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GeodSolve -- perform geodesic calculations


GeodSolve [ -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 ]


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

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.


-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.

-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.

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).

print version and exit.

-h print usage and exit.

print full documentation and exit.

read input from the file infile instead of from standard input; a file name of "-"
stands for standard input.

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.

set the line separator character to linesep. By default this is a semicolon.

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


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 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 =

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 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


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

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

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

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