RhumbSolve -- perform rhumb line calculations

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.

Author

RhumbSolve was written by Charles Karney.

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 lat1lon1azi12s12 and prints
lat2lon2S12 on standard output. This is the direct calculation.

2. With the -i option, RhumbSolve performs the inverse calculation. It reads lines containing lat1lon1lat2lon2 and prints the values of azi12s12S12 for the corresponding shortest rhumb lines.

3. Command line arguments -Llat1lon1azi12 specify a rhumb line. RhumbSolve then accepts a sequence
of s12 values (one per line) on standard input and prints lat2lon2S12 for each. This generates a
sequence of points on a rhumb line.

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.

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

History

RhumbSolve was added  to  GeographicLib,  <https://geographiclib.sourceforge.io>,  in  version  1.37  and
       substantially rewritten in version 2.2.

GeographicLib 2.5                                  2024-12-28                                      RHUMBSOLVE(1)

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).

Name

       RhumbSolve -- perform rhumb line calculations

Options

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

-Llat1lon1azi12
line mode (see 3 above); generate a sequence of points along the rhumb line specified by lat1lon1azi12. The -w flag can be used to swap the default order of the 2 geographic coordinates, provided
that it appears before -L.

-eaf
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).

-pprec
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-delimitercommentdelim
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-fileinfile
read input from the file infile instead of from standard input; a file name of "-" stands for
standard input.

--input-stringinstring
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-separatorlinesep
set the line separator character to linesep. By default this is a semicolon.

--output-fileoutfile
write output to the file outfile instead of to standard output; a file name of "-" stands for
standard output.

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.

Synopsis

RhumbSolve [ -i | -Llat1lon1azi12 ] [ -eaf ] [ -u ] [ -d | -: ] [ -w ] [ -pprec ] [ -E ] [
       --comment-delimitercommentdelim ] [ --version | -h | --help ] [ --input-fileinfile | --input-stringinstring ] [ --line-separatorlinesep ] [ --output-fileoutfile ]