATION<\/h3>\r\nNearly everyone learned latitude and longitude as a kid. But how well do you understand the geographic coordinate system, really? My experience is that while everyone who enters this class has heard of latitude and longitude, only about half can point to the location on a map that is specified by a pair of geographic coordinates. The Flash application linked below lets you test your knowledge. The application asks you to click locations on a globe as specified by randomly generated geographic coordinates.\r\n\r\nYou will notice that the application lets you choose between \u201ceasy problems\u201d and \u201chard problems.\u201d Easy problems are those in which latitude and longitude coordinates are specified in 30\u00b0 increments. Since the resolution of the graticule (the geographic coordinate system grid) used in the application is also 30\u00b0, the solution to every \u201ceasy\u201d problem occurs at the intersection of a parallel and a meridian. The \u201ceasy\u201d problems are good warm-ups.\r\n\r\n\u201cHard\u201d problems specify coordinates in 1\u00b0 increments. You have to interpolate positions between grid lines.\u00a0You can consider yourself to have a good working knowledge of the geographic coordinate system if you can solve at least six \u201chard\u201d problems consecutively and on the first click.<\/strong>\r\n\r\nClick here to download and launch the Geographic Coordinate System practice application (5.7 Mb)<\/a>. (If the globe doesn\u2019t appear after the Flash application has loaded, right-click and select \u201cPlay\u201d from the pop-up menu.)\r\n\r\n\"Screenshot\r\n\r\nNote: You will need to have the Adobe Flash player installed in order to complete this exercise. If you do not already have the Flash player, you can download it for free at the\u00a0adobe website.<\/a>\r\n

        2.12. Geographic Coordinate Formats<\/h2>\r\nGeographic coordinates may be expressed in decimal degrees, or in degrees, minutes, and seconds. Sometimes you need to convert from one form to another. Steve Kiouttis (personal communication, Spring 2002), manager of the Pennsylvania Urban Search and Rescue Program, described one such situation on the course Bulletin Board: \u201cI happened to be in the state Emergency Operations Center in Harrisburg on Wednesday evening when a call came in from the Air Force Rescue Coordination Center in Dover, DE. They had an emergency locator transmitter (ELT) activation and requested the PA Civil Air Patrol to investigate. The coordinates given to the watch officer were 39 52.5 n and -75 15.5 w. This was plotted incorrectly (treated as if the coordinates were in decimal degrees 39.525n and -75.155 w) and the location appeared to be near Vineland, New Jersey. I realized that it should have been interpreted as 39 degrees 52 minutes and 5 seconds n and -75 degrees and 15 minutes and 5 seconds w) and made the conversion (as we were taught in Chapter 2) and came up with a location on the grounds of Philadelphia International Airport, which is where the locator was found, in a parked airliner.\u201d\r\n\r\nHere\u2019s how it works:\r\n\r\nTo convert -89.40062 from decimal degrees to degrees, minutes, seconds:\r\n
          \r\n\t
        1. Subtract the number of whole degrees (89\u00b0) from the total (89.40062\u00b0). (The minus sign is used in the decimal degree format only to indicate that the value is a west longitude or a south latitude.)<\/li>\r\n\t
        2. Multiply the remainder by 60 minutes (.40062 x 60 = 24.0372).<\/li>\r\n\t
        3. Subtract the number of whole minutes (24\u2032) from the product.<\/li>\r\n\t
        4. Multiply the remainder by 60 seconds (.0372 x 60 = 2.232).<\/li>\r\n\t
        5. The result is 89\u00b0 24\u2032 2.232\u2033 W or S.<\/li>\r\n<\/ol>\r\nTo convert 43\u00b0 4\u2032 31\u2033 from degrees, minutes, seconds to decimal degrees:\r\nDD = Degrees + (Minutes\/60) + (Seconds\/3600)\r\n
            \r\n\t
          1. Divide the number of seconds by 60 (31 \u00f7 60 = 0.5166).<\/li>\r\n\t
          2. Add the quotient of step (1) to the whole number of minutes (4 + 0.5166).<\/li>\r\n\t
          3. Divide the result of step (2) by 60 (4.5166 \u00f7 60 = 0.0753).<\/li>\r\n\t
          4. Add the quotient of step (3) to the number of whole number degrees (43 + 0.0753).<\/li>\r\n\t
          5. The result is 43.0753\u00b0<\/li>\r\n<\/ol>\r\n

            PRACTICE QUIZ<\/strong><\/h3>\r\nRegistered Penn State students should return now to the Chapter 2 folder in ANGEL (via the Resources menu to the left) to take a self-assessment quiz about the Geographic Coordinate System.\r\n\r\nYou may take practice quizzes as many times as you wish. They are not scored and do not affect your grade in any way.\r\n

            2.13. Horizontal Datums<\/h2>\r\nGeographic data represent the locations and attributes of things on the Earth\u2019s surface. Locations are measured and encoded in terms of geographic coordinates (i.e., latitude and longitude) or plane coordinates (e.g., UTM). To measure and specify coordinates accurately, one first must define the geometry of the surface itself. To see what I mean, imagine a soccer ball. If you or your kids play soccer you can probably conjure up a vision of a round mosaic of 20 hexagonal (six sided) and 12 pentagonal (five sided) panels (soccer balls come in many different designs, but the 32-panel ball is used in most professional matches. Visitsoccerballworld.com<\/a>\u00a0for more than you ever wanted to know about soccer balls). Now focus on one point at an intersection of three panels. You could use spherical (e.g., geographic) coordinates to specify the position of that point. But if you deflate the ball, the position of the point in space changes, and so must its coordinates.\u00a0The absolute<\/strong>\u00a0(though not the relative)\u00a0position of a point on a surface, then, depends upon the shape of the surface<\/strong>.\r\n\r\nEvery position is determined in relation to at least one other position. Coordinates, for example, are defined relative to the origin of the coordinate system grid. A land surveyor measures the \u201ccorners\u201d of a property boundary relative to a previously-surveyed control point. Surveyors and engineers measure elevations at construction sites and elsewhere. Elevations are expressed in relation to a\u00a0vertical datum<\/strong>, a reference surface such as mean sea level. As you probably know there is also such a thing as a\u00a0horizontal datum<\/strong>, although this is harder to explain and to visualize than the vertical case.\u00a0Horizontal datums define the geometric relationship between a coordinate system grid and the Earth\u2019s surface.<\/strong>\u00a0Because the Earth\u2019s shape is complex, the relationship is too. The goal of this section is to explain the relationship.\r\n\r\nSpecifically, in this section of Chapter 2 you will learn to:\r\n
              \r\n\t
            1. Explain the concept of a horizontal datum<\/li>\r\n\t
            2. Calculate the change in a coordinate location due to a change from one horizontal datum to another<\/li>\r\n\t
            3. Estimate the magnitude of \u201cdatum shift\u201d associated with the adjustment from NAD 27 to NAD 83<\/li>\r\n<\/ol>\r\n

              2.14. Geoids<\/h2>\r\n\"Diagram\r\n\r\nThe Earth\u2019s shape is defined as a surface that closely approximates global mean sea level, but across which gravity is everywhere equal. The caricature of the geoid shown above is not drawn to scale. Irregularities are greatly exaggerated\r\n(Adapted from Smith, 1988).\r\n\r\nThe accuracy of coordinates that specify geographic locations depends upon how the coordinate system grid is aligned with the Earth\u2019s surface. Unfortunately for those who need accurate geographic data, defining the shape of the Earth\u2019s surface is a non-trivial problem. So complex is the problem that an entire profession, called\u00a0geodesy<\/strong>, has arisen to deal with it.\r\n\r\nGeodesists define the Earth\u2019s surface as\u00a0a surface that closely approximates global mean sea level, but across which gravity is everywhere equal.<\/strong>\u00a0They refer to this shape as the\u00a0geoid<\/strong>. Geoids are lumpy because gravity varies from place to place in response to local differences in terrain and variations in the density of materials in the Earth\u2019s interior. Geoids are also a little squat. Sea level gravity at the poles is greater than sea level gravity at the equator, a consequence of Earth\u2019s \u201coblate\u201d shape as well as the centrifugal force associated with its rotation.\r\n\r\nGeodesists at the\u00a0U.S. National Geodetic Survey<\/a>\u00a0describe the geoid as an \u201cequipotential surface\u201d because the potential energy associated with the Earth\u2019s gravitational pull is equivalent everywhere on the surface. Like fitting a trend line through a cluster of data points, the geoid is a three-dimensional statistical surface that fits as closely as possible gravity measurements taken at millions of locations around the world. As additional and more accurate gravity measurements become available, geodesists revise the geoid periodically. Some geoid models are solved only for limited areas; GEOID03, for instance, is calculated only for the continental U.S.\r\n\r\nRecall that horizontal datums define how coordinate system grids align with the Earth\u2019s surface. Long before geodesists calculated geoids, surveyors used much simpler surrogates called\u00a0ellipsoids<\/strong>\u00a0to model the shape of the Earth.\r\n

              2.15. Ellipsoids<\/h2>\r\n\"Diagram\r\n\r\nEllipsoids approximate the geoid (Adapted from Smith, 1988).\r\n\r\nAn ellipsoid is a three-dimensional geometric figure that resembles a sphere, but whose equatorial axis (a<\/em>\u00a0is the illustration above) is slightly longer than its polar axis (b<\/em>). The equatorial axis of the\u00a0World Geodetic System<\/a>\u00a0of 1984, for instance, is approximately 22 kilometers longer than the polar axis, a proportion that closely resembles the oblate spheroid that is planet Earth.\u00a0Ellipsoids are commonly used as surrogates for geoids so as to simplify the mathematics involved in relating a coordinate system grid with a model of the Earth\u2019s shape.<\/strong>\u00a0Ellipsoids are good, but not perfect, approximations of geoids. The map below shows differences in elevation between a geoid model called GEOID96 and the WGS84 ellipsoid. The surface of GEOID96 rises up to 75 meters above the WGS84 ellipsoid over New Guinea (where the map is colored red). In the Indian Ocean (where the map is colored purple), the surface of GEOID96 falls about 104 meters below the ellipsoid surface.\r\n\r\n\"Colored\r\n\r\nDeviations between an ellipsoid and a geoid (National Geodetic Survey, 1997).\r\n\r\nMany ellipsoids are in use around the world. (Peter Dana has published a list at\u00a0colorado.edu<\/a>) Local ellipsoids minimize differences between the geoid and the ellipsoid for individual countries or continents. The Clarke 1866 ellipsoid, for example, minimizes deviations in North America. TheNorth American Datum of 1927<\/strong>\u00a0(NAD 27) associates the geographic coordinate grid with the Clarke 1866 ellipsoid. NAD 27 involved an adjustment of the latitude and longitude coordinates of some 25,000 geodetic control point locations across the U.S. The nationwide adjustment commenced from an initial control point at Meades Ranch, Kansas, and was meant to reconcile discrepancies among the many local and regional control surveys that preceded it.\r\n\r\nThe North American Datum of 1983<\/strong>\u00a0(NAD 83) involved another nationwide adjustment, necessitated in part by the adoption of a new ellipsoid, called\u00a0GRS 80<\/strong>. Unlike Clarke 1866, GRS 80 is a global ellipsoid centered upon the Earth\u2019s center of mass. GRS 80 is essentially equivalent to WGS 84, the global ellipsoid upon which the Global Positioning System is based.\u00a0NAD 27 and NAD 83 both align coordinate system grids with ellipsoids. They differ simply in that they refer to different ellipsoids.<\/strong>\u00a0Because Clarke 1866 and GRS 80 differ slightly in shape as well as in the positions of their center points, the adjustment from NAD 27 to NAD 83 involved a shift in the geographic coordinate grid. Because a variety of datums remain in use, geospatial professionals need to understand this shift, as well as how to transform data between horizontal datums.\r\n

              2.16. Control Points and Datum Shifts<\/h2>\r\n\"Horizontal\r\n\r\nIn the U.S., high-order horizontal control point locations are marked with permanent metal \u201cmonuments\u201d like the one shown above. The physical manifestation of datum is a network of control point measurements (National Geodetic Survey, 2004).\r\n\r\nGeoids, ellipsoids, and even coordinate systems are all abstractions. The fact that \u201chorizontal datum\u201d refers to a relationship between an ellipsoid and a coordinate system, two abstractions, may explain why the concept is so frequently misunderstood. Datums do have physical manifestations, however.\r\n\r\nShown above is one of the approximately two million horizontal and vertical control points that have been established in the U.S. Although control point markers are fixed, the coordinates that specify their locations are liable to change. The\u00a0U.S. National Geodetic Survey<\/a>\u00a0maintains a database of the coordinate specifications of these control points, including historical locations as well as more recent adjustments. One occasion for adjusting control point coordinates is when new horizontal datums are adopted.\u00a0Since every coordinate system grid is aligned with an ellipsoid that approximates the Earth\u2019s shape, coordinate grids necessarily shift when one ellipsoid is replaced by another.<\/strong>\u00a0When coordinate system grids shift, the coordinates associated with fixed control points need to be adjusted. How we account for the Earth\u2019s shape makes a difference in how we specify locations.\r\n

              TRY THIS!<\/strong><\/h3>\r\nHere\u2019s a chance to calculate how much the coordinates of a control point change in response to an adjustment from North American Datum of 1927 (based on the Clarke 1866 ellipsoid) to the North American Datum of 1983 (based upon the GRS 80 ellipsoid). You\u2019ll be asked to interpret your results in an upcoming practice quiz.\r\n
                \r\n\t
              1. Find the geographic coordinates of a populated place<\/strong>\r\n
                  \r\n\t
                1. Start at the USGS\u2019 Geographic Names Information System at theU.S. Board on Geographic Names<\/a><\/li>\r\n\t
                2. Follow the links labeled\u00a0Domestic Names,<\/strong>\u00a0then\u00a0Search\u00a0<\/strong>to search place names included in the Geographic Names Information System.<\/li>\r\n\t
                3. At the Query Form, enter the name of your home town (or other named geographic feature) in the\u00a0Feature Name<\/strong>\u00a0field, as well as your home\u00a0State<\/strong>. Choose\u00a0Populated Place<\/strong>\u00a0(or other, as appropriate) for theFeature Class<\/strong>.\r\n