Understanding Map Projections

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GEOGRAPHIC TRANSFORMATION METHODSMoving your data between coordinate systemssometimes includes transforming between thegeographic coordinate systems.These include the Geocentric Translation,Molodensky, and Coordinate Frame methods.Other methods such as NADCON and NTv2 use agrid of differences and convert the longitude–latitudevalues directly.Because the geographic coordinate systems containdatums that are based on spheroids, a geographictransformation also changes the underlying spheroid.There are several methods, which have differentlevels of accuracy and ranges, for transformingbetween datums. The accuracy of a particulartransformation can range from centimeters to metersdepending on the method and the quality andnumber of control points available to define thetransformation parameters.A geographic transformation always convertsgeographic (longitude–latitude) coordinates. Somemethods convert the geographic coordinates togeocentric (X,Y,Z) coordinates, transform the X,Y,Zcoordinates, and convert the new values back togeographic coordinates.The X,Y,Z coordinate system.24 • Understanding Map Projections
EQUATION-BASED METHODSThree-parameter methodsThe simplest datum transformation method is ageocentric, or three-parameter, transformation. Thegeocentric transformation models the differencesbetween two datums in the X,Y,Z coordinate system.One datum is defined with its center at 0,0,0. Thecenter of the other datum is defined at somedistance (∆X,∆Y,∆Z) in meters away.The rotation values are given in decimal seconds,while the scale factor is in parts per million (ppm).The rotation values are defined in two differentways. It’s possible to define the rotation angles aspositive either clockwise or counterclockwise as youlook toward the origin of the X,Y,Z systems.Usually the transformation parameters are defined asgoing ‘from’ a local datum ‘to’ WGS 1984 or anothergeocentric datum.⎡X⎤ ⎡∆X⎤ ⎡X⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢Y⎥ = ⎢∆Y⎥ + ⎢Y⎥⎣⎢Z⎦⎥ Z Znew ⎣⎢∆⎦⎥⎣⎢⎦⎥originalThe three parameters are linear shifts and are alwaysin meters.Seven-parameter methodsA more complex and accurate datum transformationis possible by adding four more parameters to ageocentric transformation. The seven parameters arethree linear shifts (∆X,∆Y,∆Z), three angular rotationsaround each axis (r x,r y,r z), and scale factor(s).⎡X⎤ ⎡∆X⎤ ⎡ 1 rz− r ⎤y ⎡X⎤⎢ ⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥⎢Y⎥ = ⎢∆Y⎥ + ( 1 + s)⋅⎢−rz1 rx⎥ ⋅ ⎢Y⎥⎣⎢Z⎦⎥ Zr r Znew ⎣⎢∆⎦⎥ ⎢⎣ y −⎥x 1⎦ ⎣⎢⎦⎥originalThe Coordinate Frame (or Bursa–Wolf) definition of therotation values.The equation in the previous column is how theUnited States and Australia define the equations andis called the Coordinate Frame Rotationtransformation. The rotations are positivecounterclockwise. Europe uses a differentconvention called the Position Vector transformation.Both methods are sometimes referred to as theBursa–Wolf method. In the Projection Engine, theCoordinate Frame and Bursa–Wolf methods are thesame. Both Coordinate Frame and Position Vectormethods are supported, and it is easy to converttransformation values from one method to the othersimply by changing the signs of the three rotationvalues. For example, the parameters to convert fromthe WGS 1972 datum to the WGS 1984 datum withthe Coordinate Frame method are (in the order, ∆X,∆Y,∆Z,r x,r y,r z,s):(0.0, 0.0, 4.5, 0.0, 0.0, -0.554, 0.227)To use the same parameters with the Position Vectormethod, change the sign of the rotation so the newparameters are:(0.0, 0.0, 4.5, 0.0, 0.0, +0.554, 0.227)Geographic transformations • 25
Page 1 and 2: Understanding Map ProjectionsMelita
Page 3 and 4: ContentsCHAPTER 1: GEOGRAPHIC COORD
Page 5: State Plane Coordinate System .....
Page 8 and 9: GEOGRAPHIC COORDINATE SYSTEMSA geog
Page 10 and 11: SPHEROIDS AND SPHERESThe shape and
Page 12 and 13: DATUMSWhile a spheroid approximates
Page 15 and 16: 2ProjectedcoordinatesystemsProjecte
Page 17 and 18: WHAT IS A MAP PROJECTION?Whether yo
Page 19 and 20: PROJECTION TYPESBecause maps are fl
Page 21 and 22: maintained for both small-scale and
Page 23 and 24: Planar projectionsPlanar projection
Page 25 and 26: OTHER PROJECTIONSThe projections di
Page 27: The four parameters above are used
Page 32 and 33: Unless explicitly stated, it’s im
Page 34 and 35: National Transformation version 2Li
Page 36 and 37: LIST OF SUPPORTED MAP PROJECTIONSAi
Page 38 and 39: LoximuthalMcBryde-Thomas Flat-Polar
Page 40 and 41: AITOFFUSES AND APPLICATIONSDevelope
Page 42 and 43: ALASKA SERIES ELIMITATIONSThis proj
Page 44 and 45: AZIMUTHAL EQUIDISTANTAreaDistortion
Page 46 and 47: BIPOLAR OBLIQUE CONFORMAL CONICDESC
Page 48 and 49: CASSINI-SOLDNERAreaNo distortion al
Page 50 and 51: CRASTER PARABOLICDistanceScale is t
Page 52 and 53: DOUBLE STEREOGRAPHICOblique aspect
Page 54 and 55: ECKERT IIThe central meridian is 10
Page 56 and 57: ECKERT IVparallels. Nearer the pole
Page 58 and 59: ECKERT VIcentral meridian. Scale is
Page 60 and 61: EQUIDISTANT CYLINDRICALLINES OF CON
Page 62 and 63: GALL’S STEREOGRAPHICAreaArea is t
Page 64 and 65: GEOCENTRIC COORDINATE SYSTEMGeograp
Page 66 and 67: GNOMONICPROPERTIESShapeIncreasingly
Page 68 and 69: HAMMER-AITOFFLIMITATIONSUseful only
Page 70 and 71: KROVAKDirectionLocal angles are acc
Page 72 and 73: LAMBERT CONFORMAL CONICAreaMinimal
Page 74 and 75: LOXIMUTHALDistanceScale is true alo
Page 76 and 77: MERCATORGreenland is only one-eight
Page 78 and 79: MOLLWEIDEDistanceScale is true alon
Page 80 and 81:
ORTHOGRAPHICPROPERTIESShapeMinimal
Page 82 and 83:
PLATE CARRÉEDirectionNorth, south,
Page 84 and 85:
POLYCONICDirectionLocal angles are
Page 86 and 87:
RECTIFIED SKEWED ORTHOMORPHICDESCRI
Page 88 and 89:
SIMPLE CONICEquirectangular project
Page 90 and 91:
SPACE OBLIQUE MERCATORDESCRIPTIONTh
Page 92 and 93:
Montana—The three zones for Monta
Page 94 and 95:
TIMESLIMITATIONSUseful only for wor
Page 96 and 97:
10,000,000 meters to ensure that al
Page 98 and 99:
UNIVERSAL POLAR STEREOGRAPHICAreaTh
Page 100 and 101:
VAN DER GRINTEN IDistanceScale alon
Page 102 and 103:
WINKEL IDistanceGenerally, scale is
Page 104 and 105:
WINKEL T RIPELDistanceGenerally, sc
Page 107 and 108:
Glossaryangular unitsThe unit of me
Page 109 and 110:
High Accuracy Reference NetworkA re
Page 111:
UTMSee Universal Transverse Mercato
Page 114 and 115:
Eckert VI 52ED 1950 6Ellipse 101Ell
Page 116:
Projections (continued)planar 17, 1
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Understanding Map Projections

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