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RHUMBSOLVE(1)		    GeographicLib Utilities		 RHUMBSOLVE(1)

       RhumbSolve -- perform rhumb line	calculations

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

       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.  Also computed is 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).  A point at a pole is
       treated as a point a tiny distance away from the	pole on	the given line
       of longitude.  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 command line argument, 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.	 If the	end points are on opposite meridians, there
	   are two shortest rhumb lines	and the	east-going one is chosen.

       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.

       -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.  (-l is an alternative, deprecated,
	   spelling of this flag.)

       -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.

       -d  output angles as degrees, minutes, seconds instead of decimal

       -:  like	-d, except use : as a separator	instead	of the d, ', and "

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

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

       -s  By default, the rhumb line calculations are carried out exactly in
	   terms of elliptic integrals.	 This includes the use of the addition
	   theorem for elliptic	integrals to compute the divided difference of
	   the isometric and rectifying	latitudes.  If -s is supplied this
	   divided difference is computed using	Krueger	series for the
	   transverse Mercator projection which	is only	accurate for |f| <
	   0.01.  See "ACCURACY".

       --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).

	   print version and exit.

       -h  print usage and exit.

	   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

       --line-separator	linesep
	   set the line	separator character to linesep.	 By default this is a

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

       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).

       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.

       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.

       The algorithm used by RhumbSolve	uses exact formulas for	converting
       between the latitude, rectifying	latitude (mu), and isometric latitude
       (psi).  These formulas are accurate for any value of the	flattening.
       The computation of rhumb	lines involves the ratio (psi1 - psi2) / (mu1
       - mu2) and this is subject to large round-off errors if lat1 is close
       to lat2.	 So this ratio is computed using divided differences using one
       of two methods: by default, this	uses the addition theorem for elliptic
       integrals (accurate for all values of f); however, with the -s options,
       it is computed using the	series expansions used by
       TransverseMercatorProj(1) for the conversions between rectifying	and
       conformal latitudes (accurate for |f| < 0.01).  For the WGS84
       ellipsoid, the error is about 10	nanometers using either	method.

       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

       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
	  36:24:30.3N 051:28:26.4W
	  32:10:26.8N 030:20:57.3W
	  27:56:13.2N 010:10:54.2W
	  23:41:50.1N 009:12:45.5E
	  19:27:18.7N 027:59:22.1E
	  15:12:40.2N 046:17:01.1E
	  10:57:55.9N 064:12:52.8E
	  06:43:07.3N 081:53:28.8E
	  02:28:16.2N 099:24:54.5E
	  01:46:36.0S 116:52:59.7E

       GeoConvert(1), GeodSolve(1), TransverseMercatorProj(1).

       An online version of this utility is availbable at

       The Wikipedia page, Rhumb line,

       RhumbSolve was written by Charles Karney.

       RhumbSolve was added to GeographicLib,
       <>, in version 1.37.

GeographicLib 1.46		  2016-02-14			 RHUMBSOLVE(1)


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