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  1. 1. Spheroid, Datum, Projection and Coordinate System
  2. 2. To locate Geographic Position A model (3D) of the earth surface Spheroid How the 3D model is related to the shape of the earth? Datum A model to translate the 3D points on a 2D surface with minimal distortion Projection A coordinate system to measure the points 3D or 2D
  3. 3. Spheroid and Datum
  4. 4. Earth is not a sphere Earth mass is not distributed uniformly, so the gravitational pull is not uniform Due to rotation, equator is slightly bulged, and poles are slightly flattened (1/300) • Oblate Spheroid Terrain is not uniform
  5. 5. Highest spot on earth? What is the tallest peak on earth? Mount Everest, at 8,850 meters above MSL What is the highest spot on earth where you are the closest to the outer space? Mount Chimborazo, in the Andes, • 6,100 meters above MSL • But is sitting on a bulge which makes it 2,400 meters taller than Everest • Everest is sitting down on the lower side of the same bulge Source:
  6. 6. Deviations (undulations) between the Geoid and the WGS84 ellipsoid Source:
  7. 7. Shape of the Earth We think of the earth as a sphere ... ... when it is actually an ellipsoid/Spheroid, slightly larger in radius at the equator than at the poles.
  8. 8. Earth Models Flat Earth Survey over a short distance ~ 10km Spherical Earth Approximate global distance Ellipsoid Earth Accurate global distance Geoid Surface of equal gravitational potential
  9. 9. Earth Surfaces Sea surface Geoid Ellipsoid Topographic surface Geoid is a surface of constant gravity.
  10. 10. Taking into account all these irregularities is difficult Some irregularities can be ignored For e.g. terrain although important locally, terrain levels are minuscule in planetary scale • the tallest land peak stands less than 9km above sea level, or nearly 1/1440 of Earth diameter • the depth of the most profound sea abyss is roughly 1/1150 diameter.
  11. 11. Ellipse Z An ellipse is defined by: • Focal length = ε • Flattening ratio: f = (a-b)/a • Distance F1-P-F2 is constant for all points P on ellipse • When ε = 0 then ellipse = circle b F1 ε For the earth: •Major axis: a = 6378 km •Minor axis: b = 6357 km •Flattening ratio: f = 1/300 P P O ε a F2 X
  12. 12. Selected Ellipsoids and Datums Source:
  13. 13. Why use different spheroids? The earth's surface is not perfectly symmetrical, so the semi-major and semi-minor axes that fit one geographical region do not necessarily fit another. Satellite technology has revealed several elliptical deviations. For one thing, the most southerly point on the minor axis (the South Pole) is closer to the major axis (the equator) than is the most northerly point on the minor axis (the North Pole).
  14. 14. The earth's spheroid deviates slightly for different regions of the earth. Ignoring deviations and using the same spheroid for all locations on the earth could lead to errors of several meters, or in extreme cases hundreds of meters, in measurements on a regional scale. •EVEREST (India, Nepal, Pakistan, Bangladesh) •WGS84 (GPS World-wide) •Clark 1866 (North America) •GRS80 (North America) •International 1924 (Europe) •Bessel 1841 (Europe)
  15. 15. Ellipsoids used in India Everest – 1830 Colonel Sir George Everest is the name of the surveyor general of India • The Great Trigonometrical Survey India Sometimes referred as Indian Datum Has been modified several times since WGS – 1984 Used by GPS Need for a common global system
  16. 16. Great Trigonometrical Survey Idea • 1745 – General Watson 1802 – real commencement Baseline • 40006.4 feet, on a plane near Saint Thomas' Mount, Madras • From this a series of triangles were formed – 30 to 40 miles in length The objective is to find the curvature of the earth accurately at different latitude and longitude Everest 1830 - Spheroid
  17. 17. Datum Reference frame for locating points on Earth’s surface. Defines origin & orientation of latitude/longitude lines. Defined by spheroid and spheroid’s position relative to Earth’s center. Earth-centered Local
  18. 18. For maps covering very large areas, especially worldwide, the Earth may be assumed perfectly spherical, since any shape imprecision is dwarfed by unavoidable errors in data and media resolution. Conversely, for very small areas terrain features dominate and measurements can be based on a flat Earth
  19. 19. Datum Horizontal Datum or Geodetic Datum Reference frame for locating points on Earth’s surface. Defines origin & orientation of latitude/longitude lines. Defined by spheroid and spheroid’s position relative to Earth’s center.
  20. 20. 2 types of datums: Earth-centered Local
  21. 21. Local datum coordinate system Earth-centered datum coordinate system Earth’s S urface Earth-centered datum (WGS84) Local datum (NAD27)
  22. 22. Datum A mathematical model must be related to real-world features. a datum is a reference point or surface against which position measurements are made, and an associated model of the shape of the earth for computing positions A smooth mathematical surface that fits closely to the mean sea level surface throughout the area of interest. The surface to which the ground control measurements are referred. Provides a frame of reference for measuring locations on the surface of the earth.
  23. 23. How do I get a Datum? To determine latitude and longitude, surveyors level their measurements down to a surface called a geoid. The geoid is the shape that the earth would have if all its topography were removed. Or more accurately, the shape the earth would have if every point on the earth's surface had the value of mean sea level. Indian Datum: Kalianpur hill in Madhya Pradesh Spheroid: Everest
  24. 24. Horizontal vs Vertical Datums Horizontal datums are the reference values for a system of location measurements (E.g. Lat, Long). The horizontal datum is the model used to measure positions on the earth Vertical datums are the reference values for a system of elevation measurements. A vertical datum is used for measuring the elevations of points on the earth's surface
  25. 25. Vertical Datum Vertical data are • either tidal level – based on sea levels, – Tidal datum • gravimetric, – based on a geoid, or geodetic, • based on the same ellipsoid models of the earth used for computing horizontal datums.
  26. 26. Definition of Elevation Elevation Z P • z = zp z = 0 Land Surface Mean Sea level = Geoid Elevation is measured from the Geoid
  27. 27. Projection
  28. 28. Geographic Coordinate System Spherical Earth’s surface -radius 6371 km Meridians (lines of longitude) - passing through Greenwich, England as prime meridian or 0º longitude. Parallels (lines of latitude) - using equator as 0º latitude. degrees-minutes-seconds (DMS), decimal degrees (DD) True direction, shape, distance, and area
  29. 29. Latitude and Longitude on a Sphere Meridian of longitude Z Greenwich meridian λ=0° N Parallel of latitude ° -90 ϕ =0 P • N W λ =0 -180 X ϕ O • °W • λ Equator • R ϕ =0° λ=0-180°E °S 90 =0 ϕ E λ - Geographic longitude ϕ - Geographic latitude Y R - Mean earth radius O - Geocenter
  30. 30. Length on Meridians and Parallels (Lat, Long) = (φ, λ) Length on a Meridian: AB = Re ∆φ (same for all latitudes) Length on a Parallel: CD = R ∆λ = Re ∆λ Cos φ (varies with latitude) R ∆λ 30 N 0N Re R C ∆φ B Re A D
  31. 31. Example: What is the length of a 1º increment along on a meridian and on a parallel at 30N, 90W? Radius of the earth = 6370 km. Solution: • A 1º angle has first to be converted to radians π radians = 180 º, so 1º = π/180 = 3.1416/180 = 0.0175 radians • For the meridian, ∆L = Re ∆φ = 6370 ∗ 0.0175 = 111 km • For the parallel, ∆L = Re ∆λ Cos φ = 6370 ∗ 0.0175 ∗ Cos 30 = 96.5 km • Parallels converge as poles are approached
  32. 32. Curved Earth Distance (from A to B) Shortest distance is along a “Great Circle” Z A “Great Circle” is the intersection of a sphere with a plane going through its center. 1. Spherical coordinates converted to Cartesian coordinates. 2. Vector dot product used to calculate angle α from latitude and longitude B A α • Y X 3. Great circle distance is Rα, where R=6370 km2 R cos−1 (sin φ 1 sin φ2 + cos φ1 cos φ2 cos(λ1 − λ2 ) Longley et al. (2001)
  33. 33. Spherical and Ellipsoidal Earth Earth Centered X/Y/Z Geodetic Lat/Lon/Height
  34. 34. Projection Real-world features must be projected with minimum distortion from a round earth to a flat map; and given a grid system of coordinates. A map projection transforms latitude and longitude locations to x,y coordinates.
  35. 35. What is a Projection? Mathematical transformation of 3D objects in a 2D space with minimal distortion This two-dimensional surface would be the basis for your map.
  36. 36. Cartesian Coordinate System Planar coordinate systems are based on Cartesian coordinates. Projection: Spherical to Cartesian Coordinates with Minimal distortion
  37. 37. Why use a Projection? Can only see half the earth’s surface at a time. Unless a globe is very large it will lack detail and accuracy. Harder to represent features on a flat computer screen. Doesn’t fold, roll or transport easily.
  38. 38. Types of Projections Conic (Albers Equal Area, Lambert Conformal Conic) - good for East-West land areas Cylindrical (Transverse Mercator) good for North-South land areas Azimuthal (Lambert Azimuthal Equal Area) - good for global views
  39. 39. Conic Projections (Albers, Lambert)
  40. 40. Cylindrical Projections (Mercator) Transverse Oblique
  41. 41. Azimuthal (Lambert)
  42. 42. Map Distortions Shape Area Distance Direction
  43. 43. Map Projection & Distortion Converting a sphere to a flat surface results in distortion. Shape (conformal) - If a map preserves shape, then feature outlines (like country boundaries) look the same on the map as they do on the earth. Area (equal-area) - If a map preserves area, then the size of a feature on a map is the same relative to its size on the earth. On an equal-area map each country would take up the same percentage of map space that actual country takes up on the earth. Distance (equidistant) - An equidistant map is one that preserves true scale for all straight lines passing through a single, specified point. If a line from a to b on a map is the same distance that it is on the earth, then the map line has true scale. No map has true scale everywhere.
  44. 44. Direction/Azimuth (azimuthal) – An azimuthal projection is one that preserves direction for all straight lines passing through a single, specified point. Direction is measured in degrees of angle from the north. This means that the direction from ‘a’ to ‘b’ is the angle between the meridian on which ‘a’ lies and the great circle arc connecting ‘a’ to ‘b’. If the azimuth value from ‘a’ to ‘b’ is the same on a map as on the earth, then the map preserves direction from ‘a’ to ‘b’. No map has true direction everywhere.
  45. 45. Trade-off On an equidistant map, distances are true only along particular lines such as those radiating from a single point selected as the center of the projection. Shapes are more or less distorted on every equal-area map. Sizes of areas are distorted on conformal maps even though shapes of small areas are shown correctly.
  46. 46. Polyconic The projection is based on an infinite number of cones tangent to an infinite number of parallels. The central meridian is straight. Other meridians are complex curves. The parallels are non-concentric circles. Scale is true along each parallel and along the central meridian
  47. 47. How is projection done?
  48. 48. Geometric Projection
  49. 49. Distance Property preserved The Azimuthal Equidistant Projection North-polar aspect (Arctic at the centre) • ρ = (π / 2 − φ)R and θ = λ South-polar aspect (Antarctic at the centre) • ρ = (π / 2 + φ)R and θ = -λ Only the radial distance from the centre of the map to any object is preserved
  50. 50. Azimuthal Equidistant projection (North polar aspect)
  51. 51. Azimuthal Equidistant projection (South polar aspect)
  52. 52. Azimuthal Equal-area Projection N E N
  53. 53. Azimuthal Equal area projection (North polar aspect)
  54. 54. Lambert Azimuthal Equal Area Albers Equal Area Conic Equidistant Conic Lambert Conformal Conic
  55. 55. Universal Transverse Mercator Uses the Transverse Mercator projection. 60 six-degree-wide zones cover the earth from East to West starting at 180° West. extending from 80 degrees South latitude to 84 degrees North latitude Each zone has a Central Meridian (λo). Reference Latitude (φo) is the equator. (Xshift, Yshift) = (xo,yo) = (500,000, 0). Units are meters.
  56. 56. Transverse-secant Cylindrical (Mercator) Projection CM: central meridian AB: standard meridian DE: standard meridian -108 -105 -102
  57. 57. UTM Zone Numbers
  58. 58. India Spheroid EVEREST and WGS 1984 Projection UTM and Polyconic
  59. 59. One projection to another
  60. 60. Source:
  61. 61. Source:
  62. 62. Excellent Information
  63. 63. Additional Reading Datum l atum/datum.html
  64. 64. Additional Reading Projection ons.html ections.html approj/mapproj.html