Geometry Boost List: [geometry] Spherical area

Subject: [geometry] Spherical area
From: Vissarion Fysikopoulos (vissarion.fysikopoulos_at_[hidden])
Date: 2016-07-07 14:20:42

Hi all,

I am working on the computation of the spherical area of a polygon. I 
would like to discuss two topics.

A. Currently Boost.Geometry has a strategy called area_huiller (there is 
a typo, the right name is Huilier). Basically, this strategy adds the 
signed areas of the triangles formed by the North pole and a segment of 
the polygon.

Apart from the typo there are two more serious problems about this 
strategy.

1) Accuracy problems because of badly conditioned triangles (cf. 
https://svn.boost.org/trac/boost/ticket/11930)

2) Incorrect results in corner cases: polygons that contain a pole, 
polygons that has a pole as a vertex (cf. 
https://svn.boost.org/trac/boost/ticket/11931)

Problem 1) is fixed by using another strategy (let me call it 
area_trapezoidal) that adds the areas of the "spherical trapezoids" that 
formed by each segment of the polygon tow meridian segments starting 
from segment's endpoints and end at the equator, and the corresponding 
equator segment. See the last formula in 
https://en.wikipedia.org/wiki/Spherical_trigonometry#Area_and_spherical_excess

Problem 2) can be solved independently in both strategies, by keeping 
track whether the polygon encircles the South pole and then subtract a 
quantity---4*\pi or 2*\pi depending on the strategy---from the result.

For the case of polygons that have South pole as a vertex (sub-case of 
Problem 2), area_huiller strategy should do more work (either switch the 
computation using the North pole or count the angle of the polygon on 
South pole and subtract the corresponding quantity).

The first question here is: does it worth the effort to implement fixes 
for area_huiller since the trapezoidal one seems to be supreme? Do you 
have any test case where you need a formula such as area_huiller, i.e. 
it assumes that we decompose the polygon to triangles? My opinion is the 
area_huiller is redundant and we can remove it.

Another advantage of area_trapezoidal is that it can be used as a basis 
for geographic area, which will be my next step.

B. There is a natural question on the semantics of the spherical polygon 
area. Given a polygon on a sphere is not clear if we refer to the 
polygon with the smallest enclosing area or it's complement. Also, I 
guess that we assume that the segments of a polygon are geodesics to 
avoid cases of polygons with segments that wrap around the globe. For 
example POLYGON((1 1, -1 1, -1 -1, 1 -1, 1 1)) can be assumed to wrap 
around the sphere or not. With the assumption of geodesic segments this 
is uniquely defined.

This issue is connected to whether we allow a polygon to have negative 
area. A polygon admits two orientations. The one corresponds to positive 
area. The other either corresponds to the same value of area negated or 
to the area of the complement of the polygon. I would go for the first 
definition since it seems that will be more compatible to cartesian case.

 From your experience with Boost.Geometry what do you think is the best 
semantics?

Any comment or feedback is more than welcome!

Best regards,
Vissarion.