Coordinate Systems, Projections, and Transformations An Overview
Outline Acronyms, Terminology Coordinate System Components Conversion between Coordinate Systems
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) (www.epsg.org)
Coordinate System Projected Coordinate System Geographic Coordinate System Projection Projection Parameters Linear Unit Datum Spheroid Prime Meridian Angular Unit
Well-Known Text String PROJCS["Test",GEOGCS["GCS_WGS_1984",DATUM[ "D_WGS_1984",SPHEROID["WGS_1984",6378137,29 8.257223]],PRIMEM["Greenwich",0.0],UNIT["Degree",0. 0174532925199433]],PROJECTION["Mercator"],PARA METER["Central_Meridian",120.0],PARAMETER["Stan dard_parallel_1",0.0],parameter["false_easting",10 00000.0],PARAMETER["False_Northing",0.0],UNIT["Fo ot",0.3048]]
Well-Known Text String PROJCS[ "Test", GEOGCS[ "GCS_WGS_1984", DATUM[ "D_WGS_1984", SPHEROID[ "WGS_1984", 6378137.0, 298.257223563] ], PRIMEM[ "Greenwich", 0.0], UNIT[ "Degree", 0.0174532925199433] ], PROJECTION[ "Mercator" ], PARAMETER[ Central_Meridian, -120.0], PARAMETER[ Standard_Parallel_1, 0.0], PARAMETER[ False_Easting, 1000000.0], PARAMETER[ False_Northing, 0.0], UNIT[ "Foot", 0.3048] ]
Conversion Pathways PROJCS A1 PROJCS A2 (x, y) Projection GEOGCS A (lon, lat) (λ, φ)
Conversion Pathways PROJCS A1 PROJCS A2 PROJCS B1 (x, y) Projection GEOGCS A GEOGCS B Geographic Transformation (lon, lat) (λ, φ)
Units, Spheroids, Prime Meridians Angular - UNIT["Degree", 0.0174532925199433] UNIT[ Grad, 0.01570796326794897] The value is Radians / Unit Linear - UNIT["Foot", 0.3048] The value is Meters / Unit SPHEROID[ "WGS_1984", 6378137.0, 298.257223563] The values are Semi-Major axis length in Meters, Inverse Flattening (1 / f) Prime Meridian PRIMEM[ Paris, 2.337229166666667] PRIMEM[ Greenwich, 0.0] The value is Decimal Degrees based on Greenwich
Geographic Coordinate Systems Figure 1.2 10
Figure 1.3 11
Distances and Angular Units
Figure 1.4 13
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
Figure 1.5 15
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)
Figure 1.7 17
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)
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
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
Two types of datums Earth-centered (WGS84, NAD83) Local (NAD27, ED50)
Local datum coordinate system Earth s Surface Earth-centered datum (WGS84) Local datum (NAD27) Earth-centered datum coordinate system
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
Why so many datums? Many estimates of Earth s size and shape Improved accuracy Designed for local regions
North American Datums NAD27 Clarke 1866 spheroid Meades Ranch, KS 1880 s NAD83 GRS80 spheroid Earth-centered datum GPS-compatible
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
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
Geographic coordinate systems (gcs, geogcs) Name (European Datum 1950) Datum (European Datum 1950) Spheroid (International 1924) Prime Meridian (Greenwich) Angular unit of measure (Degrees)
Geographic transformations datum transformations Convert between GCS Includes unit, prime meridian, and spheroid changes Defined in a particular direction All are reversible
Relationship between two datums Z (145,-39,6) (0,0,0) dy dz dx Y X
Rotations Z X Y
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
Transformation example European 1950 (International 1924) a = 6378388.0 f = 1 / 297.0 e 2 = 0.006722670022 WGS 1984 (WGS 1984) a = 6378137.0 meters f = 1 / 298.257223563 e 2 = 0.0066943799901 40 different geographic transformations Geocentric Translation Position Vector, Coordinate Frame NTv2 Why so many? Areas of use Accuracy
Method Accuracies NADCON HARN/HPGN CNT (NTv1) Seven parameter Three parameter 15 cm 5 cm 10 cm 1-2 m 4-5 m
Example of GT in WKT format GEOGTRAN["ED_1950_To_WGS_1984_23", GEOGCS["GCS_European_1950", DATUM["D_European_1950", SPHEROID["International_1924",6378388.0,297.0]], PRIMEM["Greenwich",0.0], UNIT["Degree",0.0174532925199433]], GEOGCS["GCS_WGS_1984", DATUM["D_WGS_1984", SPHEROID["WGS_1984",6378137.0,298.257223563]], PRIMEM["Greenwich",0.0], UNIT["Degree",0.0174532925199433]], METHOD["Position_Vector"], PARAMETER["X_Axis_Translation",-116.641], PARAMETER["Y_Axis_Translation",-56.931], PARAMETER["Z_Axis_Translation",-110.559], PARAMETER["X_Axis_Rotation",0.893], PARAMETER["Y_Axis_Rotation",0.921], PARAMETER["Z_Axis_Rotation",-0.917], PARAMETER["Scale_Difference",-3.52]]
ED50 versus WGS84
Figure 1.17 37
Figure 1.18 38
Map Projections mathematical conversion of 3-D Earth to a 2-D surface Longitude / Latitude to X, Y (l, j) (x, y)
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 -
Visualize a light shining through the Earth onto a surface
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.
More on projections Projecting Earth s surface always involves distortion: shape area distance direction
Projection properties Conformal maintains shape Equal-area maintains area Equidistant maintains distance Direction maintains some directions
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
More on projection surfaces Tangent Secant projection surface touches spheroid surface cuts through spheroid No distortion at contact points Increases away from contact points
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.
Cylindrical projections Best for equatorial or low latitudes Rotate cylinder to reduce distortion along a line
Planar projections Best for polar or circular regions Direction always true from center Shortest distance from center to another point is a straight line
Also called azimuthal or zenithal Can be any aspect
Projection parameters Central meridian Longitude of origin Longitude of center Latitude of origin, Latitude of center Standard parallel Scale factor False easting, False northing
Y false easting = 500,000 false northing = 10,000,000 X Latitude of Origin: Y = 0 Central Meridian: X = 0
Choosing a coordinate system Often mandated by organization Thematic = equal-area Presentation = conformal (also equal-area) Navigation = Mercator, true direction or equidistant
More on choosing a coordinate system Extent Location Predominant extent Projection supports spheroids/datums?
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
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", 6378137.0, 0.0]], PRIMEM["Greenwich", 0.0], UNIT["Degree", 0.0174532925199433]], 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]] # 102113
PROJCS["WGS_1984_Web_Mercator_Auxiliary_Sphere", GEOGCS["GCS_WGS_1984", DATUM["D_WGS_1984", SPHEROID["WGS_1984",6378137.0, 298.257223563]], PRIMEM["Greenwich", 0.0], UNIT["Degree", 0.0174532925199433]], 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)
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
UTM Universal Transverse Mercator 60 zones, 6 wide Transverse Mercator zone 1, central meridian = -177 scale factor = 0.9996 false easting = 500,000 m In Southern Hemisphere, false northing = 10,000,000 m
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
Horizons PCS GCS
Spatial Domain 12,000,000 4,000,000 UTM (meters) UTM (feet) 10,000 90 Decimal Degrees 180 10,000 4,000,000
When is a foot not a foot? esriunits is limited ArcMap, Foot is US survey foot US survey foot, 1 ft = 0.3048006096012192 m Int l foot, 1 ft = 0.3048 m PE, Foot is Int l foot; Foot_US is US survey foot
9002 UNIT["Foot", 0.3048] 9003 UNIT["Foot_US", 0.3048006096012192] 9005 UNIT["Foot_Clarke", 0.304797265] 9041 UNIT["Foot_Sears", 0.3047994715386762] 9051 UNIT["Foot_Benoit_1895_A", 0.3047997333333333] 9061 UNIT["Foot_Benoit_1895_B", 0.3047997347632708] 9070 UNIT["Foot_1865", 0.3048008333333334] 9080 UNIT["Foot_Indian", 0.3047995102481469] 9081 UNIT["Foot_Indian_1937", 0.30479841] 9082 UNIT["Foot_Indian_1962", 0.3047996] 9083 UNIT["Foot_Indian_1975", 0.3047995] 9094 UNIT["Foot_Gold_Coast", 0.3047997101815088] 9095 UNIT["Foot_British_1936", 0.3048007491] 9300 UNIT["Foot_Sears_1922_Truncated", 0.3047993333333334]
Closing We ve only scratched the surface! GCS!= PCS Geographic transformations are vital Measurement in degrees in meaningless
Questions?