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GEODSOLVE(1) GeographicLib Utilities GEODSOLVE(1) NAME GeodSolve -- perform geodesic calculations SYNOPSIS GeodSolve [ -i | -L lat1 lon1 azi1 | -D lat1 lon1 azi1 s13 | -I lat1 lon1 lat3 lon3 ] [ -a ] [ -e a f ] [ -u ] [ -F ] [ -d | -: ] [ -w ] [ -b ] [ -f ] [ -p prec ] [ -E ] [ --comment-delimiter commentdelim ] [ --version | -h | --help ] [ --input-file infile | --input-string instring ] [ --line-sepa- rator linesep ] [ --output-file outfile ] DESCRIPTION The shortest path between two points on the ellipsoid at (lat1, lon1) and (lat2, lon2) is called the geodesic. Its length is s12 and the geodesic from point 1 to point 2 has forward azimuths azi1 and azi2 at the two end points. GeodSolve operates in one of three modes: 1. By default, GeodSolve accepts lines on the standard input containing lat1 lon1 azi1 s12 and prints lat2 lon2 azi2 on standard output. This is the direct geodesic calculation. 2. With the -i command line argument, GeodSolve performs the inverse geo- desic calculation. It reads lines containing lat1 lon1 lat2 lon2 and prints the corresponding values of azi1 azi2 s12. 3. Command line arguments -L lat1 lon1 azi1 specify a geodesic line. Ge- odSolve then accepts a sequence of s12 values (one per line) on stan- dard input and prints lat2 lon2 azi2 for each. This generates a se- quence of points on a single geodesic. Command line arguments -D and -I work similarly with the geodesic line defined in terms of a direct or inverse geodesic calculation, respectively. OPTIONS -i perform an inverse geodesic calculation (see 2 above). -L lat1 lon1 azi1 line mode (see 3 above); generate a sequence of points along the geo- desic specified by lat1 lon1 azi1. The -w flag can be used to swap the default order of the 2 geographic coordinates, provided that it appears before -L. -D lat1 lon1 azi1 s13 line mode (see 3 above); generate a sequence of points along the geo- desic specified by lat1 lon1 azi1 s13. The -w flag can be used to swap the default order of the 2 geographic coordinates, provided that it ap- pears before -D. Similarly, the -a flag can be used to change the in- terpretation of s13 to a13, provided that it appears before -D. -I lat1 lon1 lat3 lon3 line mode (see 3 above); generate a sequence of points along the geo- desic specified by lat1 lon1 lat3 lon3. The -w flag can be used to swap the default order of the 2 geographic coordinates, provided that it appears before -I. -a toggle the arc mode flag (it starts off); if this flag is on, then on input and output s12 is replaced by a12 the arc length (in degrees) on the auxiliary sphere. See "AUXILIARY SPHERE". -e a f specify the ellipsoid via the equatorial radius, a and the flattening, f. Setting f = 0 results in a sphere. Specify f < 0 for a prolate el- lipsoid. A simple fraction, e.g., 1/297, is allowed for f. By de- fault, the WGS84 ellipsoid is used, a = 6378137 m, f = 1/298.257223563. -u unroll the longitude. Normally, on output longitudes are reduced to lie in [-180deg,180deg). However with this option, the returned longi- tude lon2 is "unrolled" so that lon2 - lon1 indicates how often and in what sense the geodesic has encircled the earth. Use the -f option, to get both longitudes printed. -F fractional mode. This only has any effect with the -D and -I options (and is otherwise ignored). The values read on standard input are in- terpreted as fractional distances to point 3, i.e., as s12/s13 instead of s12. If arc mode is in effect, then the values denote fractional arc length, i.e., a12/a13. The fractional distances can be entered as a simple fraction, e.g., 3/4. -d output angles as degrees, minutes, seconds instead of decimal degrees. -: like -d, except use : as a separator instead of the d, ', and " delim- iters. -w toggle the longitude first flag (it starts off); if the flag is on, then on input and output, longitude precedes latitude (except that, on input, this can be overridden by a hemisphere designator, N, S, E, W). -b report the back azimuth at point 2 instead of the forward azimuth. -f full output; each line of output consists of 12 quantities: lat1 lon1 azi1 lat2 lon2 azi2 s12 a12 m12 M12 M21 S12. a12 is described in "AUX- ILIARY SPHERE". The four quantities m12, M12, M21, and S12 are de- scribed in "ADDITIONAL QUANTITIES". -p prec set the output precision to prec (default 3); prec is the precision relative to 1 m. See "PRECISION". -E use "exact" algorithms (based on elliptic integrals) for the geodesic calculations. These are more accurate than the (default) series expan- sions for |f| > 0.02. --comment-delimiter commentdelim set the comment delimiter to commentdelim (e.g., "#" or "//"). If set, the input lines will be scanned for this delimiter and, if found, the delimiter and the rest of the line will be removed prior to processing and subsequently appended to the output line (separated by a space). --version print version and exit. -h print usage and exit. --help print full documentation and exit. --input-file infile read input from the file infile instead of from standard input; a file name of "-" stands for standard input. --input-string instring read input from the string instring instead of from standard input. All occurrences of the line separator character (default is a semi- colon) in instring are converted to newlines before the reading begins. --line-separator linesep set the line separator character to linesep. By default this is a semicolon. --output-file outfile write output to the file outfile instead of to standard output; a file name of "-" stands for standard output. INPUT GeodSolve measures all angles in degrees and all lengths (s12) in meters, and all areas (S12) in meters^2. On input angles (latitude, longitude, az- imuth, arc length) can be as decimal degrees or degrees, minutes, seconds. For example, "40d30", "40d30'", "40:30", "40.5d", and 40.5 are all equiva- lent. By default, latitude precedes longitude for each point (the -w flag switches this convention); however on input either may be given first by appending (or prepending) N or S to the latitude and E or W to the longi- tude. Azimuths are measured clockwise from north; however this may be overridden with E or W. For details on the allowed formats for angles, see the "GEOGRAPHIC COORDI- NATES" section of GeoConvert(1). AUXILIARY SPHERE Geodesics on the ellipsoid can be transferred to the auxiliary sphere on which the distance is measured in terms of the arc length a12 (measured in degrees) instead of s12. In terms of a12, 180 degrees is the distance from one equator crossing to the next or from the minimum latitude to the maxi- mum latitude. Geodesics with a12 > 180 degrees do not correspond to short- est paths. With the -a flag, s12 (on both input and output) is replaced by a12. The -a flag does not affect the full output given by the -f flag (which always includes both s12 and a12). ADDITIONAL QUANTITIES The -f flag reports four additional quantities. The reduced length of the geodesic, m12, is defined such that if the ini- tial azimuth is perturbed by dazi1 (radians) then the second point is dis- placed by m12 dazi1 in the direction perpendicular to the geodesic. m12 is given in meters. On a curved surface the reduced length obeys a symmetry relation, m12 + m21 = 0. On a flat surface, we have m12 = s12. M12 and M21 are geodesic scales. If two geodesics are parallel at point 1 and separated by a small distance dt, then they are separated by a distance M12 dt at point 2. M21 is defined similarly (with the geodesics being par- allel to one another at point 2). M12 and M21 are dimensionless quanti- ties. On a flat surface, we have M12 = M21 = 1. If points 1, 2, and 3 lie on a single geodesic, then the following addition rules hold: s13 = s12 + s23, a13 = a12 + a23, S13 = S12 + S23, m13 = m12 M23 + m23 M21, M13 = M12 M23 - (1 - M12 M21) m23 / m12, M31 = M32 M21 - (1 - M23 M32) m12 / m23. Finally, S12 is the area between the geodesic from point 1 to point 2 and the equator; i.e., it is the area, measured counter-clockwise, of the geo- desic quadrilateral with corners (lat1,lon1), (0,lon1), (0,lon2), and (lat2,lon2). It is given in meters^2. PRECISION prec gives precision of the output with prec = 0 giving 1 m precision, prec = 3 giving 1 mm precision, etc. prec is the number of digits after the decimal point for lengths. For decimal degrees, the number of digits after the decimal point is prec + 5. For DMS (degree, minute, seconds) output, the number of digits after the decimal point in the seconds component is prec + 1. The minimum value of prec is 0 and the maximum is 10. ERRORS An illegal line of input will print an error message to standard output be- ginning with "ERROR:" and causes GeodSolve to return an exit code of 1. However, an error does not cause GeodSolve to terminate; following lines will be converted. ACCURACY Using the (default) series solution, GeodSolve is accurate to about 15 nm (15 nanometers) for the WGS84 ellipsoid. The approximate maximum error (expressed as a distance) for an ellipsoid with the same equatorial radius as the WGS84 ellipsoid and different values of the flattening is |f| error 0.01 25 nm 0.02 30 nm 0.05 10 um 0.1 1.5 mm 0.2 300 mm If -E is specified, GeodSolve is accurate to about 40 nm (40 nanometers) for the WGS84 ellipsoid. The approximate maximum error (expressed as a distance) for an ellipsoid with a quarter meridian of 10000 km and differ- ent values of the a/b = 1 - f is 1-f error (nm) 1/128 387 1/64 345 1/32 269 1/16 210 1/8 115 1/4 69 1/2 36 1 15 2 25 4 96 8 318 16 985 32 2352 64 6008 128 19024 MULTIPLE SOLUTIONS The shortest distance returned for the inverse problem is (obviously) uniquely defined. However, in a few special cases there are multiple az- imuths which yield the same shortest distance. Here is a catalog of those cases: lat1 = -lat2 (with neither point at a pole) If azi1 = azi2, the geodesic is unique. Otherwise there are two geo- desics and the second one is obtained by setting [azi1,azi2] = [azi2,azi1], [M12,M21] = [M21,M12], S12 = -S12. (This occurs when the longitude difference is near +/-180 for oblate ellipsoids.) lon2 = lon1 +/- 180 (with neither point at a pole) If azi1 = 0 or +/-180, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting [azi1,azi2] = [-azi1,-azi2], S12 = -S12. (This occurs when lat2 is near -lat1 for prolate ellipsoids.) Points 1 and 2 at opposite poles There are infinitely many geodesics which can be generated by setting [azi1,azi2] = [azi1,azi2] + [d,-d], for arbitrary d. (For spheres, this prescription applies when points 1 and 2 are antipodal.) s12 = 0 (coincident points) There are infinitely many geodesics which can be generated by setting [azi1,azi2] = [azi1,azi2] + [d,d], for arbitrary d. EXAMPLES Route from JFK Airport to Singapore Changi Airport: echo 40:38:23N 073:46:44W 01:21:33N 103:59:22E | GeodSolve -i -: -p 0 003:18:29.9 177:29:09.2 15347628 Equally spaced waypoints on the route: for ((i = 0; i <= 10; ++i)); do echo $i/10; done | GeodSolve -I 40:38:23N 073:46:44W 01:21:33N 103:59:22E -F -: -p 0 40:38:23.0N 073:46:44.0W 003:18:29.9 54:24:51.3N 072:25:39.6W 004:18:44.1 68:07:37.7N 069:40:42.9W 006:44:25.4 81:38:00.4N 058:37:53.9W 017:28:52.7 83:43:26.0N 080:37:16.9E 156:26:00.4 70:20:29.2N 097:01:29.4E 172:31:56.4 56:38:36.0N 100:14:47.6E 175:26:10.5 42:52:37.1N 101:43:37.2E 176:34:28.6 29:03:57.0N 102:39:34.8E 177:07:35.2 15:13:18.6N 103:22:08.0E 177:23:44.7 01:21:33.0N 103:59:22.0E 177:29:09.2 SEE ALSO GeoConvert(1). An online version of this utility is availbable at <https://geographi- clib.sourceforge.io/cgi-bin/GeodSolve>. The algorithms are described in C. F. F. Karney, Algorithms for geodesics, J. Geodesy 87, 43-55 (2013); DOI: <https://doi.org/10.1007/s00190-012-0578-z>; addenda: <https://geographi- clib.sourceforge.io/geod-addenda.html>. The Wikipedia page, Geodesics on an ellipsoid, <https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid>. AUTHOR GeodSolve was written by Charles Karney. HISTORY GeodSolve was added to GeographicLib, <https://geographiclib.source- forge.io>, in 2009-03. Prior to version 1.30, it was called Geod. (The name was changed to avoid a conflict with the geod utility in proj.4.) GeographicLib 2.2 2023-03-06 GEODSOLVE(1)
NAME | SYNOPSIS | DESCRIPTION | OPTIONS | INPUT | AUXILIARY SPHERE | ADDITIONAL QUANTITIES | PRECISION | ERRORS | ACCURACY | MULTIPLE SOLUTIONS | EXAMPLES | SEE ALSO | AUTHOR | HISTORY
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