Coordinate Systems, Projections, and Transformations. An Overview

Transcription

1 Coordinate Systems, Projections, and Transformations An Overview

2 Outline Acronyms, Terminology Coordinate System Components Conversion between Coordinate Systems

3 The PEAL (Projection Engine Acronym List) PE (Projection Engine) GCS, GEOGCS (Geographic Coordinate System) PCS, PROJCS (Projected Coordinate System) VCS, VERTCS (Vertical Coordinate System) GT, GEOGTRAN (Geographic Transformation) VT, VERTTRAN (Vertical Transformation) WKT (Well-Known Text String) EPSG (European Petroleum Survey Group) (

4 Coordinate System Projected Coordinate System Geographic Coordinate System Projection Projection Parameters Linear Unit Datum Spheroid Prime Meridian Angular Unit

5 Well-Known Text String PROJCS["Test",GEOGCS["GCS_WGS_1984",DATUM[ "D_WGS_1984",SPHEROID["WGS_1984", , ]],PRIMEM["Greenwich",0.0],UNIT["Degree", ]],PROJECTION["Mercator"],PARA METER["Central_Meridian",120.0],PARAMETER["Stan dard_parallel_1",0.0],parameter["false_easting", ],PARAMETER["False_Northing",0.0],UNIT["Fo ot",0.3048]]

6 Well-Known Text String PROJCS[ "Test", GEOGCS[ "GCS_WGS_1984", DATUM[ "D_WGS_1984", SPHEROID[ "WGS_1984", , ] ], PRIMEM[ "Greenwich", 0.0], UNIT[ "Degree", ] ], PROJECTION[ "Mercator" ], PARAMETER[ Central_Meridian, ], PARAMETER[ Standard_Parallel_1, 0.0], PARAMETER[ False_Easting, ], PARAMETER[ False_Northing, 0.0], UNIT[ "Foot", ] ]

7 Conversion Pathways PROJCS A1 PROJCS A2 (x, y) Projection GEOGCS A (lon, lat) (λ, φ)

8 Conversion Pathways PROJCS A1 PROJCS A2 PROJCS B1 (x, y) Projection GEOGCS A GEOGCS B Geographic Transformation (lon, lat) (λ, φ)

9 Units, Spheroids, Prime Meridians Angular - UNIT["Degree", ] UNIT[ Grad, ] The value is Radians / Unit Linear - UNIT["Foot", ] The value is Meters / Unit SPHEROID[ "WGS_1984", , ] The values are Semi-Major axis length in Meters, Inverse Flattening (1 / f) Prime Meridian PRIMEM[ Paris, ] PRIMEM[ Greenwich, 0.0] The value is Decimal Degrees based on Greenwich

10 Geographic Coordinate Systems Figure

11 Figure

12 Distances and Angular Units

13 Figure

14 More background geometry Rotating a circle or ellipse creates a sphere or spheroid (oblate ellipsoid of revolution) Defines the size and shape of the Earth model Sphere Spheroid

15 Figure

16 Semiminor axis (b) Background geometry Circle: all axes are the same length Ellipse: 2 axes f = (a b)/a (flattening) e 2 = (a 2 b 2 )/a 2 (ellipticity squared) Semimajor axis (a)

17 Figure

18 Earth as sphere simplifies math small-scale maps (less than 1:5,000,000) Earth as spheroid maintains accuracy for larger-scale maps (> 1:1,000,000)

19 Datums 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 Creating a Datum Pick a spheroid Pick a point on the Earth s surface All other control points are located relative to the origin point The datum s center may not coincide with the Earth s center

21 Two types of datums Earth-centered (WGS84, NAD83) Local (NAD27, ED50)

22 Local datum coordinate system Earth s Surface Earth-centered datum (WGS84) Local datum (NAD27) Earth-centered datum coordinate system

23 What is a datum? Classical geodesy (before 1960) 5 quantities Latitude of an initial point Longitude of an initial point Azimuth of a line from this point Semi-major axis length and flattening of ellipsoid Satellite geodesy (after 1960) 8 constants Three to specify the origin of the coordinate system Three to specify the orientation of the coordinate system Semi-major axis length and flattening of ellipsoid

24 Why so many datums? Many estimates of Earth s size and shape Improved accuracy Designed for local regions

25 North American Datums NAD27 Clarke 1866 spheroid Meades Ranch, KS 1880 s NAD83 GRS80 spheroid Earth-centered datum GPS-compatible

26 North American Datums HPGN / HARN GPS readjustment of NAD83 in the US Also known as NAD91 or NAD93 27 states & 2 territories (42 states in PE) NAD27 (1976) & CGQ77 Redefinitions for Ontario and Quebec NAD83 (CSRS98) GPS readjustment

27 International Datums Defined for countries, regions, or the world World: WGS84, WGS72 Regional: ED50 (European Datum 1950) Arc 1950 (Africa) Countries: GDA 1994 (Australia) Tokyo

28 Geographic coordinate systems (gcs, geogcs) Name (European Datum 1950) Datum (European Datum 1950) Spheroid (International 1924) Prime Meridian (Greenwich) Angular unit of measure (Degrees)

29 Geographic transformations datum transformations Convert between GCS Includes unit, prime meridian, and spheroid changes Defined in a particular direction All are reversible

30 Relationship between two datums Z (145,-39,6) (0,0,0) dy dz dx Y X

31 Rotations Z X Y

32 Transformation methods Equation-based Molodensky, Abridged Molodensky, Geocentric Translation Coordinate Frame, Position Vector, Molodensky- Badekas Longitude Rotation, 2D lat / lon offsets File-based NADCON, HARN, NTv2

33 Transformation example European 1950 (International 1924) a = f = 1 / e 2 = WGS 1984 (WGS 1984) a = meters f = 1 / e 2 = different geographic transformations Geocentric Translation Position Vector, Coordinate Frame NTv2 Why so many? Areas of use Accuracy

34 Method Accuracies NADCON HARN/HPGN CNT (NTv1) Seven parameter Three parameter 15 cm 5 cm 10 cm 1-2 m 4-5 m

35 Example of GT in WKT format GEOGTRAN["ED_1950_To_WGS_1984_23", GEOGCS["GCS_European_1950", DATUM["D_European_1950", SPHEROID["International_1924", ,297.0]], PRIMEM["Greenwich",0.0], UNIT["Degree", ]], GEOGCS["GCS_WGS_1984", DATUM["D_WGS_1984", SPHEROID["WGS_1984", , ]], PRIMEM["Greenwich",0.0], UNIT["Degree", ]], METHOD["Position_Vector"], PARAMETER["X_Axis_Translation", ], PARAMETER["Y_Axis_Translation", ], PARAMETER["Z_Axis_Translation", ], PARAMETER["X_Axis_Rotation",0.893], PARAMETER["Y_Axis_Rotation",0.921], PARAMETER["Z_Axis_Rotation",-0.917], PARAMETER["Scale_Difference",-3.52]]

36 ED50 versus WGS84

37 Figure

38 Figure

39 Map Projections mathematical conversion of 3-D Earth to a 2-D surface Longitude / Latitude to X, Y (l, j) (x, y)

40 Projected coordinate system Linear units Shape, area, and distance may be X - Y + Y Data X + Y + usually here X distorted X - Y - X + Y -

41 Visualize a light shining through the Earth onto a surface

42 Fitting sphere to plane causes stretching or shrinking of features This much earth surface has to fit onto this much map surface... projection plane therefore, much of the Earth surface has to be represented smaller than the nominal scale.

43 More on projections Projecting Earth s surface always involves distortion: shape area distance direction

44 Projection properties Conformal maintains shape Equal-area maintains area Equidistant maintains distance Direction maintains some directions

45 Projection surfaces Cones, Cylinders, Planes Can be flattened without distortion A point or line of contact is created when surface is combined with a sphere or spheroid

46 More on projection surfaces Tangent Secant projection surface touches spheroid surface cuts through spheroid No distortion at contact points Increases away from contact points

47 Conic Projections Best for mid-latitudes with an East-West orientation. Tangent or secant along 1 or 2 lines of latitude known as standard parallels.

48 Cylindrical projections Best for equatorial or low latitudes Rotate cylinder to reduce distortion along a line

49 Planar projections Best for polar or circular regions Direction always true from center Shortest distance from center to another point is a straight line

50 Also called azimuthal or zenithal Can be any aspect

51 Projection parameters Central meridian Longitude of origin Longitude of center Latitude of origin, Latitude of center Standard parallel Scale factor False easting, False northing

52 Y false easting = 500,000 false northing = 10,000,000 X Latitude of Origin: Y = 0 Central Meridian: X = 0

53 Choosing a coordinate system Often mandated by organization Thematic = equal-area Presentation = conformal (also equal-area) Navigation = Mercator, true direction or equidistant

54 More on choosing a coordinate system Extent Location Predominant extent Projection supports spheroids/datums?

55 Web Mercator Online mapping services use a sphere-only Mercator Two ways to emulate it Sphere-based GCS Projection that can force sphere equations Mathematically EQUAL

56 PROJCS["WGS_1984_Web_Mercator", GEOGCS["GCS_WGS_1984_Major_Auxiliary_Sphere", DATUM["D_WGS_1984_Major_Auxiliary_Sphere", SPHEROID["WGS_1984_Major_Auxiliary_Sphere", , 0.0]], PRIMEM["Greenwich", 0.0], UNIT["Degree", ]], PROJECTION["Mercator"], PARAMETER["False_Easting", 0.0] PARAMETER["False_Northing", 0.0], PARAMETER["Central_Meridian", 0.0], PARAMETER["Standard_Parallel_1", 0.0], UNIT["Meter", 1.0]] #

57 PROJCS["WGS_1984_Web_Mercator_Auxiliary_Sphere", GEOGCS["GCS_WGS_1984", DATUM["D_WGS_1984", SPHEROID["WGS_1984", , ]], PRIMEM["Greenwich", 0.0], UNIT["Degree", ]], PROJECTION["Mercator_Auxiliary_Sphere"], PARAMETER["False_Easting", 0.0], PARAMETER["False_Northing", 0.0], PARAMETER["Central_Meridian", 0.0], PARAMETER["Standard_Parallel_1", 0.0], PARAMETER["Auxiliary_Sphere_Type", 0.0], UNIT["Meter", 1.0]] #3857 (old #102100)

58 Mercator Equations (std parallel at equator) Sphere x = R(λ λ 0 ) y = ln(tan(π/4 + φ/2)) Spheroid x = a(λ λ 0 ) y = ln(tan(π/4 + χ/2) where conformal latitude χ = 2 arctan{tan(π/4 + φ/2)[(1-e sin φ)/(1+e sin φ)] e/2 } - π/2

59 UTM Universal Transverse Mercator 60 zones, 6 wide Transverse Mercator zone 1, central meridian = -177 scale factor = false easting = 500,000 m In Southern Hemisphere, false northing = 10,000,000 m

60 SPCS State Plane Coordinate System States have 1 or more zones Uses either NAD27 or NAD83 datums Uses Lambert Conic, Transverse Mercator, and Oblique Mercator

61 Horizons PCS GCS

62 Spatial Domain 12,000,000 4,000,000 UTM (meters) UTM (feet) 10, Decimal Degrees ,000 4,000,000

63 When is a foot not a foot? esriunits is limited ArcMap, Foot is US survey foot US survey foot, 1 ft = m Int l foot, 1 ft = m PE, Foot is Int l foot; Foot_US is US survey foot

64 9002 UNIT["Foot", ] 9003 UNIT["Foot_US", ] 9005 UNIT["Foot_Clarke", ] 9041 UNIT["Foot_Sears", ] 9051 UNIT["Foot_Benoit_1895_A", ] 9061 UNIT["Foot_Benoit_1895_B", ] 9070 UNIT["Foot_1865", ] 9080 UNIT["Foot_Indian", ] 9081 UNIT["Foot_Indian_1937", ] 9082 UNIT["Foot_Indian_1962", ] 9083 UNIT["Foot_Indian_1975", ] 9094 UNIT["Foot_Gold_Coast", ] 9095 UNIT["Foot_British_1936", ] 9300 UNIT["Foot_Sears_1922_Truncated", ]

65 Closing We ve only scratched the surface! GCS!= PCS Geographic transformations are vital Measurement in degrees in meaningless

66 Questions?