Understanding Map Projections

More documents

Recommendations

Info

GEOGRAPHIC TRANSFORMATION METHODS Moving your data between coordinate systems sometimes includes transforming between the geographic coordinate systems. These include the Geocentric Translation, Molodensky, and Coordinate Frame methods. Other methods such as NADCON and NTv2 use a grid of differences and convert the longitude–latitude values directly. Because the geographic coordinate systems contain datums that are based on spheroids, a geographic transformation also changes the underlying spheroid. There are several methods, which have different levels of accuracy and ranges, for transforming between datums. The accuracy of a particular transformation can range from centimeters to meters depending on the method and the quality and number of control points available to define the transformation parameters. A geographic transformation always converts geographic (longitude–latitude) coordinates. Some methods convert the geographic coordinates to geocentric (X,Y,Z) coordinates, transform the X,Y,Z coordinates, and convert the new values back to geographic coordinates. The X,Y,Z coordinate system. 24 • Understanding Map Projections
EQUATION-BASED METHODS Three-parameter methods The simplest datum transformation method is a geocentric, or three-parameter, transformation. The geocentric transformation models the differences between two datums in the X,Y,Z coordinate system. One datum is defined with its center at 0,0,0. The center of the other datum is defined at some distance (∆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 different ways. It’s possible to define the rotation angles as positive either clockwise or counterclockwise as you look toward the origin of the X,Y,Z systems. Usually the transformation parameters are defined as going ‘from’ a local datum ‘to’ WGS 1984 or another geocentric datum. ⎡X ⎤ ⎡∆X ⎤ ⎡X ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢Y ⎥ = ⎢∆Y ⎥ + ⎢Y ⎥ ⎣ ⎢Z ⎦ ⎥ Z Z new ⎣ ⎢∆ ⎦ ⎥ ⎣ ⎢ ⎦ ⎥ original The three parameters are linear shifts and are always in meters. Seven-parameter methods A more complex and accurate datum transformation is possible by adding four more parameters to a geocentric transformation. The seven parameters are three linear shifts (∆X,∆Y,∆Z), three angular rotations around each axis (r x ,r y ,r z ), and scale factor(s). ⎡X ⎤ ⎡∆X ⎤ ⎡ 1 rz − r ⎤ y ⎡X ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢Y ⎥ = ⎢∆Y ⎥ + ( 1 + s) ⋅⎢− rz 1 rx ⎥ ⋅ ⎢Y ⎥ ⎣ ⎢Z ⎦ ⎥ Z r r Z new ⎣ ⎢∆ ⎦ ⎥ ⎢ ⎣ y − ⎥ x 1 ⎦ ⎣ ⎢ ⎦ ⎥ original The Coordinate Frame (or Bursa–Wolf) definition of the rotation values. The equation in the previous column is how the United States and Australia define the equations and is called the Coordinate Frame Rotation transformation. The rotations are positive counterclockwise. Europe uses a different convention called the Position Vector transformation. Both methods are sometimes referred to as the Bursa–Wolf method. In the Projection Engine, the Coordinate Frame and Bursa–Wolf methods are the same. Both Coordinate Frame and Position Vector methods are supported, and it is easy to convert transformation values from one method to the other simply by changing the signs of the three rotation values. For example, the parameters to convert from the WGS 1972 datum to the WGS 1984 datum with the 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 Vector method, change the sign of the rotation so the new parameters are: (0.0, 0.0, 4.5, 0.0, 0.0, +0.554, 0.227) Geographic transformations • 25
Page 1 and 2: Understanding Map Projections GIS b
Page 3 and 4: Contents CHAPTER 1: GEOGRAPHIC COOR
Page 5: State Plane Coordinate System .....
Page 8 and 9: GEOGRAPHIC COORDINATE SYSTEMS A geo
Page 10 and 11: SPHEROIDS AND SPHERES The shape and
Page 12 and 13: DATUMS While a spheroid approximate
Page 15 and 16: 2 Projected coordinate systems Proj
Page 17 and 18: WHAT IS A MAP PROJECTION? Whether y
Page 19 and 20: PROJECTION TYPES Because maps are f
Page 21 and 22: maintained for both small-scale and
Page 23 and 24: Planar projections Planar projectio
Page 25 and 26: OTHER PROJECTIONS The projections d
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 2 L
Page 36 and 37: LIST OF SUPPORTED MAP PROJECTIONS A
Page 38 and 39: Loximuthal McBryde-Thomas Flat-Pola
Page 40 and 41: AITOFF USES AND APPLICATIONS Develo
Page 42 and 43: ALASKA SERIES E LIMITATIONS This pr
Page 44 and 45: AZIMUTHAL EQUIDISTANT Area Distorti
Page 46 and 47: BIPOLAR OBLIQUE CONFORMAL CONIC DES
Page 48 and 49: CASSINI-SOLDNER Area No distortion
Page 50 and 51: CRASTER PARABOLIC Distance Scale is
Page 52 and 53: DOUBLE STEREOGRAPHIC Oblique aspect
Page 54 and 55: ECKERT II The central meridian is 1
Page 56 and 57: ECKERT IV parallels. Nearer the pol
Page 58 and 59: ECKERT VI central meridian. Scale i
Page 60 and 61: EQUIDISTANT CYLINDRICAL LINES OF CO
Page 62 and 63: GALL’S STEREOGRAPHIC Area Area is
Page 64 and 65: GEOCENTRIC COORDINATE SYSTEM Geogra
Page 66 and 67: GNOMONIC PROPERTIES Shape Increasin
Page 68 and 69: HAMMER-AITOFF LIMITATIONS Useful on
Page 70 and 71: KROVAK Direction Local angles are a
Page 72 and 73: LAMBERT CONFORMAL CONIC Area Minima
Page 74 and 75: LOXIMUTHAL Distance Scale is true a
Page 76 and 77: MERCATOR Greenland is only one-eigh
Page 78 and 79: MOLLWEIDE Distance Scale is true al
Page 80 and 81:
ORTHOGRAPHIC PROPERTIES Shape Minim
Page 82 and 83:
PLATE CARRÉE Direction North, sout
Page 84 and 85:
POLYCONIC Direction Local angles ar
Page 86 and 87:
RECTIFIED SKEWED ORTHOMORPHIC DESCR
Page 88 and 89:
SIMPLE CONIC Equirectangular projec
Page 90 and 91:
SPACE OBLIQUE MERCATOR DESCRIPTION
Page 92 and 93:
Montana—The three zones for Monta
Page 94 and 95:
TIMES LIMITATIONS Useful only for w
Page 96 and 97:
10,000,000 meters to ensure that al
Page 98 and 99:
UNIVERSAL POLAR STEREOGRAPHIC Area
Page 100 and 101:
VAN DER GRINTEN I Distance Scale al
Page 102 and 103:
WINKEL I Distance Generally, scale
Page 104 and 105:
WINKEL T RIPEL Distance Generally,
Page 107 and 108:
Glossary angular units The unit of
Page 109 and 110:
High Accuracy Reference Network A r
Page 111:
UTM See Universal Transverse Mercat
Page 114 and 115:
Eckert VI 52 ED 1950 6 Ellipse 101
Page 116:
Projections (continued) planar 17,
show all

Understanding Map Projections

Create successful ePaper yourself

Delete template?

Save as template?