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PLANIMETER(1) | GeographicLib Utilities | PLANIMETER(1) |

# NAME¶

Planimeter -- compute the area of geodesic polygons

# 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¶

An online version of this utility is availbable 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*, Technical Report,
SRI International (2023); URL: <https://arxiv.org/abs/2303.03219>.

# AUTHOR¶

**Planimeter** was written by Charles Karney.

# HISTORY¶

**Planimeter** was added to GeographicLib,
<https://geographiclib.sourceforge.io>, in version 1.4.

2023-07-25 | GeographicLib 2.3 |