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I wrote previously:
"But sometimes a miracle happens. Even when the objects aren't perfectly
aligned, the same almost trivial math applies. The math doesn't care whether
the objects are really aligned so long as there is an equivalent case where
they are in fact aligned. For example, the two objects could be separated in
azimuth by 170 or even 165 degrees instead of 180 degrees, and under the
right conditions we can "pretend" that they are separated in azimuth by 180
degrees, and it all works out correctly.

How can this be?! Some kind of crazy voodoo? No, just good old rigorous
math. It works because the altitude of the Moon doesn't matter much,
especially when the observed lunar distance is close to 90 degrees or the
Moon's altitude is near the zenith."

I thought a picture might help to see the variety of cases where this works
properly. In the attached graphic, we see the surprisingly large region of
the celestial sphere where "trig-free" clearing of a lunar distance would
yield near-perfect results. Note that this diagram applies when the Moon's
altitude is 40 degrees. At lower altitudes, the zones are a bit smaller. At
higher altitudes, they expand greatly. When the Moon's altitude is 80
degrees, they cover half the sky.

 -FER
www.HistoricalAtlas.com/lunars



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