NAME¶
RhumbSolve -- perform rhumb line calculations
SYNOPSIS¶
RhumbSolve [
-i |
-l lat1 lon1 azi12 ]
[
-e a f ] [
-d |
-: ] [
-f ] [
-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. 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 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 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 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
for the corresponding shortest rhumb lines
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.
- -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. (Also, if f > 1,
the flattening is set to 1/ 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.
- -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.
RhumbSolve measures all angles in degrees and all lengths (
s12) in
meters. On input angles (latitude, longitude, azimuth) can be as decimal
degrees or degrees (d), minutes ('), seconds ("). A decimal point can
only appear in the least significant component and the designator (d, ', or
") for this component is optional; thus "40d30",
"40d30'", "40.5d", and 40.5 are all equivalent. By
default, latitude precedes longitude for each point; 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.
See the "QUOTING" section of
GeoConvert(1) for how to quote the
DMS designators ' and ".
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
SEE ALSO¶
GeoConvert(1),
GeodSolve(1),
TransverseMercatorProj(1).
AUTHOR¶
RhumbSolve was written by Charles Karney.
HISTORY¶
RhumbSolve was added to GeographicLib,
<
http://geographiclib.sf.net>, in version 1.37.