# GeodSolve man page

GeodSolve -- perform geodesic calculations

## Synopsis

`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 ]`

## 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. 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.

## Options

-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.

## Input

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).

## 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).

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 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``````

## 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``````

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``````

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>.