Incorrect formula for centroid of spherical triangle (plus inaccurate area formula)

[cffk]

The formula used for the centroid of a spherical triangle in gmtlib_geo_centroid_area (take the 3d mean of the three vertices and normalize). An example where you clearly get the wrong answer is given by whuber in

https://gis.stackexchange.com/questions/43505/calculating-a-spherical-polygon-centroid

The right formula is given by Brock (1975)

https://doi.org/10.1115/1.3423535

This easily generalizes to a polygon (without the need to break the polygon into triangles).

While on the subject of gmtlib_geo_centroid_area, I'll note that the method of computing the area (a direct calculation of the spherical excess) is terribly inaccurate for small areas. I recommend instead the formula given in

https://en.wikipedia.org/wiki/Spherical_trigonometry#From_latitude_and_longitude

Here again this formula can naturally be applied to a polygon (although care needs to be taken if the polygon encircles a pole).

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[PaulWessel]

Thanks for pointing this out. I will have a look once able to come up for air.