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@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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