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mapproject.1gmt

NAME

mapproject - Forward and Inverse map transformation of 2-D coordinates

SYNOPSIS

mapproject infiles -Jparameters -Rwest/east/south/north[r] [ -Ab|B|f|F[lon0/lat0] ] [ -C[dx/dy] ] [ -Dc|i|m|p ] [ -E[datum] ] [ -F[k|m|n|i|c|p] ] [ -G[x0/y0][+|-][/unit] ] [ -H[i][nrec] ] [ -I ] [ -Lline.xy[/unit] ] [ -Q[d|e] [ -S ] [ -T[h]from[/to] ] [ -V ] [ -:[i|o] ] [ -b[i|o][s|S|d|D[ncol]|c[var1/...]] ] [ -f[i|o]colinfo ] [ -g[a]x|y|d|X|Y|D|[col]z[+|-]gap[u] ] [ -m[i|o][flag] ]

DESCRIPTION

mapproject reads (longitude, latitude) positions from infiles [or standard input] and computes (x,y) coordinates using the specified map projection and scales. Optionally, it can read (x,y) positions and compute (longitude, latitude) values doing the inverse transformation. This can be used to transform linear (x,y) points obtained by digitizing a map of known projection to geographical coordinates. May also calculate distances along track, to a fixed point, or closest approach to a line. Finally, can be used to perform various datum conversions. Additional data fields are permitted after the first 2 columns which must have (longitude,latitude) or (x,y). See option -: on how to read (latitude,longitude) files.

infiles

Data file(s) to be transformed. If not given, standard input is read.

-J

Selects the map projection. The following character determines the projection. If the character is upper case then the argument(s) supplied as scale(s) is interpreted to be the map width (or axis lengths), else the scale argument(s) is the map scale (see its definition for each projection). UNIT is cm, inch, or m, depending on the MEASURE_UNIT setting in .gmtdefaults4, but this can be overridden on the command line by appending c, i, or m to the scale or width values. Append h, +, or - to the given width if you instead want to set map height, the maximum dimension, or the minimum dimension, respectively [Default is w for width].
In case the central meridian is an optional parameter and it is being omitted, then the center of the longitude range given by the -R option is used. The default standard parallel is the equator.
The ellipsoid used in the map projections is user-definable by editing the .gmtdefaults4 file in your home directory. 73 commonly used ellipsoids and spheroids are currently supported, and users may also specify their own custum ellipsoid parameters [Default is WGS-84]. Several GMT parameters can affect the projection: ELLIPSOID, INTERPOLANT, MAP_SCALE_FACTOR, and MEASURE_UNIT; see the gmtdefaults man page for details.
Choose one of the following projections (The E or C after projection names stands for Equal-Area and Conformal, respectively):

CYLINDRICAL PROJECTIONS:

-Jclon0/lat0/scale or -JClon0/lat0/width (Cassini).

Give projection center lon0/lat0 and scale (1:xxxx or UNIT/degree).

-Jcyl_stere/[lon0/[lat0/]]scale or -JCyl_stere/[lon0/[lat0/]]width (Cylindrical Stereographic).

Give central meridian lon0 (optional), standard parallel lat0 (optional), and scale along parallel (1:xxxx or UNIT/degree). The standard parallel is typically one of these (but can be any value):

66.159467 - Miller's modified Gall
55 - Kamenetskiy's First
45 - Gall's Stereographic
30 - Bolshoi Sovietskii Atlas Mira or Kamenetskiy's Second
0 - Braun's Cylindrical

-Jj[lon0/]scale or -JJ[lon0/]width (Miller Cylindrical Projection).

Give the central meridian lon0 (optional) and scale (1:xxxx or UNIT/degree).

-Jm[lon0/[lat0/]]scale or -JM[lon0/[lat0/]]width

Give central meridian lon0 (optional), standard parallel lat0 (optional), and scale along parallel (1:xxxx or UNIT/degree).

-Joparameters (Oblique Mercator [C]).

Specify one of:

-Jo[a]lon0/lat0/azimuth/scale or -JO[a]lon0/lat0/azimuth/width

Set projection center lon0/lat0, azimuth of oblique equator, and scale.

-Jo[b]lon0/lat0/lon1/lat1/scale or -JO[b]lon0/lat0/lon1/lat1/scale

Set projection center lon0/lat0, another point on the oblique equator lon1/lat1, and scale.

-Joclon0/lat0/lonp/latp/scale or -JOclon0/lat0/lonp/latp/scale

Set projection center lon0/lat0, pole of oblique projection lonp/latp, and scale.

Give scale along oblique equator (1:xxxx or UNIT/degree).

-Jq[lon0/[lat0/]]scale or -JQ[lon0/[lat0/]]width (Cylindrical Equidistant).

Give the central meridian lon0 (optional), standard parallel lat0 (optional), and scale (1:xxxx or UNIT/degree). The standard parallel is typically one of these (but can be any value):

61.7 - Grafarend and Niermann, minimum linear distortion
50.5 - Ronald Miller Equirectangular
43.5 - Ronald Miller, minimum continental distortion
42 - Grafarend and Niermann
37.5 - Ronald Miller, minimum overall distortion
0 - Plate Carree, Simple Cylindrical, Plain/Plane Chart

-Jtlon0/[lat0/]scale or -JTlon0/[lat0/]width

Give the central meridian lon0, central parallel lat0 (optional), and scale (1:xxxx or UNIT/degree).

-Juzone/scale or -JUzone/width (UTM - Universal Transverse Mercator [C]).

Give the UTM zone (A,B,1-60[C-X],Y,Z)) and scale (1:xxxx or UNIT/degree).
Zones: If C-X not given, prepend - or + to enforce southern or northern hemisphere conventions [northern if south > 0].

-Jy[lon0/[lat0/]]scale or -JY[lon0/[lat0/]]width (Cylindrical Equal-Area [E]).

Give the central meridian lon0 (optional), standard parallel lat0 (optional), and scale (1:xxxx or UNIT/degree). The standard parallel is typically one of these (but can be any value):

50 - Balthasart
45 - Gall-Peters
37.0666 - Caster
37.4 - Trystan Edwards
37.5 - Hobo-Dyer
30 - Behrman
0 - Lambert (default)

CONIC PROJECTIONS:

-Jblon0/lat0/lat1/lat2/scale or -JBlon0/lat0/lat1/lat2/width (Albers [E]).

Give projection center lon0/lat0, two standard parallels lat1/lat2, and scale (1:xxxx or UNIT/degree).

-Jdlon0/lat0/lat1/lat2/scale or -JDlon0/lat0/lat1/lat2/width (Conic Equidistant)

Give projection center lon0/lat0, two standard parallels lat1/lat2, and scale (1:xxxx or UNIT/degree).

-Jllon0/lat0/lat1/lat2/scale or -JLlon0/lat0/lat1/lat2/width (Lambert [C])

Give origin lon0/lat0, two standard parallels lat1/lat2, and scale along these (1:xxxx or UNIT/degree).

-Jpoly/[lon0/[lat0/]]scale or -JPoly/[lon0/[lat0/]]width ((American) Polyconic).

Give the central meridian lon0 (optional), reference parallel lat0 (optional, default = equator), and scale along central meridian (1:xxxx or UNIT/degree).

AZIMUTHAL PROJECTIONS:

Except for polar aspects, -Rw/e/s/n will be reset to -Rg. Use -R<...>r for smaller regions.

-Jalon0/lat0[/horizon]/scale or -JAlon0/lat0[/horizon]/width (Lambert [E]).

lon0/lat0 specifies the projection center. horizon specifies the max distance from projection center (in degrees, <= 180, default 90). Give scale as 1:xxxx or radius/lat, where radius is distance in UNIT from origin to the oblique latitude lat.

-Jelon0/lat0[/horizon]/scale or -JElon0/lat0[/horizon]/width (Azimuthal Equidistant).

lon0/lat0 specifies the projection center. horizon specifies the max distance from projection center (in degrees, <= 180, default 180). Give scale as 1:xxxx or radius/lat, where radius is distance in UNIT from origin to the oblique latitude lat.

-Jflon0/lat0[/horizon]/scale or -JFlon0/lat0[/horizon]/width (Gnomonic).

lon0/lat0 specifies the projection center. horizon specifies the max distance from projection center (in degrees, < 90, default 60). Give scale as 1:xxxx or radius/lat, where radius is distance in UNIT from origin to the oblique latitude lat.

-Jglon0/lat0[/horizon]/scale or -JGlon0/lat0[/horizon]/width (Orthographic).

lon0/lat0 specifies the projection center. horizon specifies the max distance from projection center (in degrees, <= 90, default 90). Give scale as 1:xxxx or radius/lat, where radius is distance in UNIT from origin to the oblique latitude lat.

-Jglon0/lat0/altitude/azimuth/tilt/twist/Width/Height/scale or -JGlon0/lat0/altitude/azimuth/tilt/twist/Width/Height/width (General Perspective).

lon0/lat0 specifies the projection center. altitude is the height (in km) of the viewpoint above local sea level. If altitude is less than 10, then it is the distance from the center of the earth to the viewpoint in earth radii. If altitude has a suffix r then it is the radius from the center of the earth in kilometers. azimuth is measured to the east of north of view. tilt is the upward tilt of the plane of projection. If tilt is negative, then the viewpoint is centered on the horizon. Further, specify the clockwise twist, Width, and Height of the viewpoint in degrees. Give scale as 1:xxxx or radius/lat, where radius is distance in UNIT from origin to the oblique latitude lat.

-Jslon0/lat0[/horizon]/scale or -JSlon0/lat0[/horizon]/width (General Stereographic [C]).

lon0/lat0 specifies the projection center. horizon specifies the max distance from projection center (in degrees, < 180, default 90). Give scale as 1:xxxx (true at pole) or lat0/1:xxxx (true at standard parallel lat0) or radius/lat (radius in UNIT from origin to the oblique latitude lat). Note if 1:xxxx is used then to specify horizon you must also specify the lat0 as +-90 to avoid ambiguity.

MISCELLANEOUS PROJECTIONS:

-Jh[lon0/]scale or -JH[lon0/]width (Hammer [E]).

Give the central meridian lon0 (optional) and scale along equator (1:xxxx or UNIT/degree).

-Ji[lon0/]scale or -JI[lon0/]width (Sinusoidal [E]).

Give the central meridian lon0 (optional) and scale along equator (1:xxxx or UNIT/degree).

-Jkf[lon0/]scale or -JKf[lon0/]width (Eckert IV) [E]).

Give the central meridian lon0 (optional) and scale along equator (1:xxxx or UNIT/degree).

-Jk[s][lon0/]scale or -JK[s][lon0/]width (Eckert VI) [E]).

Give the central meridian lon0 (optional) and scale along equator (1:xxxx or UNIT/degree).

-Jn[lon0/]scale or -JN[lon0/]width (Robinson).

Give the central meridian lon0 (optional) and scale along equator (1:xxxx or UNIT/degree).

-Jr[lon0/]scale -JR[lon0/]width (Winkel Tripel).

Give the central meridian lon0 (optional) and scale along equator (1:xxxx or UNIT/degree).

-Jv[lon0/]scale or -JV[lon0/]width (Van der Grinten).

Give the central meridian lon0 (optional) and scale along equator (1:xxxx or UNIT/degree).

-Jw[lon0/]scale or -JW[lon0/]width (Mollweide [E]).

Give the central meridian lon0 (optional) and scale along equator (1:xxxx or UNIT/degree).

NON-GEOGRAPHICAL PROJECTIONS:

-Jp[a]scale[/origin][r|z] or -JP[a]width[/origin][r|z] (Polar coordinates (theta,r))

Optionally insert a after -Jp [ or -JP] for azimuths CW from North instead of directions CCW from East [Default]. Optionally append /origin in degrees to indicate an angular offset [0]). Finally, append r if r is elevations in degrees (requires s >= 0 and n <= 90) or z if you want to annotate depth rather than radius [Default]. Give scale in UNIT/r-unit.

-Jxx-scale[/y-scale] or -JXwidth[/height] (Linear, log, and power scaling)

Give x-scale (1:xxxx or UNIT/x-unit) and/or y-scale (1:xxxx or UNIT/y-unit); or specify width and/or height in UNIT. y-scale=x-scale if not specified separately and using 1:xxxx implies that x-unit and y-unit are in meters. Use negative scale(s) to reverse the direction of an axis (e.g., to have y be positive down). Set height or width to 0 to have it recomputed based on the implied scale of the other axis. Optionally, append to x-scale, y-scale, width or height one of the following:

d

Data are geographical coordinates (in degrees).

l

Take log10 of values before scaling.

ppower

Raise values to power before scaling.

t

Input coordinates are time relative to TIME_EPOCH.

T

Input coordinates are absolute time.

Default axis lengths (see gmtdefaults) can be invoked using -JXh (for landscape); -JXv (for portrait) will swap the x- and y-axis lengths. The default unit for this installation is either cm or inch, as defined in the file share/gmt.conf. However, you may change this by editing your .gmtdefaults4 file(s).

-R

xmin, xmax, ymin, and ymax specify the Region of interest. For geographic regions, these limits correspond to west, east, south, and north and you may specify them in decimal degrees or in [+-]dd:mm[:ss.xxx][W|E|S|N] format. Append r if lower left and upper right map coordinates are given instead of w/e/s/n. The two shorthands -Rg and -Rd stand for global domain (0/360 and -180/+180 in longitude respectively, with -90/+90 in latitude). Alternatively, specify the name of an existing grid file and the -R settings (and grid spacing, if applicable) are copied from the grid. For calendar time coordinates you may either give (a) relative time (relative to the selected TIME_EPOCH and in the selected TIME_UNIT; append t to -JX|x), or (b) absolute time of the form [date]T[clock] (append T to -JX|x). At least one of date and clock must be present; the T is always required. The date string must be of the form [-]yyyy[-mm[-dd]] (Gregorian calendar) or yyyy[-Www[-d]] (ISO week calendar), while the clock string must be of the form hh:mm:ss[.xxx]. The use of delimiters and their type and positions must be exactly as indicated (however, input, output and plot formats are customizable; see gmtdefaults). Special case for the UTM projection: If -C is used and -R is not given then the region is set to coincide with the given UTM zone so as to preserve the full ellipsoidal solution (See RESTRICTIONS for more information).

OPTIONS

No space between the option flag and the associated arguments.

infile(s)

input file(s) with 2 or more columns. If no file(s) is given, mapproject will read the standard input.

-A[f|b]

-A calculates the (forward) azimuth from fixed point lon/lat to each data point. Use -Ab to get back-azimuth from data points to fixed point. Upper case F or B will convert from geodetic to geocentric latitudes and estimate azimuth of geodesics (assuming the current ellipsoid is not a sphere). If no fixed point is given then we compute the azimuth (or back-azimuth) from the previous point.

-C

Set center of projected coordinates to be at map projection center [Default is lower left corner]. Optionally, add offsets in the projected units to be added (or subtracted when -I is set) to (from) the projected coordinates, such as false eastings and northings for particular projection zones [0/0]. The unit used for the offsets is the plot distance unit in effect (see MEASURE_UNIT) unless -F is used, in which case the offsets are in meters.

-D

Temporarily override MEASURE_UNIT and use c (cm), i (inch), m (meter), or p (points) instead. Cannot be used with -F.

-E

Convert from geodetic (lon, lat, height) to Earth Centered Earth Fixed (ECEF) (x,y,z) coordinates (add -I for the inverse conversion). Append datum ID (see -Qd) or give ellipsoid:dx,dy,dz where ellipsoid may be an ellipsoid ID (see -Qe) or given as a[,inv_f], where a is the semi-major axis and inv_f is the inverse flattening (0 if omitted). If datum is - or not given we assume WGS-84.

-F

Force 1:1 scaling, i.e., output (or input, see -I) data are in actual projected meters. To specify other units, append k (km), m (mile), n (nautical mile), i (inch), c (cm), or p (points). Without -F, the output (or input, see -I) are in the units specified by MEASURE_UNIT (but see -D).

-G

Calculate distances along track OR to the optional point set with -Gx0/y0. Append the distance unit; choose among e (m), k (km), m (mile), n (nautical mile), d (spherical degree), c (Cartesian distance using input coordinates) or C (Cartesian distance using projected coordinates). The last unit requires -R and -J to be set. Upper case E, K, M, N, or D will use exact methods for geodesic distances (Rudoe's method for distances in length units and employing geocentriclatitudes in degree calculations, assuming the current ellipsoid is not spherical). With no fixed point we calculate cumulate distances along track. To obtain incremental distance between successive points, use -G-. To specify the 2nd point via two extra columns in the input file, choose -G+.

-H

Input file(s) has header record(s). If used, the default number of header records is N_HEADER_RECS. Use -Hi if only input data should have header records [Default will write out header records if the input data have them]. Blank lines and lines starting with # are always skipped.

-I

Do the Inverse transformation, i.e. get (longitude,latitude) from (x,y) data.

-L

Determine the shortest distance from the input data points to the line(s) given in the ASCII multi-segment file line.xy. The distance and the coordinates of the nearest point will be appended to the output as three new columns. Append the distance unit; choose among e (m), k (km), m (mile), n (nautical mile), d (spherical degree), c (Cartesian distance using input coordinates) or C (Cartesian distance using projected coordinates). The last unit requires -R and -J to be set. A spherical approximation is used for geographic data.

-Q

List all projection parameters. To only list datums, use -Qd. To only list ellipsoids, use -Qe.

-S

Suppress points that fall outside the region.

-T

Coordinate conversions between datums from and to using the standard Molodensky transformation. Use -Th if 3rd input column has height above ellipsoid [Default assumes height = 0, i.e., on the ellipsoid]. Specify datums using the datum ID (see -Qd) or give ellipsoid:dx,dy,dz where ellipsoid may be an ellipsoid ID (see -Qe) or given as a[,inv_f], where a is the semi-major axis and inv_f is the inverse flattening (0 if omitted). If datum is - or not given we assume WGS-84. -T may be used in conjunction with -R -J to change the datum before coordinate projection (add -I to apply the datum conversion after the inverse projection). Make sure that the ELLIPSOID setting is correct for your case.

-V

Selects verbose mode, which will send progress reports to stderr [Default runs "silently"].

-:

Toggles between (longitude,latitude) and (latitude,longitude) input and/or output. [Default is (longitude,latitude)]. Append i to select input only or o to select output only. [Default affects both].

-bi

Selects binary input. Append s for single precision [Default is d (double)]. Uppercase S or D will force byte-swapping. Optionally, append ncol, the number of columns in your binary input file if it exceeds the columns needed by the program. Or append c if the input file is netCDF. Optionally, append var1/var2/... to specify the variables to be read. [Default is 2 input columns].

-bo

Selects binary output. Append s for single precision [Default is d (double)]. Uppercase S or D will force byte-swapping. Optionally, append ncol, the number of desired columns in your binary output file. [Default is same as input].

-f

Special formatting of input and/or output columns (time or geographical data). Specify i or o to make this apply only to input or output [Default applies to both]. Give one or more columns (or column ranges) separated by commas. Append T (absolute calendar time), t (relative time in chosen TIME_UNIT since TIME_EPOCH), x (longitude), y (latitude), or f (floating point) to each column or column range item. Shorthand -f[i|o]g means -f[i|o]0x,1y (geographic coordinates).

-g

Examine the spacing between consecutive data points in order to impose breaks in the line. Append x|X or y|Y to define a gap when there is a large enough change in the x or y coordinates, respectively, or d|D for distance gaps; use upper case to calculate gaps from projected coordinates. For gap-testing on other columns use [col]z; if col is not prepended the it defaults to 2 (i.e., 3rd column). Append [+|-]gap and optionally a unit u. Regarding optional signs: -ve means previous minus current column value must exceed |gap to be a gap, +ve means current minus previous column value must exceed gap, and no sign means the absolute value of the difference must exceed gap. For geographic data (x|y|d), the unit u may be meter [Default], kilometer, miles, or nautical miles. For projected data (X|Y|D), choose from inch, centimeter, meter, or points [Default unit set by MEASURE_UNIT]. Note: For x|y|z with time data the unit is instead controlled by TIME_UNIT. Repeat the option to specify multiple criteria, of which any can be met to produce a line break. Issue an additional -ga to indicate that all criteria must be met instead.

-m

Multiple segment file(s). Segments are separated by a special record. For ASCII files the first character must be flag [Default is '>']. For binary files all fields must be NaN and -b must set the number of output columns explicitly. By default the -m setting applies to both input and output. Use -mi and -mo to give separate settings to input and output.

ASCII FORMAT PRECISION

The ASCII output formats of numerical data are controlled by parameters in your .gmtdefaults4 file. Longitude and latitude are formatted according to OUTPUT_DEGREE_FORMAT, whereas other values are formatted according to D_FORMAT. Be aware that the format in effect can lead to loss of precision in the output, which can lead to various problems downstream. If you find the output is not written with enough precision, consider switching to binary output (-bo if available) or specify more decimals using the D_FORMAT setting.

EXAMPLES

To transform a file with (longitude,latitude) into (x,y) positions in cm on a Mercator grid for a given scale of 0.5 cm per degree, run

mapproject lonlatfile -R20/50/12/25 -Jm0.5c > xyfile

To transform several 2-column, binary, double precision files with (latitude,longitude) into (x,y) positions in inch on a Transverse Mercator grid (central longitude 75W) for scale = 1:500000 and suppress those points that would fall outside the map area, run

mapproject tracks.* -R-80/-70/20/40 -Jt-75/1:500000 -: -S -Di -bo -bi2 > tmfile.b

To convert the geodetic coordinates (lon, lat, height) in the file old.dat from the NAD27 CONUS datum (Datum ID 131 which uses the Clarke-1866 ellipsoid) to WGS 84, run

mapproject old.dat -Th131 > new.dat

To compute the closest distance (in km) between each point in the input file quakes.dat and the line segments given in the multi-segment ASCII file coastline.xy, run

mapproject quakes.dat -Lcoastline.xy/k > quake_dist.dat

RESTRICTIONS

The rectangular input region set with -R will in general be mapped into a non-rectangular grid. Unless -C is set, the leftmost point on this grid has xvalue = 0.0, and the lowermost point will have yvalue = 0.0. Thus, before you digitize a map, run the extreme map coordinates through mapproject using the appropriate scale and see what (x,y) values they are mapped onto. Use these values when setting up for digitizing in order to have the inverse transformation work correctly, or alternatively, use awk to scale and shift the (x,y) values before transforming.
For some projection, a spherical solution may be used despite the user having selected an ellipsoid. This occurs when the users -R setting implies a region that exceeds the domain in which the ellipsoidal series expansions are valid. These are the conditions: (1) Lambert Conformal Conic (-JL)and Albers Equal-Area (-JB) will use the spherical solution when the map scale exceeds 1.0E7. (2) Transverse Mercator (-JT) and UTM (-JU) will will use the spherical solution when either the west or east boundary given in -R is more than 10 degrees from the central meridian, and (3) same for Cassini (-JC) but with a limit of only 4 degrees.

ELLIPSOIDS AND SPHEROIDS

GMT will use ellipsoidal formulae if they are implemented and the user have selected an ellipsoid as the reference shape (see ELLIPSOID in gmtdefaults). The user needs to be aware of a few potential pitfalls: (1) For some projections, such as Transverse Mercator, Albers, and Lamberts conformal conic we use the ellipsoidal expressions when the areas mapped are small, and switch to the spherical expressions (and substituting the appropriate auxiliary latitudes) for larger maps. The ellipsoidal formulae are used as follows: (a) Transverse Mercator: When all points are within 10 degrees of central meridian, (b) Conic projections when longitudinal range is less than 90 degrees, (c) Cassini projection when all points are within 4 degrees of central meridian. (2) When you are trying to match some historical data (e.g., coordinates obtained with a certain projection and a certain reference ellipsoid) you may find that GMT gives results that are slightly different. One likely source of this mismatch is that older calculations often used less significant digits. For instance, Snyder's examples often use the Clarke 1866 ellipsoid (defined by him ashaving a flattening f = 1/294.98). From f we get the eccentricity squared to be 0.00676862818 (this is what GMT uses), while Snyder rounds off and uses 0.00676866. This difference can give discrepancies of several tens of cm. If you need to reproduce coordinates projected with this slightly different eccentricity, you should specify your own ellipsoid with the same parameters as Clarke 1866, but with f = 1/294.97861076. Also, be aware that older data may be referenced to different datums, and unless you know which datum was used and convert all data to a common datum you may experience mismatches of tens to hundreds of meters. (3) Finally, be aware that MAP_SCALE_FACTOR have certain default values for some projections so you may have to override the setting in order to match results produced with other settings.

SEE ALSO

gmtdefaults(1), GMT(1), project(1)

REFERENCES

Bomford, G., 1952, Geodesy, Oxford U. Press.
Snyder, J. P., 1987, Map Projections - A Working Manual, U.S. Geological Survey Prof. Paper 1395.
Vanicek, P. and Krakiwsky, E, 1982, Geodesy - The Concepts, North-Holland Publ., ISBN: 0 444 86149 1.