# Representing The Shape of the Earth

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1 Representing The Shape of the Earth Geoids Ellipsoids Datums Coordinate Systems Projections The Shape of the Earth 3 ways to model it Topographic surface the land/air interface complex (rivers, valleys, etc) and difficult to model Geoid a theoretical, continuous surface for the earth which is perpendicular at every point to the direction of gravity (surface to which plumb line is perpendicular) approximates mean sea level in open ocean without tides, waves or swell satellite observation (after 1957) showed it to be quite irregular because of local variations in gravity. Spheres and spheroids (3 dimensional circle and ellipse) mathematical models which can be used to approximate the geoid and provide the basis for accurate location (horizontal) and elevation (vertical) measurement sphere (3 dimensional circle) with radius of 6,370,997m considered close enough for small scale maps (1:5,000,000 and below e.g. 1:7,500,000) spheroid (3 dimensional ellipse) should be used for larger scale maps of 1:1,000,000 or more (e.g. 1:24,000) 1

2 Spheroid, Ellipsoid, and Geoid Spheroid is a solid generated by rotating an ellipse about either the major or minor axis Ellipsoid is a solid for which all plane sections through one axis are ellipses and through the other are ellipses or circles If any two of the three axes of that ellipsoid are equal, the figure becomes a spheroid (ellipsoid of revolution) If all three are equal, it becomes a sphere Geoid is the equipotential gravity surface of the earth at mean sea level. At any point it is perpendicular to the direction of gravity Reference: Smith, James R., Introduction to Geodesy: The history and concepts of modern geodesy. John Wiley & Sons, 1997 What is an Oblate Ellipsoid (Spheroid)? 2

3 Which Spheroid to use? Hundreds have been defined depending upon: Available measurement technology Area of the globe e.g North America, Africa Map extent country, continent or global Political issues e.g Warsaw pact versus NATO ArcGIS supports 26 different spheroids! conversions via math formulae Most commonly encountered are: Clarke 1866 for North America basis for USGS 7.5 Quads a=6,378,206.4m b=6,356,583.8m GRS80 (Geodetic Ref. System, 1980) current North America mapping a=6,378,137m b=6,356, m WGS84 (World Geodetic Survey, 1984) current global choice a=6,378,137 b=6,356, Latitude and Longitude: location on the spheroid Lat / long coordinates for a location change depending on spheroid chosen! Longitude meridians Prime meridian is zero: Greenwich, U.K. International Date Line is 180 E&W 1 degree=69.17 mi at Equator mi at 40N/S 0 mi at 90N/S Latitude parallels equator is zero 1 degree=68.70 mi at equator mi at poles (1 mile= km=5280 feet) 3

4 Latitude and Longitude Graticule graticule: network of lines on globe or map representing latitude and longitude. Origin is at Equator/Prime Meridian intersection (0,0) grid: set of uniformly spaced straight lines intersecting at right angles. (XY Cartesian coordinate system) Latitude normally listed first (lat,long), the reverse of the convention for X,Y Cartesian coordinates Latitude and Longitude The most comprehensive and powerful method of georeferencing Metric, standard, stable, unique Uses a well defined and fixed reference frame Based on the Earth s rotation and center of mass, and the Greenwich Meridian 4

5 Definition of longitude. The Earth is seen here from above the North Pole, looking along the Axis, with the Equator forming the outer circle. The location of Greenwich defines the Prime Meridian. The longitude of the point at the center of the red cross is determined by drawing a plane through it and the axis, and measuring the angle between this plane and the Prime Meridian. Definition of Latitude Requires a model of the Earth s shape The Earth is somewhat elliptical The N S diameter is roughly 1/300 less than the E W diameter More accurately modeled as an ellipsoid than a sphere An ellipsoid is formed by rotating an ellipse about its shorter axis (the Earth s axis in this case) 5

6 The Public Land Survey System (PLSS) 6

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11 The (Brief) History of Ellipsoids Because the Earth is not shaped precisely as an ellipsoid, initially each country felt free to adopt its own as the most accurate approximation to its own part of the Earth Today an international standard has been adopted known as WGS 84 Its US implementation is the North American Datum of 1983 (NAD 83) Many US maps and data sets still use the North American Datum of 1927 (NAD 27) Differences can be as much as 200 m 11

12 Latitude and the Ellipsoid Latitude (of the red point) is the angle between a perpendicular to the surface and the plane of the Equator WGS 84 Radius of the Earth at the Equator km Flattening 1 part in Geoid A geoid is a representation of the Earth which would coincide exactly with the mean ocean surface of the Earth, if the oceans were to be extended through the continents A smooth but highly irregular surface that corresponds but to a surface which can only be known through extensive gravitational measurements and calculations, not to the actual surface of the Earth's crust 12

13 Geoid Geoid vs. an Ellipsoid 1. Ocean 2. Ellipsoid 3. Local plumb 4. Continent 5. Geoid 13

14 Datums: all location measurement is relative to a specific datum For the Geodesist a set of parameters defining a coordinate system, including: the spheroid (earth model) a point of origin (ties spheroid to earth) For the Local Surveyor a set of points whose precise location and /or elevation has been determined, which serve as reference points from which other point s locations can be determined (horizontal datum) a surface to which elevations are referenced, usually mean sea level (vertical datum) points usually marked with brass plates called survey markers or monuments whose identification codes and precise locations (usually in lat/long) are published North American Datums NAD27 Clark 1866 spheroid Meades Ranch origin visual triangulation 25,000 stations (250,000 by 1970) NAVD29 (North American Vertical Datum, 1929) provided elevation basis for most USGS 7.5 minute quads NAD83 satellite (since 1957) and laser distance data showed inaccuracy of NAD National Academy of Sciences report recommended new datum used GRS80 spheroid (functionally equivalent to WGS84) origin: Mass center of Earth 275,000 stations Helmert blocking least squares technique fitted 2.5 million other fed, state and local agency points. NAVD88 provides vertical datum points can differ up to 160m from NAD27, but seldom more than 30m, and data from digit. map more inaccurate than datum diff. no universal mathematical formulae for conversion from NAD27: See USGS Survey Bulletin # 1875 for conversion tables (in ARC/INFO). Transformations are preformed to local coordinates bin/nadcon.prl 14

15 Ground zero for Geo nerds everywhere The NGS data sheet is here: Meades Ranch, KS (12 miles north of Lucas, KS) is the designated geodedetic base point for the North American Datum 1927 (NAD 27) Owner of ranch is now Mr. Kyle Brant access with permission only State Plane & the NAD 27 Calculations for map projections are performed using the parameters of the ellipsoid 15

16 Roadside Marker for the Geodetic Center of North America Meades Ranch, KS (12 miles north of Lucas, KS) is the designated geodedetic base point for the North American Datum 1927 (NAD 27) 16

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18 Map Projections: the concept A method by which the curved 3 D surface of the earth is represented on a flat 2 D map surface a two dimensional representation, using a plane coordinate system, of the earth s three dimensional sphere/spheroid location on the 3 D earth is measured by latitude and longitude location on the 2 D map is measured by x,y Cartesian coordinates unlike choice of spheroid, choice of map projection does not change a location s lat/long coords, only its XY coords. Map projections Earth spherical maps flat! Thus all maps have distortions A good map has distortions that are predictable and systematic 18

19 Map projection Why is it called a projection? Because we project the earth s spherical surface on a flat surface As if we were shining a light from center of earth: Maps can be: 1. Conformal: Shape correct 2. Equivalent (Equal area): Area correct 3. Azimuthal: Direction correct 19

20 Physical Surface Use a physical surface to project the sphere 1. Plane 2. Cone 3. Cylinder Note differences between projections by comparing distortions of the lines of latitude and longitude. 1. Plane projection Hold plane against the surface of the globe (typically the pole) Lines of longitude straight, radiating Lines of latitudes are circles 20

21 1. Plane projection 4. Distortion increases away from the center ("principal point") 5. Good for polar regions 6. But can't show more than half the world. 2. Conic projection Hold cone over pole 21

22 2. Conic projection 2. Lines of latitude curve 3. Line of longitude are straight, and converge towards top 4. Distortion increases away from standard parallel 5. Good for Mid latitudes 3. Cylindrical projection 1. Wrap cylinder around the earth 2. Lines of latitude and longitude are straight, intersect at Distortion greatest at poles 22

23 3. Cylindrical projection 4. Good for lowlatitude areas 5. Poor representation of poles Good for navigation Mercator s Projection Other mathematical Condensed and interrupted projections. 23

24 Projections and Coordinates There are many reasons for wanting to project the Earth s surface onto a plane, rather than deal with the curved surface The paper used to output GIS maps is flat Flat maps are scanned and digitized to create GIS databases Rasters are flat, it s impossible to create a raster on a curved surface The Earth has to be projected to see all of it at once It s much easier to measure distance on a plane Map Projections: the inevitability of distortion because we are trying to represent a 3 D sphere on a 2 D plane, distortion is inevitable thus, every two dimensional map is inaccurate with respect to at least one of the following: area shape distance direction We are trying to represent this amount of the earth on this amount of map space. 24

25 Distortions Any projection must distort the Earth in some way Two types of projections are important in GIS Conformal property: Shapes of small features are preserved: anywhere on the projection the distortion is the same in all directions Equal area property: Shapes are distorted, but features have the correct area Both types of projections will generally distort distances! Map Projections: classifications or How can we think about map projections? Classified by Property Preserved or by Geometrical Model Property Preserved Equal area projections preserve the area of features (popular in GIS) Conformal projections preserve the shape of small features (good for presentations), and show local directions (bearings) correctly (useful for coastal navigation) Equidistant projections preserve distances (scale) to places from one point, or along a one or more lines True direction projections preserve bearings (azimuths) either locally (in which case they are also conformal) or from center of map. Geometric Model Used Planar/Azimuthal/Zenithal: image of spherical globe is projected onto a map plane which is tangent to (touches) globe at single point Conical: image of spherical globe is projected onto a cone which is tangent along a line(s) (usually a parallel of latitude) cone is then unfolded to create flat map Cylindrical: image of spherical globe is projected onto a cylinder which also is tangent along a line(s) again, cylinder is unfolded to create a flat map 25

26 Geometric Model Choosing a Map Projection Issues to Consider: Extent of area to map: city, state, country, world? Location: polar, mid latitude, equatorial? Predominant extent of area to map: E W, N S, oblique? Rules of thumb Always record lat/long coords not projected X,Y coords in the GIS database if possible Check project specifications; does it specify a required projection? State Plane or UTM often specified for US gov. work. Use equal area projections for thematic or distribution maps, and as a general choice for GIS work Use conformal projections in presentations For navigational applications, need true distance or direction 26

27 Commonly Encountered Map Projections in GIS Albers Conic Equal Area often used for US base maps showing all of the lower 48 states standard parallels set at 29 1/2N and 45 1/2N Lambert Conformal Conic often used for US Base map of all 50 states (including Alaska and Hawaii), with standard parallels set at 37N and 65N also for State Base Map series, with standard parallels at 33N and 45N also used in State Plane Coordinate System (SPCS) Transverse Mercator used in SPCS for States with major N/S extent Universal Transverse Mercator (UTM) used for world wide military (and other) large scale mapping Most commonly, you encounter these 3 projections, along with the SPCS and UTM projections systems which use them. Cylindrical Projections Normal Mercator Transverse Transverse Mercator Oblique Oblique Mercator Parallels are line of tangency Meridians are line of tangency 27

28 Preserve local shape Misrepresent areas Conformal Projections Graticule lines (lat and long) on globe are perpendicular Best used for large scale reference maps Preserve angles and shapes at points Do not use for data distribution maps will distort area, and therefore, misrepresent densities Better for showing routes and locations Lambert Conformal Conic Conterminous U.S. 28

29 Equal Area Projections Preserve the area of displayed features Graticule lines (lat and long) on globe may not intersect at right angles In some instances (especially maps of smaller regions) it will not be obvious that shape has been distorted Thus, it will not be obvious that shape has been distorted and distinguishing from an equal area projection from a conformal projection is difficult (unless documented) Choose equal area when making thematic maps Remember if you are mapping data distributions, choose an equal area projection! Albers Equal Area is customized for the continental United States Has two lines of standard parallel (two lines that are true ) Albers Equal Area Conic Conterminous U.S. 29

30 The Unprojected Projection Assign latitude to the y axis and longitude to the x axis A type of cylindrical projection Is neither conformal nor equal area As latitude increases, lines of longitude are much closer together on the Earth, but are the same distance apart on the projection Also known as the Plate Carrée or Cylindrical Equidistant Projection The Universal Transverse Mercator (UTM) Projection A type of cylindrical projection Implemented as an internationally standard coordinate system Initially devised as a military standard Uses a system of 60 zones Maximum distortion is 0.04% Transverse Mercator because the cylinder is wrapped around the Poles, not the Equator 30

31 Zones are each six degrees of longitude, numbered as shown at the top, from W to E Universal Transverse Mercator (UTM) First adopted by US Army in 1947 for large scale maps worldwide Used from lat. 84 N to 80 S; Universal Polar Stereographic (UPS) used for polar areas Globe divided into 60 N/S zones, each 6 wide; these are numbered from one to sixty going east from 180th meridian Each zone divided into 20 E/W belts, each 8 high lettered from the south pole using C thru X (O and I omitted) The meridian halfway between the two boundary meridians for each zone is designated as the central meridian and a cylindrical projection is done for each zone 31

32 Universal Transverse Mercator (UTM) Coordinate origins are at the intersection of the equator and the zone s central meridian; This origin given a value of 0 meters north, 500,000m east, thus no negative values a false origin at 10,000,000 meters south used for southern hemishere Military uses a different system for coordinate location dividing each UTM primary grid zone into 10km by 10km squares and designates each by a double letter UTM in Pennsylvania: 2 zones: Zone 17 (84W-78W), Zone 18 (78W-72W) Universal Transverse Mercator (UTM) 32

33 Implications of the Zone System Each zone defines a different projection Two maps of adjacent zones will not fit along their common border Jurisdictions that span two zones must make special arrangements Use only one of the two projections, and accept the greaterthan normal distortions in the other zone Use a third projection spanning the jurisdiction E.g. Italy is spans UTM zones 32 and 33 UTM Coordinates In the N Hemisphere define the Equator as 0 mn The central meridian of the zone is given a false easting of 500,000 me Eastings and northings are both in meters allowing easy estimation of distance on the projection A UTM georeference consists of a zone number, a hemisphere, a six digit easting and a sevendigit northing E.g., 14, N, E, N 33

34 State Plane Coordinates Defined in the US by each state Some states use multiple zones Several different types of projections are used by the system Provides less distortion than UTM Preferred for applications needing very high accuracy, such as surveying State Plane Coordinate System (SPCS) Began in 1930s for public works projects; popular with interstate designers States divided into 1 or more zones (~130 total for US) each zone designed to maintain scale distortion to less than 1 part per 10,000 Pennsylvania has 2 zones running E/W: north (3701), south (3702) Different projections used: transverse mercator (conformal) for States with large N/S extent Lambert conformal conic for rest (incl. Pennsylvania) some states use both projections (NY, FL, AK) oblique mercator used for Alaska panhandle Each zone also has: unique standard parallels (2 for Lambert) or central meridian (1 for mercator) false coordinate origins which differ between zones, and use feet for NAD27 and meters for NAD83 (1m=39.37 in. exact used for conversion) See Snyder, 1982 USGS Bulletin # 1532, p for details 34

35 State Plane Coordinate System (SPCS) State Plane Coordinate System (SPCS) 35

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