/usr/bin/RhumbSolve

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SYNOPSIS
       RhumbSolve [ -i | -L lat1 lon1 azi12 ] [ -e a f ] [ -u ] [ -d | -: ] [
       -w ] [ -p prec ] [ -E ] [ --comment-delimiter commentdelim ] [
       --version | -h | --help ] [ --input-file infile | --input-string
       instring ] [ --line-separator linesep ] [ --output-file outfile ]

DESCRIPTION
       The path with constant heading between two points on the ellipsoid at
       (lat1, lon1) and (lat2, lon2) is called the rhumb line or loxodrome.
       Its length is s12 and the rhumb line has a forward azimuth azi12 along
       its length.  The quantity S12 is the area between the rhumb line 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).  The longitude
       becomes indeterminate when a rhumb line passes through a pole, and
       RhumbSolve reports NaNs for the longitude and the area in this case.

       NOTE: the rhumb line is not the shortest path between two points; that
       is the geodesic and it is calculated by GeodSolve(1).

       RhumbSolve operates in one of three modes:

       1.  By default, RhumbSolve accepts lines on the standard input
           containing lat1 lon1 azi12 s12 and prints lat2 lon2 S12 on standard
           output.  This is the direct calculation.

       2.  With the -i option, RhumbSolve performs the inverse calculation.
           It reads lines containing lat1 lon1 lat2 lon2 and prints the values
           of azi12 s12 S12 for the corresponding shortest rhumb lines.

       3.  Command line arguments -L lat1 lon1 azi12 specify a rhumb line.
           RhumbSolve then accepts a sequence of s12 values (one per line) on
           standard input and prints lat2 lon2 S12 for each.  This generates a
           sequence of points on a rhumb line.

OPTIONS
       -i  perform an inverse calculation (see 2 above).

       -L lat1 lon1 azi12
           line mode (see 3 above); generate a sequence of points along the
           rhumb line specified by lat1 lon1 azi12.  The -w flag can be used
           to swap the default order of the 2 geographic coordinates, provided
           that it appears before -L.

       -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.

       -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  on input and output, longitude precedes latitude (except that on
           input this can be overridden by a hemisphere designator, N, S, E,
           W).

       -p prec
           set the output precision to prec (default 3); prec is the precision
           relative to 1 m.  See "PRECISION".

       -E  By default, the rhumb line calculations are carried out using
           series expansions valid for |f| < 0.01.  If -E is supplied, exact
           equations for the rhumb line are used and the area integral is
           computed with an accurate fit based on this exact equations; these
           are valid for arbitrary eccentricities.

       --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
       RhumbSolve 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).

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 RhumbSolve to return an exit code of
       1.  However, an error does not cause RhumbSolve to terminate; following
       lines will be converted.

ACCURACY
       The algorithm used by RhumbSolve uses either series expansions or (if
       -E is specified) exact formulas for computing the rhumb line and the
       area.  These series are formulas are accurate for |f| < 0.01 and the
       exact formulas apply for any value of the flattening.  The computation
       of rhumb lines and the area involves the ratio of differences and, for
       nearly east- or west-going rhumb lines, this might result in a large
       loss of accuracy.  However, this problem is avoided by the use of
       divided differences. For the WGS84 ellipsoid, the error is about 10
       nanometers using either method.

EXAMPLES
       Route from JFK Airport to Singapore Changi Airport:

          echo 40:38:23N 073:46:44W 01:21:33N 103:59:22E |
          RhumbSolve -i -: -p 0

          103:34:58.2 18523563 45921660958919

       N.B. This is not the route typically taken by aircraft because it's
       considerably longer than the geodesic given by GeodSolve(1).

       Waypoints on the route at intervals of 2000km:

          for ((i = 0; i <= 20; i += 2)); do echo ${i}000000;done |
          RhumbSolve -L 40:38:23N 073:46:44W 103:34:58.2 -: -p 0

          40:38:23.0N 073:46:44.0W 0
          36:24:30.3N 051:28:26.4W 9817078307821
          32:10:26.8N 030:20:57.3W 18224745682005
          27:56:13.2N 010:10:54.2W 25358020327741
          23:41:50.1N 009:12:45.5E 31321269267102
          19:27:18.7N 027:59:22.1E 36195163180159
          15:12:40.2N 046:17:01.1E 40041499143669
          10:57:55.9N 064:12:52.8E 42906570007050
          06:43:07.3N 081:53:28.8E 44823504180200
          02:28:16.2N 099:24:54.5E 45813843358737
          01:46:36.0S 116:52:59.7E 45888525219677

SEE ALSO
       GeoConvert(1), GeodSolve(1).

       An online version of this utility is available at
       <https://geographiclib.sourceforge.io/cgi-bin/RhumbSolve>.

       An online version of this utility is available at
       <https://geographiclib.sourceforge.io/cgi-bin/RhumbSolve>.

       This solution for rhumb line is described in C. F. F. Karney, The area
       of rhumb polygons, Stud. Geophys. Geod. 68(3--4), 99--120 (2024); DOI
       <https://doi.org/10.1007/s11200-024-0709-z>.

       The Wikipedia page, Rhumb line,
       <https://en.wikipedia.org/wiki/Rhumb_line>.

AUTHOR
       RhumbSolve was written by Charles Karney.

HISTORY
       RhumbSolve was added to GeographicLib,