=head1 NAME

GeodSolve -- perform geodesic calculations

=head1 SYNOPSIS

B<GeodSolve>
[ B<-i> | B<-L> I<lat1> I<lon1> I<azi1> |
B<-D> I<lat1> I<lon1> I<azi1> I<s13> | B<-I> I<lat1> I<lon1> I<lat3> I<lon3> ]
[ B<-a> ] [ B<-e> I<a> I<f> ] [ B<-u> ] [ B<-F> ]
[ B<-d> | B<-:> ] [ B<-w> ] [ B<-b> ] [ B<-f> ] [ B<-p> I<prec> ] [ B<-E> ]
[ B<--comment-delimiter> I<commentdelim> ]
[ B<--version> | B<-h> | B<--help> ]
[ B<--input-file> I<infile> | B<--input-string> I<instring> ]
[ B<--line-separator> I<linesep> ]
[ B<--output-file> I<outfile> ]

=head1 DESCRIPTION

The shortest path between two points on the ellipsoid at (I<lat1>,
I<lon1>) and (I<lat2>, I<lon2>) is called the geodesic.  Its length is
I<s12> and the geodesic from point 1 to point 2 has forward azimuths
I<azi1> and I<azi2> at the two end points.

B<GeodSolve> operates in one of three modes:

=over

=item 1.

By default, B<GeodSolve> accepts lines on the standard input containing
I<lat1> I<lon1> I<azi1> I<s12> and prints I<lat2> I<lon2> I<azi2>
on standard output.  This is the direct geodesic calculation.

=item 2.

With the B<-i> command line argument, B<GeodSolve> performs the inverse
geodesic calculation.  It reads lines containing I<lat1> I<lon1> I<lat2>
I<lon2> and prints the corresponding values of I<azi1> I<azi2> I<s12>.

=item 3.

Command line arguments B<-L> I<lat1> I<lon1> I<azi1> specify a geodesic
line.  B<GeodSolve> then accepts a sequence of I<s12> values (one per
line) on standard input and prints I<lat2> I<lon2> I<azi2> for each.
This generates a sequence of points on a single geodesic.  Command line
arguments B<-D> and B<-I> work similarly with the geodesic line defined
in terms of a direct or inverse geodesic calculation, respectively.

=back

=head1 OPTIONS

=over

=item B<-i>

perform an inverse geodesic calculation (see 2 above).

=item B<-L> I<lat1> I<lon1> I<azi1>

line mode (see 3 above); generate a sequence of points along the
geodesic specified by I<lat1> I<lon1> I<azi1>.  The B<-w> flag can be
used to swap the default order of the 2 geographic coordinates, provided
that it appears before B<-L>.  (B<-l> is an alternative, deprecated,
spelling of this flag.)

=item B<-D> I<lat1> I<lon1> I<azi1> I<s13>

line mode (see 3 above); generate a sequence of points along the
geodesic specified by I<lat1> I<lon1> I<azi1> I<s13>.  The B<-w> flag
can be used to swap the default order of the 2 geographic coordinates,
provided that it appears before B<-D>.  Similarly, the B<-a> flag can be
used to change the interpretation of I<s13> to I<a13>, provided that it
appears before B<-D>.

=item B<-I> I<lat1> I<lon1> I<lat3> I<lon3>

line mode (see 3 above); generate a sequence of points along the
geodesic specified by I<lat1> I<lon1> I<lat3> I<lon3>.  The B<-w> flag
can be used to swap the default order of the 2 geographic coordinates,
provided that it appears before B<-I>.

=item B<-a>

toggle the arc mode flag (it starts off); if this flag is on, then on
input I<and> output I<s12> is replaced by I<a12> the arc length (in
degrees) on the auxiliary sphere.  See L</AUXILIARY SPHERE>.

=item B<-e> I<a> I<f>

specify the ellipsoid via the equatorial radius, I<a> and
the flattening, I<f>.  Setting I<f> = 0 results in a sphere.  Specify
I<f> E<lt> 0 for a prolate ellipsoid.  A simple fraction, e.g., 1/297,
is allowed for I<f>.  By default, the WGS84 ellipsoid is used, I<a> =
6378137 m, I<f> = 1/298.257223563.

=item B<-u>

unroll the longitude.  Normally, on output longitudes are reduced to lie
in [-180deg,180deg).  However with this option, the returned longitude
I<lon2> is "unrolled" so that I<lon2> - I<lon1> indicates how often and
in what sense the geodesic has encircled the earth.  Use the B<-f>
option, to get both longitudes printed.

=item B<-F>

fractional mode.  This only has any effect with the B<-D> and B<-I>
options (and is otherwise ignored).  The values read on standard input
are interpreted as fractional distances to point 3, i.e., as
I<s12>/I<s13> instead of I<s12>.  If arc mode is in effect, then the
values denote fractional arc length, i.e., I<a12>/I<a13>.  The
fractional distances can be entered as a simple fraction, e.g., 3/4.

=item B<-d>

output angles as degrees, minutes, seconds instead of decimal degrees.

=item B<-:>

like B<-d>, except use : as a separator instead of the d, ', and "
delimiters.

=item B<-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, I<N>, I<S>, I<E>,
I<W>).

=item B<-b>

report the I<back> azimuth at point 2 instead of the forward azimuth.

=item B<-f>

full output; each line of output consists of 12 quantities: I<lat1>
I<lon1> I<azi1> I<lat2> I<lon2> I<azi2> I<s12> I<a12> I<m12> I<M12>
I<M21> I<S12>.  I<a12> is described in L</AUXILIARY SPHERE>.  The four
quantities I<m12>, I<M12>, I<M21>, and I<S12> are described in
L</ADDITIONAL QUANTITIES>.

=item B<-p> I<prec>

set the output precision to I<prec> (default 3); I<prec> is the
precision relative to 1 m.  See L</PRECISION>.

=item B<-E>

use "exact" algorithms (based on elliptic integrals) for the geodesic
calculations.  These are more accurate than the (default) series
expansions for |I<f>| E<gt> 0.02.

=item B<--comment-delimiter> I<commentdelim>

set the comment delimiter to I<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).

=item B<--version>

print version and exit.

=item B<-h>

print usage and exit.

=item B<--help>

print full documentation and exit.

=item B<--input-file> I<infile>

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

=item B<--input-string> I<instring>

read input from the string I<instring> instead of from standard input.
All occurrences of the line separator character (default is a semicolon)
in I<instring> are converted to newlines before the reading begins.

=item B<--line-separator> I<linesep>

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

=item B<--output-file> I<outfile>

write output to the file I<outfile> instead of to standard output; a
file name of "-" stands for standard output.

=back

=head1 INPUT

B<GeodSolve> measures all angles in degrees and all lengths (I<s12>) in
meters, and all areas (I<S12>) in meters^2.  On input angles (latitude,
longitude, azimuth, arc length) can be as decimal degrees or degrees,
minutes, seconds.  For example, C<40d30>, C<40d30'>, C<40:30>, C<40.5d>,
and C<40.5> are all equivalent.  By default, latitude precedes longitude
for each point (the B<-w> flag switches this convention); however on
input either may be given first by appending (or prepending) I<N> or
I<S> to the latitude and I<E> or I<W> to the longitude.  Azimuths are
measured clockwise from north; however this may be overridden with I<E>
or I<W>.

For details on the allowed formats for angles, see the C<GEOGRAPHIC
COORDINATES> section of GeoConvert(1).

=head1 AUXILIARY SPHERE

Geodesics on the ellipsoid can be transferred to the I<auxiliary sphere>
on which the distance is measured in terms of the arc length I<a12>
(measured in degrees) instead of I<s12>.  In terms of I<a12>, 180
degrees is the distance from one equator crossing to the next or from
the minimum latitude to the maximum latitude.  Geodesics with I<a12>
E<gt> 180 degrees do not correspond to shortest paths.  With the B<-a>
flag, I<s12> (on both input and output) is replaced by I<a12>.  The
B<-a> flag does I<not> affect the full output given by the B<-f> flag
(which always includes both I<s12> and I<a12>).

=head1 ADDITIONAL QUANTITIES

The B<-f> flag reports four additional quantities.

The reduced length of the geodesic, I<m12>, is defined such that if the
initial azimuth is perturbed by dI<azi1> (radians) then the second point
is displaced by I<m12> dI<azi1> in the direction perpendicular to the
geodesic.  I<m12> is given in meters.  On a curved surface the
reduced length obeys a symmetry relation, I<m12> + I<m21> = 0.  On a
flat surface, we have I<m12> = I<s12>.

I<M12> and I<M21> are geodesic scales.  If two geodesics are parallel at
point 1 and separated by a small distance I<dt>, then they are separated
by a distance I<M12> I<dt> at point 2.  I<M21> is defined similarly
(with the geodesics being parallel to one another at point 2).  I<M12>
and I<M21> are dimensionless quantities.  On a flat surface, we have
I<M12> = I<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, I<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 (I<lat1>,I<lon1>), (0,I<lon1>),
(0,I<lon2>), and (I<lat2>,I<lon2>).  It is given in meters^2.

=head1 PRECISION

I<prec> gives precision of the output with I<prec> = 0 giving 1 m
precision, I<prec> = 3 giving 1 mm precision, etc.  I<prec> is the
number of digits after the decimal point for lengths.  For decimal
degrees, the number of digits after the decimal point is I<prec> + 5.
For DMS (degree, minute, seconds) output, the number of digits after the
decimal point in the seconds component is I<prec> + 1.  The minimum
value of I<prec> is 0 and the maximum is 10.

=head1 ERRORS

An illegal line of input will print an error message to standard output
beginning with C<ERROR:> and causes B<GeodSolve> to return an exit code
of 1.  However, an error does not cause B<GeodSolve> to terminate;
following lines will be converted.

=head1 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 B<-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 I<a/b> = 1 - I<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

=head1 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:

=over

=item I<lat1> = -I<lat2> (with neither point at a pole)

If I<azi1> = I<azi2>, the geodesic is unique.  Otherwise there are two
geodesics and the second one is obtained by setting [I<azi1>,I<azi2>] =
[I<azi2>,I<azi1>], [I<M12>,I<M21>] = [I<M21>,I<M12>], I<S12> = -I<S12>.
(This occurs when the longitude difference is near +/-180 for oblate
ellipsoids.)

=item I<lon2> = I<lon1> +/- 180 (with neither point at a pole)

If I<azi1> = 0 or +/-180, the geodesic is unique.  Otherwise there are
two geodesics and the second one is obtained by setting
[I<azi1>,I<azi2>] = [-I<azi1>,-I<azi2>], I<S12> = -I<S12>.  (This occurs
when I<lat2> is near -I<lat1> for prolate ellipsoids.)

=item Points 1 and 2 at opposite poles

There are infinitely many geodesics which can be generated by setting
[I<azi1>,I<azi2>] = [I<azi1>,I<azi2>] + [I<d>,-I<d>], for arbitrary
I<d>.  (For spheres, this prescription applies when points 1 and 2 are
antipodal.)

=item I<s12> = 0 (coincident points)

There are infinitely many geodesics which can be generated by setting
[I<azi1>,I<azi2>] = [I<azi1>,I<azi2>] + [I<d>,I<d>], for arbitrary I<d>.

=back

=head1 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

=head1 SEE ALSO

GeoConvert(1).

An online version of this utility is availbable at
L<https://geographiclib.sourceforge.io/cgi-bin/GeodSolve>.

The algorithms are described in C. F. F. Karney,
I<Algorithms for geodesics>, J. Geodesy 87, 43-55 (2013); DOI:
L<https://doi.org/10.1007/s00190-012-0578-z>;
addenda: L<https://geographiclib.sourceforge.io/geod-addenda.html>.

The Wikipedia page, Geodesics on an ellipsoid,
L<https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid>.

=head1 AUTHOR

B<GeodSolve> was written by Charles Karney.

=head1 HISTORY

B<GeodSolve> was added to GeographicLib,
L<https://geographiclib.sourceforge.io>, in 2009-03.  Prior to version
1.30, it was called B<Geod>.  (The name was changed to avoid a conflict
with the B<geod> utility in I<proj.4>.)
