[R-sig-Geo] Computing polygon area with decimal degree coordinates
White.Denis at epamail.epa.gov
White.Denis at epamail.epa.gov
Fri Jun 2 18:33:16 CEST 2006
Computing area on the spheroid:
Kimerling AJ. 1984. Area computation from geodetic coordinates on the
spheroid. Surveying and Mapping 44(4):343-351.
Measuring distortion:
Many cartography texts cover this. Analytical formulas for computing
distortion were developed by Tissot in the 19th century. A recent paper
using Tissot measures in a sampling framework is available at:
http://www.epa.gov/wed/pages/staff/white/cagis.95.pdf
Kimerling AJ, Overton WS, White D. 1995. Statistical comparison of map
projection distortions within irregular areas. Cartography and
Geographic Information Systems 22(3):205-221.
Denis
-----r-sig-geo-bounces at stat.math.ethz.ch wrote: -----
To: adrian at maths.uwa.edu.au
From: Tim Keitt <tkeitt at gmail.com>
Sent by: r-sig-geo-bounces at stat.math.ethz.ch
Date: 02.06.2006 08:56
cc: r-sig-geo at stat.math.ethz.ch
Subject: Re: [R-sig-Geo] Computing polygon area with decimal degree
coordinates
Ah, that is very useful. I'd always assumed the hard part is computing
the
area of the triangles on the surface of a sphere/elipse, but perhaps
that is
not so hard after all. Anyone have a reference to some simple formulas?
The
other question is how much error is introduced if one computes areas
after
projecting to UTM or other map projections? (Probably depends a lot on
how
large an area one is talking about.)
THK
On 6/1/06, Adrian Baddeley <adrian at maths.uwa.edu.au> wrote:
>
>
> I may be misunderstanding the discussion about polygon area,
> but:
>
> At the risk of restating a well-known fact, it's not mathematically
> complicated
> to compute the area of a polygonal region (consisting of one or more
> separate
> regions bounded by polygons, possibly including holes) on any oriented
> two-dimensional
> surface (a plane, a sphere, a blob) using any measure of area,
provided we
> have the following information:
> - a list of all edges, in any order, oriented so that the
interior is
> to the left of each edge
> (the edges are traversed anticlockwise on the exterior,
> clockwise for a hole, etc);
> - the area (in the sense you want) of the trapezium bounded by an
> edge,
> meridians through the two endpoints, and a fixed baseline which
is
> outside the polygon.
>
> No information is required about how the edges fit together.
> No decomposition or geometric manipulation of the polygon is required,
> because area is additive with respect to sets.
>
> For example in the flat plane, we can define meridians to be vertical
> lines
> and the baseline to be the x-axis. Just shift the polygon so that it
is
> above
> the x-axis. Then associate with each edge the trapezium bounded by the
> edge,
> two vertical lines that go through the endpoints, and the x-axis.
Compute
> the area
> of each trapezium. Add these areas with the appropriate sign, and you
get
> the
> polygon area. This is just the `discrete Green's formula'. [This
algorithm
> is
> implemented in package 'spatstat'.]
>
> On a sphere, or a sphere-like surface, in any coordinate system,
> define meridians to be the lines of constant longitude coordinate, and
the
> baseline can be
> the south pole (a point, but that's OK). Then all we need to know is
the
> area (in the sense we want) of the triangular region delineated by an
edge
> and the two meridians that stretch from the edge's endpoints to the
> south pole. Sum these areas (with the appropriate sign) and we get the
> polygon area. [Other choices of baseline can be used if you only have
> a local coordinate system.]
>
> regards
> Adrian Baddeley
>
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>
--
Timothy H. Keitt
http://www.keittlab.org/
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