NAME¶
GeodSolve -- perform geodesic calculations
SYNOPSIS¶
GeodSolve [
-i |
-l lat1 lon1 azi1 ] [
-a ] [
-e a f ] [
-d |
-: ] [
-b ] [
-f ] [
-p prec ] [
-E ] [
--comment-delimiter commentdelim ] [
--version |
-h |
--help ] [
--input-file infile |
--input-string instring ] [
--line-separator
linesep ] [
--output-file outfile ]
DESCRIPTION¶
The shortest path between two points on the ellipsoid at (
lat1,
lon1) and (
lat2,
lon2) is called the geodesic. Its
length is
s12 and the geodesic from point 1 to point 2 has forward
azimuths
azi1 and
azi2 at the two end points.
GeodSolve operates in one of three modes:
- 1.
- By default, GeodSolve accepts lines on the standard input
containing lat1 lon1 azi1 s12 and prints
lat2 lon2 azi2 on standard output. This is the direct
geodesic calculation.
- 2.
- Command line arguments -l lat1 lon1 azi1
specify a geodesic line. GeodSolve then accepts a sequence of
s12 values (one per line) on standard input and prints lat2
lon2 azi2 for each. This generates a sequence of points on a
single geodesic.
- 3.
- With the -i command line argument, GeodSolve performs the
inverse geodesic calculation. It reads lines containing lat1
lon1 lat2 lon2 and prints the corresponding values of
azi1 azi2 s12.
OPTIONS¶
- -i
- perform an inverse geodesic calculation (see 3 above).
- -l
- line mode (see 2 above); generate a sequence of points along the geodesic
specified by lat1 lon1 azi1.
- -a
- arc mode; on input and output s12 is replaced by a12
the arc length (in degrees) on the auxiliary sphere. See "AUXILIARY
SPHERE".
- -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.
- -b
- report the back azimuth at point 2 instead of the forward
azimuth.
- -f
- full output; each line of output consists of 12 quantities: lat1
lon1 azi1 lat2 lon2 azi2 s12
a12 m12 M12 M21 S12. a12 is
described in "AUXILIARY SPHERE". The four quantities m12,
M12, M21, and S12 are described in "ADDITIONAL
QUANTITIES".
- -p
- set the output precision to prec (default 3); prec is the
precision relative to 1 m. See "PRECISION".
- -E
- use "exact" algorithms (based on elliptic integrals) for the
geodesic calculations. These are more accurate than the (default) series
expansions for | f| > 0.02.
- --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.
GeodSolve measures all angles in degrees and all lengths (
s12) in
meters. On input angles (latitude, longitude, azimuth, arc length) 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 ".
AUXILIARY SPHERE¶
Geodesics on the ellipsoid can be transferred to the
auxiliary sphere on
which the distance is measured in terms of the arc length
a12 (measured
in degrees) instead of
s12. In terms of
a12, 180 degrees is the
distance from one equator crossing to the next or from the minimum latitude to
the maximum latitude. Geodesics with
a12 > 180 degrees do not
correspond to shortest paths. With the
-a flag,
s12 (on both
input and output) is replaced by
a12. The
-a flag does
not affect the full output given by the
-f flag (which always
includes both
s12 and
a12).
ADDITIONAL QUANTITIES¶
The
-f flag reports four additional quantities.
The reduced length of the geodesic,
m12, is defined such that if the
initial azimuth is perturbed by d
azi1 (radians) then the second point
is displaced by
m12 d
azi1 in the direction perpendicular to the
geodesic.
m12 is given in meters. On a curved surface the reduced
length obeys a symmetry relation,
m12 +
m21 = 0. On a flat
surface, we have
m12 =
s12.
M12 and
M21 are geodesic scales. If two geodesics are parallel at
point 1 and separated by a small distance
dt, then they are separated
by a distance
M12 dt at point 2.
M21 is defined similarly
(with the geodesics being parallel to one another at point 2).
M12 and
M21 are dimensionless quantities. On a flat surface, we have
M12
=
M21 = 1.
If points 1, 2, and 3 lie on a single geodesic, then the following addition
rules hold,
s13 =
s12 +
s23,
a13 =
a12 +
a23,
S13 =
S12 +
S23,
m13 =
m12
M23 +
m23 M21,
M13 =
M12 M23 - (1 -
M12 M21)
m23 /
m12, and
M31 =
M32
M21 - (1 -
M23 M32)
m12 /
m23.
Finally,
S12 is the area between the geodesic 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). It is given in meters^2.
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
GeodSolve to return an
exit code of 1. However, an error does not cause
GeodSolve to
terminate; following lines will be converted.
ACCURACY¶
Using the (default) series solution, GeodSolve is accurate to about 15 nm (15
nanometers) for the WGS84 ellipsoid. The approximate maximum error (expressed
as a distance) for an ellipsoid with the same major radius as the WGS84
ellipsoid and different values of the flattening is
|f| error
0.01 25 nm
0.02 30 nm
0.05 10 um
0.1 1.5 mm
0.2 300 mm
If
-E is specified, GeodSolve is accurate to about 40 nm (40 nanometers)
for the WGS84 ellipsoid. The approximate maximum error (expressed as a
distance) for an ellipsoid with a quarter meridian of 10000 km and different
values of the
a/b = 1 -
f is
1-f error (nm)
1/128 387
1/64 345
1/32 269
1/16 210
1/8 115
1/4 69
1/2 36
1 15
2 25
4 96
8 318
16 985
32 2352
64 6008
128 19024
MULTIPLE SOLUTIONS¶
The shortest distance returned for the inverse problem is (obviously) uniquely
defined. However, in a few special cases there are multiple azimuths which
yield the same shortest distance. Here is a catalog of those cases:
- lat1 = -lat2 (with neither point at a pole)
- If azi1 = azi2, the geodesic is unique. Otherwise there are
two geodesics and the second one is obtained by setting [
azi1,azi2] = [ azi2,azi1],
[M12,M21] = [ M21,M12], S12 =
-S12. (This occurs when the longitude difference is near +/-180 for
oblate ellipsoids.)
- lon2 = lon1 +/- 180 (with neither point at a pole)
- If azi1 = 0 or +/-180, the geodesic is unique. Otherwise there are
two geodesics and the second one is obtained by setting [
azi1,azi2] = [- azi1,-azi2], S12 =
-S12. (This occurs when lat2 is near -lat1 for
prolate ellipsoids.)
- Points 1 and 2 at opposite poles
- There are infinitely many geodesics which can be generated by setting [
azi1,azi2] = [azi1,azi2] +
[d,-d], for arbitrary d. (For spheres, this
prescription applies when points 1 and 2 are antipodal.)
- s12 = 0 (coincident points)
- There are infinitely many geodesics which can be generated by setting [
azi1,azi2] = [azi1,azi2] +
[d,d], for arbitrary d.
EXAMPLES¶
Route from JFK Airport to Singapore Changi Airport:
echo 40:38:23N 073:46:44W 01:21:33N 103:59:22E |
GeodSolve -i -: -p 0
003:18:29.9 177:29:09.2 15347628
Waypoints on the route at intervals of 2000km:
for ((i = 0; i <= 16; i += 2)); do echo ${i}000000;done |
GeodSolve -l 40:38:23N 073:46:44W 003:18:29.9 -: -p 0
40:38:23.0N 073:46:44.0W 003:18:29.9
58:34:45.1N 071:49:36.7W 004:48:48.8
76:22:28.4N 065:32:17.8W 010:41:38.4
84:50:28.0N 075:04:39.2E 150:55:00.9
67:26:20.3N 098:00:51.2E 173:27:20.3
49:33:03.2N 101:06:52.6E 176:07:54.3
31:34:16.5N 102:30:46.3E 177:03:08.4
13:31:56.0N 103:26:50.7E 177:24:55.0
04:32:05.7S 104:14:48.7E 177:28:43.6
SEE ALSO¶
GeoConvert(1). The algorithms are described in C. F. F. Karney,
Algorithms for geodesics, J. Geodesy 87, 43-55 (2013); DOI:
<
http://dx.doi.org/10.1007/s00190-012-0578-z>; addenda:
<
http://geographiclib.sf.net/geod-addenda.html>. The wikipedia page,
Geodesics on an ellipsoid,
<
http://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid>.
AUTHOR¶
GeodSolve was written by Charles Karney.
HISTORY¶
GeodSolve was added to GeographicLib,
<
http://geographiclib.sf.net>, in 2009-03. Prior to version 1.30, it was
called
Geod. (The name was changed to avoid a conflict with the
geod utility in
proj.4.)