/usr/bin/Planimeter

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SYNOPSIS
       Planimeter [ -r ] [ -s ] [ -l ] [ -e a f ] [ -w ] [ -p prec ] [ -G | -Q
       | -R ] [ -E ] [ --geoconvert-input ] [ --comment-delimiter commentdelim
       ] [ --version | -h | --help ] [ --input-file infile | --input-string
       instring ] [ --line-separator linesep ] [ --output-file outfile ]

DESCRIPTION
       Measure the area of a geodesic polygon.  Reads polygon vertices from
       standard input, one per line.  Vertices are be given as latitude and
       longitude.  By default latitude precedes longitude, however this
       convention is reversed with the -w flag and a hemisphere designator (N,
       S, E, W) can be used to disambiguate the coordinates.  The end of
       input, a blank line, or a line which can't be interpreted as a vertex
       signals the end of one polygon and the start of the next.  For each
       polygon print a summary line with the number of points, the perimeter
       (in meters), and the area (in meters^2).

       The edges of the polygon are given by the shortest geodesic (or rhumb
       line) between consecutive vertices.  In certain cases, there may be two
       or many such shortest path, and in that case, the polygon is not
       uniquely specified by its vertices.  For geodesics, this only happens
       with very long edges (for the WGS84 ellipsoid, any edge shorter than
       19970 km is uniquely specified by its end points).  In such cases,
       insert an additional vertex near the middle of the long edge to define
       the boundary of the polygon.

       By default, polygons traversed in a counter-clockwise direction return
       a positive area and those traversed in a clockwise direction return a
       negative area.  This sign convention is reversed if the -r option is
       given.

       Of course, encircling an area in the clockwise direction is equivalent
       to encircling the rest of the ellipsoid in the counter-clockwise
       direction.  The default interpretation used by Planimeter is the one
       that results in a smaller magnitude of area; i.e., the magnitude of the
       area is less than or equal to one half the total area of the ellipsoid.
       If the -s option is given, then the interpretation used is the one that
       results in a positive area; i.e., the area is positive and less than
       the total area of the ellipsoid.

       Arbitrarily complex polygons are allowed.  In the case of
       self-intersecting polygons the area is accumulated "algebraically",
       e.g., the areas of the 2 loops in a figure-8 polygon will partially
       cancel.  Polygons may include one or both poles.  There is no need to
       close the polygon.

OPTIONS
       -r  toggle whether counter-clockwise traversal of the polygon returns a
           positive (the default) or negative result.

       -s  toggle whether to return a signed result (the default) or not.

       -l  toggle whether the vertices represent a polygon (the default) or a
           polyline.  For a polyline, the number of points and the length of
           the path joining them is returned; the path is not closed and the
           area is not reported.

       -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.  If entering vertices as UTM/UPS or
           MGRS coordinates, use the default ellipsoid, since the conversion
           of these coordinates to latitude and longitude always uses the
           WGS84 parameters.

       -w  toggle the longitude first flag (it starts off); if the flag is on,
           then when reading geographic coordinates, longitude precedes
           latitude (this can be overridden by a hemisphere designator, N, S,
           E, W).

       -p prec
           set the output precision to prec (default 6); the perimeter is
           given (in meters) with prec digits after the decimal point; the
           area is given (in meters^2) with (prec - 5) digits after the
           decimal point.

       -G  the edges joining the vertices are geodesics.  This is the default
           option and is recommended for terrestrial applications.  This
           option, -G, and the following two options, -Q and -R, are mutually
           exclusive.

       -Q  map the points to the authalic sphere and compute the area of the
           resulting spherical polygon.  The area will be reasonable accurate
           provided that the edges are sufficiently short.  The perimeter
           calculation is not accurate.

       -R  the edges joining the vertices are rhumb lines instead of
           geodesics.

       -E  use the exact equations for the geodesic -G, authalic -Q, and rhumb
           -R calculations instead of series expansions.  For the geodesic and
           rhumb methods, the area is computed by applying discrete sine
           transforms to the integrand in the expression for the area.  These
           are more accurate, albeit slower, than the (default) series
           expansions for |f| > 0.02 and will give high accuracy for -99 < f <
           0.99.  It is not necessary to specify this option for terrestrial
           applications.

       --geoconvert-input
           The input lines are interpreted in the same way as GeoConvert(1)
           allowing the coordinates for the vertices to be given as UTM/UPS or
           MGRS coordinates, as well as latitude and longitude.  CAUTION:
           GeoConvert assumes the coordinates refer to the WGS84 ellipsoid
           (disregarding the -e flag) and MGRS coordinates signify the center
           of the corresponding MGRS square.

       --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.  For a given polygon, the last such string found
           will be 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.

EXAMPLES
       Example (the area of the 100km MGRS square 18SWK)

          Planimeter --geoconvert-input <<EOF
          18n 500000 4400000
          18n 600000 4400000
          18n 600000 4500000
          18n 500000 4500000
          EOF
          => 4 400139.532959 10007388597.2

       The following code takes the output from gdalinfo and reports the area
       covered by the data (assuming the edges of the image are geodesics).

          #! /bin/sh
          grep -E '^((Upper|Lower) (Left|Right)|Center) ' |
          sed -e 's/d /d/g' -e "s/' /'/g" | tr -s '(),\r\t' ' ' | awk '{
              if ($1 $2 == "UpperLeft")
                  ul = $6 " " $5;
              else if ($1 $2 == "LowerLeft")
                  ll = $6 " " $5;
              else if ($1 $2 == "UpperRight")
                  ur = $6 " " $5;
              else if ($1 $2 == "LowerRight")
                  lr = $6 " " $5;
              else if ($1 == "Center") {
                  printf "%s\n%s\n%s\n%s\n\n", ul, ll, lr, ur;
                  ul = ll = ur = lr = "";
              }
          }
          ' | Planimeter | cut -f3 -d' '

ACCURACY
       Using the -G option (the default), the accuracy was estimated by
       computing the error in the area for 10^7 approximately regular polygons
       on the WGS84 ellipsoid.  The centers and the orientations of the
       polygons were uniformly distributed, the number of vertices was
       log-uniformly distributed in [3, 300], and the center to vertex
       distance log-uniformly distributed in [0.1 m, 9000 km].

       The maximum error in the perimeter was 200 nm, and the maximum error in
       the area was

          0.0013 m^2 for perimeter < 10 km
          0.0070 m^2 for perimeter < 100 km
          0.070 m^2 for perimeter < 1000 km
          0.11 m^2 for all perimeters

SEE ALSO
       GeoConvert(1), GeodSolve(1).

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

       The algorithm for the area of geodesic polygon is given in Section 6 of
       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 algorithm for the area of a rhumb polygon is given in Section 3 of
       C. F. F. Karney, The area of rhumb polygons, Stud. Geophys. Geod.
       68(3--4), 99--120 (2024); DOI
       <https://doi.org/10.1007/s11200-024-0709-z>.

AUTHOR
       Planimeter was written by Charles Karney.

HISTORY