GEODSOLVE(1) GeographicLib Utilities GEODSOLVE(1)NAMEGeodSolve-- perform geodesic calculations
SYNOPSISGeodSolve [ -i | -l lat1 lon1 azi1 ] [ -a ] [ -e a f ] -u ] [ -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 ]
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. The -w flag can be used to
swap the default order of the 2 geographic coordinates, provided
that it appears before -l.
-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. 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.
-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).
-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.
INPUTGeodSolve 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).
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 dazi1 (radians) then the second point
is displaced by m12 dazi1 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.
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 ALSOGeoConvert(1).
An online version of this utility is availbable at
<http://geographiclib.sourceforge.net/cgi-bin/GeodSolve>.
The algorithms are described in C. F. F. Karney, Algorithms for
geodesics, J. Geodesy 87, 43-55 (2013); DOI:
<https://dx.doi.org/10.1007/s00190-012-0578-z>; addenda:
<http://geographiclib.sf.net/geod-addenda.html>.
The Wikipedia page, Geodesics on an ellipsoid,
<https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid>.
AUTHORGeodSolve was written by Charles Karney.
HISTORYGeodSolve 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.)
GeographicLib 1.45 2015-09-30 GEODSOLVE(1)
