Thanks to all for the suggestions so far. Here's some more information from
further experimenting.

The problem seems to be restricted to objects that cross or touch the 180
degree line in any way. If you click on the triangles and drag them one
pixel away from the 180degree line and the pole, the calculated areas change
to be (presumably) correct. In the objects in the sample I sent out, the
correct size is approx 2.3 million sq km. The size of 5600 sq km is
incorrect. 

I have tried creating the objects using different "renditions" of those
geographic points, i.e. using all positive longitudes, all negative, mixed
positive and negative, and using lambert azimuthal coordinates (metres).
None of these changes make any difference.

For the work I'm doing, mapping fishing activity in the seas around
Antarctica, this is quite a problem. Obviously the Antarctic continent
unavoidably crosses the 180 meridian, and so do various other areas we are
monitoring (fisheries, treaty areas, etc).

I suspect this problem is related to (it may even be the same problem?) one
I came across last year, where the Intersects operator fails in certain
situations, I think objects crossing the 180 degree line was one condition.

Hypothesis: even when working entirely in rectangular coordinates (metres)
in polar projections, all of MI's geographic functionality (geographic
operators e.g. "intersects", geographic functions e.g. Area(), geographic
procedures e.g. "Overlay Nodes") is liable (likely?) to fail when working
with objects that cross the 180 degree meridian. Any takers? Maybe this
should be called "Mark's theory of circular failure", that sounds fun!

** Note to MapInfo corp ** : If you could document and publish any
limitations (such as these) that we need to be aware of, it would be
appreciated, saving us having to go through the painful trial-and-error
process of testing pieces of functionality to see what works and what
doesn't. 

Cheers
Mark O


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