RhumbSolve - Online in the Cloud

PROGRAM:

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
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
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

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