GeodSolve -- perform geodesic calculations
GeodSolve [
-i |
-L lat1 lon1 azi1 |
-D lat1 lon1 azi1 s13 |
-I
lat1 lon1 lat3 lon3 ] [
-a ] [
-e
a f ] [
-u ] [
-F ] [
-d |
-: ] [
-w ] [
-b ] [
-f ] [
-p prec ] [
-E
] [
--comment-delimiter commentdelim ] [
--version |
-h |
--help ] [
--input-file infile |
--input-string instring ] [
--line-separator
linesep ] [
--output-file outfile ]
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.
- 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.
- 3.
- 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. Command line arguments -D and
-I work similarly with the geodesic line defined in terms of a
direct or inverse geodesic calculation, respectively.
- -i
- perform an inverse geodesic calculation (see 2 above).
-
-L lat1 lon1 azi1
- line mode (see 3 above); generate a sequence of points
along the geodesic specified by lat1 lon1 azi1. 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.)
-
-D lat1 lon1 azi1
s13
- line mode (see 3 above); generate a sequence of points
along the geodesic specified by lat1 lon1 azi1
s13. The -w flag can be used to swap the default order of
the 2 geographic coordinates, provided that it appears before -D.
Similarly, the -a flag can be used to change the interpretation of
s13 to a13, provided that it appears before -D.
-
-I lat1 lon1 lat3
lon3
- line mode (see 3 above); generate a sequence of points
along the geodesic specified by lat1 lon1 lat3
lon3. The -w flag can be used to swap the default order of
the 2 geographic coordinates, provided that it appears before
-I.
- -a
- toggle the arc mode flag (it starts off); if this flag is
on, then on input and output s12 is replaced by a12
the arc length (in degrees) on the auxiliary sphere. See "AUXILIARY
SPHERE".
-
-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.
- -u
- unroll the longitude. Normally, on output longitudes are
reduced to lie in [-180deg,180deg). However with this option, the returned
longitude lon2 is "unrolled" so that lon2 -
lon1 indicates how often and in what sense the geodesic has
encircled the earth. Use the -f option, to get both longitudes
printed.
- -F
- fractional mode. This only has any effect with the
-D and -I options (and is otherwise ignored). The values
read on standard input are interpreted as fractional distances to point 3,
i.e., as s12/s13 instead of s12. If arc mode is in
effect, then the values denote fractional arc length, i.e.,
a12/a13. The fractional distances can be entered as a simple
fraction, e.g., 3/4.
- -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
- toggle the longitude first flag (it starts off); if the
flag is on, then on input and output, longitude precedes latitude (except
that, on input, this can be overridden by a hemisphere designator,
N, S, E, W).
- -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 prec
- 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 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).
- --version
- print version and exit.
- -h
- print usage and exit.
- --help
- 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 begins.
-
--line-separator linesep
- set the line separator character to linesep. By
default this is a semicolon.
-
--output-file outfile
- 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, 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).
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).
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,
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.
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
GeodSolve to return an
exit code of 1. However, an error does not cause
GeodSolve to
terminate; following lines will be converted.
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 equatorial 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
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.
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
Equally spaced waypoints on the route:
for ((i = 0; i <= 10; ++i)); do echo $i/10; done |
GeodSolve -I 40:38:23N 073:46:44W 01:21:33N 103:59:22E -F -: -p 0
40:38:23.0N 073:46:44.0W 003:18:29.9
54:24:51.3N 072:25:39.6W 004:18:44.1
68:07:37.7N 069:40:42.9W 006:44:25.4
81:38:00.4N 058:37:53.9W 017:28:52.7
83:43:26.0N 080:37:16.9E 156:26:00.4
70:20:29.2N 097:01:29.4E 172:31:56.4
56:38:36.0N 100:14:47.6E 175:26:10.5
42:52:37.1N 101:43:37.2E 176:34:28.6
29:03:57.0N 102:39:34.8E 177:07:35.2
15:13:18.6N 103:22:08.0E 177:23:44.7
01:21:33.0N 103:59:22.0E 177:29:09.2
GeoConvert(1).
An online version of this utility is availbable at
<
https://geographiclib.sourceforge.io/cgi-bin/GeodSolve>.
The algorithms are described in C. F. F. Karney,
Algorithms for
geodesics, J. Geodesy 87, 43-55 (2013); DOI:
<
https://doi.org/10.1007/s00190-012-0578-z>; addenda:
<
https://geographiclib.sourceforge.io/geod-addenda.html>.
The Wikipedia page, Geodesics on an ellipsoid,
<
https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid>.
GeodSolve was written by Charles Karney.
GeodSolve was added to GeographicLib,
<
https://geographiclib.sourceforge.io>, 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.)