# What are map projections?

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1 Page 1 of 155 What are map projections? ArcGIS 10 Within ArcGIS, every dataset has a coordinate system, which is used to integrate it with other geographic data layers within a common coordinate framework such as a map. Coordinate systems enable you to integrate datasets within maps as well as to perform various integrated analytical operations such as overlaying data layers from disparate sources and coordinate systems. What is a coordinate system? Coordinate systems enable geographic datasets to use common locations for integration. A coordinate system is a reference system used to represent the locations of geographic features, imagery, and observations such as GPS locations within a common geographic framework. Each coordinate system is defined by: Its measurement framework which is either geographic (in which spherical coordinates are measured from the earth's center) or planimetric (in which the earth's coordinates are projected onto a two-dimensional planar surface). Unit of measurement (typically feet or meters for projected coordinate systems or decimal degrees for latitude longitude). The definition of the map projection for projected coordinate systems. Other measurement system properties such as a spheroid of reference, a datum, and projection parameters like one or more standard parallels, a central meridian, and possible shifts in the x- and y-directions. Types of coordinate systems There are two common types of coordinate systems used in GIS: A global or spherical coordinate system such as latitude longitude. These are often referred to

5 Page 5 of 155 To more accurately represent locations on the earth's surface, mapmakers studied the shape of the earth (geodesy) and created the concept of the spheroid. A datum links a spheroid to a particular portion of the earth's surface. Recent datums are designed to fit the entire earth's surface well. These are the most commonly used datums in North America: North American Datum (NAD) 1927 using the Clarke 1866 spheroid NAD 1983 using the Geodetic Reference System (GRS) 1980 spheroid World Geodetic System (WGS) 1984 using the WGS 1984 spheroid Newer spheroids are developed from satellite measurements and are more accurate than those developed in the 19th and early 20th centuries. You will find that the terms "geographic coordinate system" and "datum" are used interchangeably. Learn more about datums. The coordinates for a location will change depending on the datum and spheroid on which those coordinates are based, even if using the same map projection and projection parameters. For example, the geographic coordinates below are for the city of Bellingham, Washington, using three different datums: Datum Latitude Longitude NAD NAD WGS A principle of good data management is to obtain the coordinate system information from the data source providing the data. Do not guess about the coordinate system of data because this will result in an inaccurate GIS database. The necessary parameters are the following: Geographic coordinate system (Datum) Unit of measure Zone (for UTM or State Plane) Projection Projection parameters Projection parameters may be required, depending on the map projection. For example, the Albers and Lambert conic projections require the following parameters: 1st standard parallel 2nd standard parallel Central meridian Latitude of origin False easting False northing Unit of measure You can define a coordinate system for data with the Define Projection tool in the Data Management toolbox. If the data has a coordinate system definition, but it does not match the typical coordinate system

6 Page 6 of 155 used by an organization, you can reproject the data. You can reproject data in a geodatabase feature dataset, feature class, shapefile, or raster dataset using the Project tool or Project Raster tool in the Data Management toolbox. Identifying an unknown coordinate system The geoid, ellipsoid, spheroid and datum What are map projections? The geoid, ellipsoid, spheroid and datum, and how they are related The geoid is defined as the surface of the earth's gravity field, which is approximately the same as mean sea level. It is perpendicular to the direction of gravity pull. Since the mass of the earth is not uniform at all points, and the direction of gravity changes, the shape of the geoid is irregular. Click on the link below to access a website maintained by the National Oceanographic & Atmospheric Administration (NOAA). The website has links to images showing interpretations of the geoid under North America. To simplify the model, various spheroids or ellipsoids have been devised. These terms are used interchangeably. For the remainder of this article, the term spheroid will be used. A spheroid is a three-dimensional shape created from a two-dimensional ellipse. The ellipse is an oval, with a major axis (the longer axis), and a minor axis (the shorter axis). If you rotate the ellipse, the shape of the rotated figure is the spheroid. The semimajor axis is half the length of the major axis. The semiminor axis is half the length of the minor axis. For the earth, the semimajor axis is the radius from the center of the earth to the equator, while the semiminor axis is the radius from the center of the earth to the pole. One particular spheroid is distinguished from another by the lengths of the semimajor and semiminor axes. For example, compare the Clarke 1866 spheroid with the GRS 1980 and the WGS 1984 spheroids, based on the measurements (in meters) below. Spheroid Semimajor axis (m) Semiminor axis (m) Clarke GRS WGS Spheroid comparison A particular spheroid can be selected for use in a specific geographic area, because that particular spheroid does an exceptionally good job of mimicking the geoid for that part of the world. For North America, the spheroid of choice is GRS 1980, on which the North American Datum 1983 (NAD83) is based. A datum is built on top of the selected spheroid, and can incorporate local variations in elevation. With the spheroid, the rotation of the ellipse creates a totally smooth surface across the world. Because this doesn't reflect reality very well, a local datum can incorporate local variations in elevation.

7 Page 7 of 155 The underlying datum and spheroid to which coordinates for a dataset are referenced can change the coordinate values. An illustrative example using the city of Bellingham, Washington follows. Compare the coordinates in decimal degrees for Bellingham using NAD27, NAD83 and WGS84. It is apparent that while NAD83 and WGS84 express coordinates that are nearly identical, NAD27 is quite different, because the underlying shape of the earth is expressed differently by the datums and spheroids used. Datum Longitude Latitude NAD NAD WGS Geographic coordinates below are for the city of Bellingham, Washington using 3 different datums The longitude is the measurement of the angle from the prime meridian at Greenwich, England, to the center of the earth, then west to the longitude of Bellingham, Washington. The latitude is the measurement of the angle formed from the equator to the center of the earth, then north to the latitude of Bellingham, Washington. If the surface of the earth at Bellingham is bulged out, the angular measurements in decimal degrees from Greenwich and the equator will become slightly larger. If the surface at Bellingham is lowered, the angles will become slightly smaller. These are two examples of how the coordinates change based on the datum. Datums Projection basics for GIS professionals Spheroids and spheres Identifying an unknown coordinate system Coordinate system information is usually obtained from the data source, but not always, as with legacy data. The technique described below helps identify the correct coordinate system. If the coordinate system is unknown, you will receive this warning message when trying to add the layer to ArcMap: The following data sources you added are missing spatial reference information. This data can be drawn in ArcMap, but cannot be projected. The term coordinate system can refer to data expressed in decimal degrees, or a projected coordinate system expressed in meters or feet. The term projection, or PRJ, is an older term that is also used, but it is not as precise. If a data source has a defined coordinate system, ArcMap can project it on the fly to a different coordinate system. If data does not have a defined coordinate system, ArcMap cannot project it on the fly. ArcMap simply will draw it. If you change the data frame's coordinate system, all layers that have coordinate systems will be projected on the fly to the new coordinate system. If you set the data frame's coordinate system and the data with a known coordinate system lines up with the unknown data, the data frame's coordinate system is that of the unknown data's. Steps: 1. Start ArcMap with a new empty map and add the data with the unknown coordinate

10 Page 10 of 155 DD = (Seconds/3600) + (Minutes/60) + Degrees The conversion must be handled differently if the degrees value is negative. Here's one way: DD = - (Seconds/3600) - (Minutes/60) + Degrees In the instructions below you will convert one field in a table of latitude or longitude values in Degrees, Minutes, and Seconds to Decimal Degrees using the Field Calculator. The code is in VBA, but is easily converted to other programming languages. It is assumed that the Degrees, Minutes, and Seconds are stored as a string (text), with spaces between the numbers and not containing any symbols. For example the data would be stored as: where 25 is degrees, 35 is minutes, and 22.3 is seconds. The output will be stored in a number field. Steps: 1. Add the table to ArcMap. 2. Right-click the table in the table of contents and click open. 3. Click the Options button and click Add Field. 4. Type "Lat2" in the Name field. 5. Click the Type drop-down arrow and click Double from the list. If "Lat2" is already used as a field name, select a name that is not used. 6. Click OK. 7. Right-click the Lat2 field and click Field Calculator. 8. Click Yes if presented with a message box. 9. Check the Advanced check box. 10. Paste the following code into the expression box: Dim Degrees as Double Dim Minutes as Double Dim Seconds as Double Dim DMS as Variant Dim DD as Double DMS = Split([Latitude]) Degrees = CDbl(DMS(0)) Minutes = CDbl(DMS(1)) Seconds = CDbl(DMS(2)) If Degrees < 0 Then DD = -(Seconds/3600) - (Minutes/60) + Degrees Else DD = (Seconds/3600) + (Minutes/60) + Degrees End If In the sixth line, beginning with 'DMS...' the text within the brackets [ ] should be the name of the field holding the latitude values. Replace the word Latitude in the code with

11 Page 11 of 155 the name of the field (in your table) that stores the DMS latitude values in the table. 11. Paste the following code into the 'Lat2 =' box at the bottom of the dialog box. CDbl(DD) 12. Click OK. 13. Repeat steps 3 through 12 for the longitude values. What are geographic coordinate systems? A geographic coordinate system (GCS) uses a three-dimensional spherical surface to define locations on the earth. A GCS is often incorrectly called a datum, but a datum is only one part of a GCS. A GCS includes an angular unit of measure, a prime meridian, and a datum (based on a spheroid). A point is referenced by its longitude and latitude values. Longitude and latitude are angles measured from the earth's center to a point on the earth's surface. The angles often are measured in degrees (or in grads). The following illustration shows the world as a globe with longitude and latitude values. In the spherical system, horizontal lines, or east west lines, are lines of equal latitude, or parallels. Vertical lines, or north south lines, are lines of equal longitude, or meridians. These lines encompass the globe and form a gridded network called a graticule. The line of latitude midway between the poles is called the equator. It defines the line of zero latitude. The line of zero longitude is called the prime meridian. For most geographic coordinate systems, the prime meridian is the longitude that passes through Greenwich, England. Other countries use longitude lines that pass through Bern, Bogota, and Paris as prime meridians. The origin of the graticule (0,0) is defined by where the equator and prime meridian intersect. The globe is then divided into four geographical quadrants that are based on compass bearings from the origin. North and south are above and below the equator, and west and east are to the left and right of the prime meridian.

12 Page 12 of 155 Latitude and longitude values are traditionally measured either in decimal degrees or in degrees, minutes, and seconds (DMS). Latitude values are measured relative to the equator and range from - 90 at the South Pole to +90 at the North Pole. Longitude values are measured relative to the prime meridian. They range from -180 when traveling west to 180 when traveling east. If the prime meridian is at Greenwich, then Australia, which is south of the equator and east of Greenwich, has positive longitude values and negative latitude values. It may be helpful to equate longitude values with X and latitude values with Y. Data defined on a geographic coordinate system is displayed as if a degree is a linear unit of measure. This method is basically the same as the Plate Carrée projection. Learn more about the Plate Carrée projection Although longitude and latitude can locate exact positions on the surface of the globe, they are not uniform units of measure. Only along the equator does the distance represented by one degree of longitude approximate the distance represented by one degree of latitude. This is because the equator is the only parallel as large as a meridian. (Circles with the same radius as the spherical earth are called great circles. The equator and all meridians are great circles.) Above and below the equator, the circles defining the parallels of latitude get gradually smaller until they become a single point at the North and South Poles where the meridians converge. As the meridians converge toward the poles, the distance represented by one degree of longitude decreases to zero. On the Clarke 1866 spheroid, one degree of longitude at the equator equals km, while at 60 latitude it is only km. Because degrees of latitude and longitude don't have a standard length, you can t measure distances or areas accurately or display the data easily on a flat map or computer screen. Tables of the supported geographic coordinate systems, datums, and so on are available in a geographic_coordinate_systems.pdf file in the ArcGIS Documentation folder. This illustration shows the parallels and meridians that form a graticule. Datums Fundamentals of vertical coordinate systems North American datums Spheroids and spheres What are projected coordinate systems? Spheroids and spheres The shape and size of a geographic coordinate system's surface is defined by a sphere or spheroid.

13 Page 13 of 155 Although the earth is best represented by a spheroid, it is sometimes treated as a sphere to make mathematical calculations easier. The assumption that the earth is a sphere is possible for small-scale maps (smaller than 1:5,000,000). At this scale, the difference between a sphere and a spheroid is not detectable on a map. However, to maintain accuracy for larger-scale maps (scales of 1:1,000,000 or larger), a spheroid is necessary to represent the shape of the earth. Between those scales, choosing to use a sphere or spheroid will depend on the map's purpose and the accuracy of the data. Definition of a spheroid A sphere is based on a circle, while a spheroid (or ellipsoid) is based on an ellipse. A spheroid, or ellipsoid, is a sphere flattened at the poles. The shape of an ellipse is defined by two radii. The longer radius is called the semimajor axis, and the shorter radius is called the semiminor axis. The semimajor axis, or equatorial radius, is half the major axis. the semiminor axis, or polar radius, is half the minor axis. Rotating the ellipse around the semiminor axis creates a spheroid. A spheroid is also known as an oblate ellipsoid of revolution. The following graphic shows the semimajor and semiminor axes of a spheroid. The semimajor axis is in the equatorial plane, while the semiminor axis is perpendicular to the equatorial plane. A spheroid is defined by either the semimajor axis, a, and the semiminor axis, b, or by a and the flattening. The flattening is the difference in length between the two axes expressed as a fraction or a decimal. The flattening, f, is derived as follows: f = (a - b) / a The flattening is a small value, so usually the quantity 1/f is used instead. These are the spheroid parameters for the World Geodetic System of 1984 (WGS 1984 or WGS84):

14 Page 14 of 155 a = meters b = meters 1/f = The flattening ranges from 0 to 1. A flattening value of 0 means the two axes are equal, resulting in a sphere. The flattening of the earth is approximately Another quantity that, like the flattening, describes the shape of a spheroid is the square of the eccentricity, e 2. It is represented by the following: Defining different spheroids for accurate mapping The earth has been surveyed many times to better understand its surface features and their peculiar irregularities. The surveys have resulted in many spheroids that represent the earth. Generally, a spheroid is chosen to fit one country or a particular area. A spheroid that best fits one region is not necessarily the same one that fits another region. Until recently, North American data used a spheroid determined by Clarke in The semimajor axis of the Clarke 1866 spheroid is 6,378,206.4 meters, and the semiminor axis is 6,356,583.8 meters. Because of gravitational and surface feature variations, the earth is neither a perfect sphere nor a perfect spheroid. Satellite technology has revealed several elliptical deviations; for example, the South Pole is closer to the equator than the North Pole. Satellite-determined spheroids are replacing the older ground-measured spheroids. For example, the new standard spheroid for North America is the Geodetic Reference System of 1980 (GRS 1980), whose radii are 6,378,137.0 and 6,356, meters. The GRS 1980 spheroid parameters were set by the International Union for Geodesy and Geophysics in Because changing a coordinate system's spheroid will change all feature coordinate values, many organizations haven't switched to newer (and more accurate) spheroids. Datums North American datums What are geographic coordinate systems? Datums While a spheroid approximates the shape of the earth, a datum defines the position of the spheroid relative to the center of the earth. A datum provides a frame of reference for measuring locations on the surface of the earth. It defines the origin and orientation of latitude and longitude lines. Learn more about spheroids and spheres. Whenever you change the datum, or more correctly, the geographic coordinate system, the coordinate values of your data will change. Here are the coordinates in degrees/minutes/seconds (DMS) of a control point in Redlands, California, on the North American Datum of 1983 (NAD 1983 or NAD83):

15 Page 15 of Here's the same point on the North American Datum of 1927 (NAD 1927 or NAD27): The longitude value differs by approximately 3 seconds, while the latitude value differs by about 0.05 seconds. NAD 1983 and the World Geodetic System of 1984 (WGS 1984) are identical for most applications. Here are the coordinates for the same control point based on WGS 1984: Geocentric datums In the last 15 years, satellite data has provided geodesists with new measurements to define the best earth-fitting spheroid, which relates coordinates to the earth's center of mass. An earthcentered, or geocentric, datum uses the earth's center of mass as the origin. The most recently developed and widely used datum is WGS It serves as the framework for locational measurement worldwide. Local datums A local datum aligns its spheroid to closely fit the earth's surface in a particular area. A point on the surface of the spheroid is matched to a particular position on the surface of the earth. This point is known as the origin point of the datum. The coordinates of the origin point are fixed, and all other points are calculated from it. The coordinate system origin of a local datum is not at the center of the earth. The center of the spheroid of a local datum is offset from the earth's center. NAD 1927 and the European Datum of 1950 (ED 1950) are local datums. NAD 1927 is designed to fit North America reasonably well, while ED 1950 was created for use in Europe. Because a local datum aligns its spheroid so closely to a particular area on the earth's surface, it's not suitable for use outside the area for which it was designed. Learn more about North American datums. North American datums Spheroids and spheres What are geographic coordinate systems?

18 Page 18 of 155 and -180 lines are separated, a feature that crosses outside the horizon will be clipped to the horizon. If the data frame is using a projected coordinate system, with the +/-180 line inside, a feature that crosses it will be split into multiple parts. Some clients instead use a or -360 to 0 longitude range, although this does just move the issue to the Greenwich meridian. It's also possible to store data in a projected coordinate system that is centered in the Pacific, but again, that will mean that there is a "break" somewhere else in the world. GCS_Assumed_Geographic_1 The GCS_Assumed_Geographic_1 coordinate system definition was created for ArcGIS 8 to permit ArcMap to infer the coordinate system for certain shapefiles when the coordinate system is undefined. The shapefile's extent must have values that look like decimal degrees. For instance, if longitude, or x, values are between -180 and +180, ArcMap "assumes" the coordinate system is in degrees and assigns GCS_Assumed_Geographic_1. ArcMap then can project the data on the fly, although there are no valid geographic (datum) transformations. The coordinate system is not permanently attached to the shapefile. If you look at the shapefile's metadata, the coordinate system is undefined. In version 9.2, GCS_Assumed_Geographic_1 was removed. Shapefiles, even if their extents fit decimal degrees, will have an unknown coordinate system in ArcMap. Shapefiles that did line up, because they were projected on the fly before, will no longer line up with datasets in other coordinate systems. If you have a shapefile that no longer lines up in version 9.2, check its coordinate system. Occasionally, data has been defined with the GCS_Assumed_Geographic_1 coordinate system. If you see that a shapefile or other data type displays the GCS_Assumed_Geographic_1 coordinate system, take steps to identify the data's true coordinate system. What are projected coordinate systems? A projected coordinate system is defined on a flat, two-dimensional surface. Unlike a geographic coordinate system, a projected coordinate system has constant lengths, angles, and areas across the two dimensions. A projected coordinate system is always based on a geographic coordinate system that is based on a sphere or spheroid. In a projected coordinate system, locations are identified by x,y coordinates on a grid, with the origin at the center of the grid. Each position has two values that reference it to that central location. One specifies its horizontal position and the other its vertical position. The two values are called the x- coordinate and y-coordinate. Using this notation, the coordinates at the origin are x = 0 and y = 0. On a gridded network of equally spaced horizontal and vertical lines, the horizontal line in the center is called the x-axis and the central vertical line is called the y-axis. Units are consistent and equally spaced across the full range of x and y. Horizontal lines above the origin and vertical lines to the right of the origin have positive values; those below or to the left have negative values. The four quadrants represent the four possible combinations of positive and negative X and Y coordinates.

19 Page 19 of 155 When working with data in a geographic coordinate system, it is sometimes useful to equate the longitude values with the X axis and the latitude values with the Y axis. Lists of the supported projected coordinate systems are available in a projected_coordinate_systems.pdf file in the ArcGIS Documentation folder. About map projections Projection types About map projections Whether you treat the earth as a sphere or a spheroid, you must transform its three-dimensional surface to create a flat map sheet. This mathematical transformation is commonly referred to as a map projection. One easy way to understand how map projections alter spatial properties is to visualize shining a light through the earth onto a surface, called the projection surface. Imagine the earth's surface is clear with the graticule drawn on it. Wrap a piece of paper around the earth. A light at the center of the earth will cast the shadows of the graticule onto the piece of paper. You can now unwrap the paper and lay it flat. The shape of the graticule on the flat paper is different from that on the earth. The map projection has distorted the graticule. A spheroid can't be flattened to a plane any more easily than a piece of orange peel can be flattened it will rip. Representing the earth's surface in two dimensions causes distortion in the shape, area, distance, or direction of the data. A map projection uses mathematical formulas to relate spherical coordinates on the globe to flat, planar coordinates. Different projections cause different types of distortions. Some projections are designed to minimize the distortion of one or two of the data's characteristics. A projection could maintain the area of a feature but alter its shape. In the graphic below, data near the poles is stretched.

20 Page 20 of 155 The following diagram shows how three-dimensional features are compressed to fit onto a flat surface. Map projections are designed for specific purposes. One map projection might be used for large-scale data in a limited area, while another is used for a small-scale map of the world. Map projections designed for small-scale data are usually based on spherical rather than spheroidal geographic coordinate systems. Conformal projections Conformal projections preserve local shape. To preserve individual angles describing the spatial relationships, a Conformal projection must show the perpendicular graticule lines intersecting at 90-degree angles on the map. A map projection accomplishes this by maintaining all angles. The drawback is that the area enclosed by a series of arcs may be greatly distorted in the process. No map projection can preserve shapes of larger regions. Equal area projections Equal area projections preserve the area of displayed features. To do this, the other properties shape, angle, and scale are distorted. In Equal area projections, the meridians and parallels may not intersect at right angles. In some instances, especially maps of smaller regions, shapes are not obviously distorted, and distinguishing an Equal area projection from a Conformal projection is difficult unless documented or measured. Equidistant projections Equidistant maps preserve the distances between certain points. Scale is not maintained correctly by any projection throughout an entire map. However, there are in most cases, one or more lines on a map along which scale is maintained correctly. Most Equidistant projections have one or more lines in which the length of the line on a map is the same length (at map scale) as the same line on the globe, regardless of whether it is a great or small circle, or straight or curved. Such distances are said to be true. For example, in the Sinusoidal projection, the equator and all parallels are their true lengths. In other Equidistant projections, the equator and all meridians are

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