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```RHUMBSOLVE(1)               GeographicLib Utilities              RHUMBSOLVE(1)

NAME
RhumbSolve -- perform rhumb line calculations

SYNOPSIS
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 ]

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.  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.  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.

3.  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.

OPTIONS
-i  perform an inverse calculation (see 3 above).

-l  line mode (see 2 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  specify the ellipsoid via a f; the equatorial radius is a and the
flattening is 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
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  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
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
read input from the file infile instead of from standard input; a
file name of "-" stands for standard input.

--input-string
All occurrences of the line separator character (default is a
semicolon) in instring are converted to newlines before the reading
begins.

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

--output-file
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 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.

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

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
<http://geographiclib.sourceforge.net/cgi-bin/RhumbSolve>.