# Map Projection, Datum and Plane Coordinate Systems

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1 Map Projection, Datum and Plane Coordinate Systems Geodetic Control Network Horizontal Control Network Datum A datum is a set of parameters defining a coordinate system, and a set of control points whose geometric relationships are known, either through measurement or calculation. A datum is defined by an ellipsoid, which approximates the shape of the Earth, and the spheroid s position relative to the center of the Earth. A horizontal 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. North American Network Historically, control networks have been created using the method of triangulation. Until the development of GPS methods, each point was measured on the geoid, and then adjusted for its elevation as to lie on the surface of the ellipsoid. In the US, a network of over 200,000 points is precisely measured. A small bronze survey monument marks the physical location of each control point. The control points have three different accuracy levels: first-, second-, and third-order. North American Datum of 1927 (NAD27): Based on a highly accurate base line beginning at Meades Ranch, Kansas The reference ellipsoid is the Clarke 1866 ellipsoid. A local datum aligns its ellipsoid to closely fit the Earth s surface in a particular area and its origin point is located on the surface of the Earth. The coordinates of the origin point are fixed and all other points are calculated from this control point. The coordinate system origin of a local datum is not at the center of the Earth. North American Datum of 1983 (NAD83) An adjustment of the 1927 datum through a least squares adjustment of nearly two million geodetic control points. With recent advances in satellite geodesy with higher degree of accuracy. The North American Datum is referenced to WGS84 ellipsoid. Recent satellite data has provided geodesists with new measurements to define the best Earth-fitting ellipsoid, which relates coordinates to the Earth s center of mass. An Earth-centered, or geocentric, datum does not have an initial point of origin like a local datum. As a result of adopting the new datum, almost all places in North America will have a slightly different latitude and longitude. Some of these shifts can be as much as 300 m. Vertical Control Network For centuries, elevations have been measured relative to the surface of the geoid at mean sea level. In the US, the first level line was run in 1856 from a tidal gauge in New York harbor up the Hudson River to Albany. 1

2 Differential leveling is previously used to measure the elevation difference between start and ending points. In general, the elevation control points did not coincide with the horizontal control network points. National Geodetic Vertical Datum of 1929 (NGVD29) Were adjusted based on 500,000 vertical control points in National Vertical Datum of 1983 (NVD83): A readjustment with additional vertical control points, elevations are now accurately determined from GPS measurements relative to the WGS84 ellipsoid. The ellipsoidal elevations must be adjusted so as to be relative the mean sea level datum: H = h N where H is elevation, h is the ellipsoidal height, and N is the vertical geoidal undulation. Map Projection Map projection: A systematic transformation of the 3-D spherical surface onto a 2-D plane surface of a flat map. Projection formulas are mathematical expressions that convert data from a geographical location (latitude and longitude) on a sphere or ellipsoid to a representative location on a flat surface. This projection process inevitably distorts at least one of these properties - shape, area, distance, direction - and often more. So no perfect projection! Forward map projection 3D spherical coordinates to 2D map Cartesian coordinates. Inverse map projection 2D map Cartesian coordinates to 3D spherical coordinates. Projection Surface 1) Plane (flat) surface -> Azimuthal projection family The spherical grid is projected onto a flat planar surface touching the globe. also known as azimuthal projection, zenithal projection or perspective projection. In normal orientation, parallels of latitude are concentric circles centered on the pole, and meridians are straight lines that intersect at the pole with their true angles of orientation 2) Cylinder -> Cylindrical projection family A cylinder is developed by cutting along its length and unrolling it. 3) Cone -> Conic projection family Features from the reference globe are transferred onto a conic surface, and then developed by making a cut along a meridian before unwrapping it. The meridian opposite to the cut line is known as the central meridian. 2

3 In normal orientation, the projected graticule has straight converging meridians and concentric circular arcs for parallels. Light-Source Position 1) Gnomonic: light source is at the center of the globe. 2) Stereographic: light source is at the point exactly opposite the point of tangency. 3) Orthographic: at a considerable distance (infinite point). Light rays are parallel. Orientation of Projection Surface 1) Normal (regular) projection The normal orientation for a plane is tangent at the pole (polar azimuthal); A cylinder is normally orientated so that it is tangent along equator (equtorial). A cone is normally oriented so that it is tangent along a parallel, with its apex over the pole, in alignment with the axis of rotation. 2) Transverse projection The projection surface is turned 90 degree from normal. In a transverse projection on a plane, the plane is tangent at the equator (equatorial). In a transverse cylindrical, the cylinder is tangent along a meridian; A transverse conic is not frequently seen. 3) Oblique projection: The project surface lies at an angle somewhere between the normal and transverse position. Tangency of Projection Surface Tangent projection The projection surface is tangent to the globe. A planar surface is tangent to globe at one point only. Tangential cones and cylinders contact the globe along a line. Secant projection The projection surface intersects the globe instead of merely touching its surface. A planar surface intersects the globe, forming a small circle along the intersection line. A cone or cylinder intersects the globe, resulting in two small circles along the intersection lines. Standard point The point at which a planar surface touches the globe. Directions from the standard point are accurate. Distortions of area and angle around the standard point have a circular pattern, and increases with the distance to the standard point. Standard line (line of true scale) The line along which projection surface touches or intersects the globe are called standard lines There is one standard line when a planar surface intersects the globe, or a cone or cylinder is tangential to the globe. 3

4 There are two standard lines when a cone or cylinder intersects the globe. Along the standard line, the map has no distortion, and the map scale is identical to the nominal globe. In general, geometric distortion increases with the distance to the standard lines. Desired Geometric Properties for Projection Distortions: A globe is the only correct representation of the geometric properties of the spherical earth s surface. No map projection can maintain all the geometric properties of the nominal globe (distance, area, shape, and direction) simultaneously. Major properties 1) Equal-area (Equivalence) 2) Conformal (orthomorphic) Minor properties 3) Equi-distant 4) Azimuthal (direction) Factors Influencing the Projection Choice 1) The primary use of the map and desired geometric properties to be preserved Equal-area cylindrical projections are often used to show the world-wide distribution of a variety of geographic phenomena. Presentation maps are usually conformal projections, although compromise and equal-area projections can also be used. Navigational maps are usually Mercator, true direction and/or equidistant. 2) The locations of the area to be mapped Azimuthal projection is often used to map polar region Conic projection is often used to map mid-latitude regions; Cylindrical projection is often used to map equatorial region. 3) Predominant orientation of the area to be mapped Albers Equal-area Conic, Lambert Conformal Conic are often used to map the countries that are mainly east-west in extent. Transverse Mercator is used to map the states that are mainly north-south in extent Azimuthal projection is better for mapping the area with a circular shape. 4) The Extent of the area to be mapped As the mapped area increases to sub-continental, distortion becomes a significant problem. As mapped areas become smaller in extent, the selection of the projection becomes less critical. Potential scale errors begin to drop off considerably. Projections Commonly Used in the US Albers Equal-area Conic: 4

5 This is an equal-area conic projection having two standard parallels, but not conformal, perspective, or equidistant. Well suited for large countries that mainly east-west in extent. USGS uses 29.5N and 45.5N as the standard parallels to map the conterminous US. Maximum scale error is approximately 1.25% over an area the size of the US. Meridians are straight lines that intersect parallels at right angle. Parallels are concentric circles, making construction relatively easy. This was exclusively used in the National Atlas of the United States of America at scales of 1:7,500,000, 1:17,000,000 and 1:34,000,000. I is also used by the Census Bureau for their base maps. Lambert Conformal Conic Map is conformal, but not perspective, equal-area, or equidistant. Distances are true only along standard lines and reasonably accurate elsewhere in limited regions. Directions are reasonably accurate, and distortion of shapes and areas minimal at the standard lines. One of the most widely used map projections in the US. Used by USGS for many 7.5 and 1.5-minute topographic maps and for the State Base Map series for the 48 conterminous states with standard parallels of 33N and 45N. Also used to show other countries or regions that is mainly east-west in extent. Transverse Mercator Transverse cylindrical projection; The map is conformal, the shapes and angles within any small area are essentially true. Distortion of distances and size of areas increases outside the 15 degree of the central meridian. Graticule spacing increases away from central meridian. Used by USGS for many quadrangle maps at scales from 1:24,000 to 1:250,000. Also used for mapping large areas that are mainly north-south in extent. Common Map Projections Used in Texas Texas State Mapping System (TSMS) Projection: Lambert Conformal Conic Spheroid: Clarke GRS 80 Datum: North American Datum of 1983 (NAD83) Longitude of Origin: 100 degrees West (-100) Latitude of Origin: 31 degrees 10 minutes North (31.16_) Standard Parallel #1: 27 degrees 25 minutes North (27.416_) Standard Parallel #2: 34 degrees 55 minutes North (34.916_) False Easting: 1,000,000 meters False Northing: 1,000,000 meters Units of Measure: meters Shackelford Projection: Lambert Conformal Conic 5

6 Spheroid: Clarke 1866 Datum: North American Datum of 1927 (NAD27) Longitude of Origin: 100 degrees West (-100) Latitude of Origin: 31 degrees 10 minutes North (31.16_) Standard Parallel #1: 27 degrees 25 minutes North (27.416_) Standard Parallel #2: 34 degrees 55 minutes North (34.916_) False Easting: 3,000,000 feet (914,400 meters) False Northing: 3,000,000 feet (914,400 meters) Units of Measure: feet (international) 6

7 Texas Centric Mapping System/Lambert Conformal (TCMS/LC) Projection: Lambert Conformal Conic Spheroid: Clarke GRS 80 Datum: North American Datum of 1983 (NAD83) Longitude of Origin: 100 degrees West (-100) Latitude of Origin: 18 degrees North (18) Standard Parallel #1: 27 degrees 30 minutes (27.5) Standard Parallel #2: 35 degrees (35) False Easting: 1,500,000 meters False Northing: 5,000,000 meters Units of Measure: meters Texas Centric Mapping System/Albers Equal Area (TCMS/AEA) Projection: Albers Equal Area Conic Spheroid: Clarke GRS 80 Datum: North American Datum of 1983 (NAD83) Longitude of Origin: 100 degrees West (-100) Latitude of Origin: 18 degrees North (18) Standard Parallel #1: 27 degrees 30 minutes (27.5) Standard Parallel #2: 35 degrees (35) False Easting: 1,500,000 meters False Northing: 6,000,000 meters Units of Measure: meters UTM Zones in Texas Zone 13 (west), Zone 14 (central), Zone 15 (east) II. Rectangular (Plane) Coordinate Systems The Needs for Rectangular Coordinates Systems Rectangular Coordinate System is used to locate positions on a large-scale map. It is evolved from Cartesian coordinates applied for military needs. Easy plane geometry and trigonometry could be applied. A rectangular grid is superimposed over the projected graticule. Accurate conversions at points can be made from the plane coordinate system (UTM and SPCS grid) to the projection grid. Plane coordinate systems are only used on large-scale maps. Most plane coordinate systems are based on only three projections: the transverse Mercator, the polar stereographic, and Lambert conformal conic. The USGS uses three conformal projections to map the states: the Lambert conformal conic for states with long east-west dimensions, the transverse Mercator for states with long northsouth dimensions, and oblique Mercator for portion of Alaska. In each case, over small areas these projections cause little or no areal and distance distortion. Universal Transversal Mercator (UTM) Grid System 7

8 1) UTM Coordinate System In UTM grid system, the area of the earth between 84 N and 80 S latitude is divided into north-south columns 6 of longitude wide. These columns are called UTM zones. They are numbered from 1 to 60 eastward, beginning at the 180th meridian. For UTM grid system, each of the 60 zones is mapped onto one Transverse Mercator projection. Y-axis is coincides with the central meridian of the 6 longitudal zone. Each zone has its own coordinate system. Each column is divided into quadrilaterals of 8 of latitude. The rows quadrilaterals are assigned letters C to X consecutively. Each quadrilateral is uniquely identified by a number-letter combination. Each 6 by 8 quadrilateral is divided into 100,000-meter zones. Within each 100,000 m square, you can specify an easting and a northing. The SF is constant along each north-south coordinate grid line, but it varies in the east-west direction. A secant case of the transverse Mercator projection is constructed for each zone to minimize variations in the SF over the entire projection. It is a metric system. The meter is the basic unit for measurement. This reference system is theoretically correct to 1-meter resolution. You can calculate distance and directions between two points in a UTM zone to accuracy of one meter in 2,500 meters. Widely adopted by topographic maps, satellite imagery, natural resource databases, etc in the US and other countries. 2) UGSU Modification of UTM Coordinate System Only the zone number (1 through 60) and an easting and northing value are used to designate a point. Within each UTM zone covering part of the United States, the meridian in the zone s center is given a false easting value of 500,000 meters. The equator is given a northing value of 0 meters for the northern hemisphere coordinates and an arbitrary northing value of 10 million meters for the southern hemisphere. Universal Polar Stereographic (UPS) Grid System UPS grid system is used to cover the polar regions (south of 80 S and north of 84 N). Each circular polar zone is divided in half by the line representing the 0 and 180 meridians. In the north polar zone, the west half is designated grid zone Y, the east half as grid zone Z. In south polar zone, the west half is designated A, and the east half B. In the polar areas, the false northings and false eastings of the poles are given 2,000,000 m in both zones. The 2,000,000 easting coincides with the meridian line. The 2,000,000 northing coincides with the 90 E-90 W meridian line. UPS grid zones are divided into 100,000 meter squares as in UMT grid system. 8

9 A secant stereographic projection is used, and standard parallel is 81 latitude. The SFs vary from parallel to parallel. At the pole, the SF is set at (shrinked). In the vicinity of 80 latitude, the SF increases to State Plane Coordinate (SPC) System Devised by the US Coast and Geodetic Survey (now US Chart and Geodetic Survey) for each of the 50 states. The purpose is to permanently record original land survey monument locations. Based on the transverse Mercator or Lambert conformal conic projection. For SPC zones mapped on the transverse Mercator projection, scale is constant north-south along the meridian and varies east-west along the parallels. When Lambert conformal conic is used, scale varies north and south of the curved standard parallels of the projection and is constant east and west along those parallels. To keep the unavoidable scale variation to less than one part in 10,000, a state may have two or more overlapping zones. Each of these zones has its own projection and coordinate grid system. The units used are feet. Tick marks on USGS large-scale topographic maps show locations of the 10,000-foot grids for the SPC zones. To specify a location using SPC notation, you need to give the state, zone name, easting (X value), and northing (Y value) in feet. 9

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