/usr/bin/GeodSolve

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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-separator 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 option, GeodSolve performs the inverse geodesic
           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.
           GeodSolve then accepts a sequence of s12 values (one per line) on
           standard input and prints lat2 lon2 azi2 for each.  This generates
           a sequence 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
           geodesic 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
           geodesic 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 appears before -D.  Similarly, the -a flag can be used to
           change the interpretation 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
           geodesic 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 ellipsoid.  A simple fraction, e.g., 1/297, is
           allowed for f.  By default, 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
           longitude 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 interpreted 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 "
           delimiters.

       -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 "AUXILIARY SPHERE".  The four quantities m12, M12, M21, and S12
           are described 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 expansions 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
           semicolon) 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, all lengths (s12) in meters,
       and all areas (S12) in meters^2.  On input angles (latitude, longitude,
       azimuth, arc length) can be as decimal degrees or degrees, minutes,
       seconds.  For example, "40d30", "40d30'", "40:30", "40.5d", and 40.5
       are all equivalent.  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 longitude.  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
       COORDINATES" 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 maximum latitude.  Geodesics with a12 > 180 degrees do
       not correspond to shortest 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
       initial azimuth is perturbed by dazi1 (radians) then the second point
       is displaced 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 parallel to one another at point 2).  M12 and M21 are
       dimensionless quantities.  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 geodesic 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
       beginning 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 different 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
       azimuths 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
           geodesics 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 available at
       <https://geographiclib.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>.

       The exact routines are covered in C. F. F. Karney, Geodesics on an
       arbitrary ellipsoid of revolution, J. Geodesy 98, 4:1-14 (2024); DOI:
       <https://doi.org/10.1007/s00190-023-01813-2>.

       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.sourceforge.io>, in 2009-03.  Prior to version