[Proj] Re: Transverse or oblique width-adjusted Aitoff or Hammer-Aitoff for oblong regions

[strebe]

Mr. Ossipoff:

If the oblong region is on the order of 180° of extent, a meridian-multiplied Aitoff may well perform better than a cylindric equidistant. I would expect the benefit to decrease as the extent increased, eventually turning worse. This is just a (reasonably well-educated) guess. With the assistance of symbolic math programs (Mathematica, Maple, &c.), it should not be terribly difficult to derive closed-form expresssions for scale variation around a given point on the transverse Aitoff, being the simple projection that it is. Cylindric equidistant being even simpler, the comparison would not be so onerous, either. That would answer the question of transverse Aitoff versus cylindric equidistant.

A much more difficult problem is to show that you have come anywhere near some theoretical "best" for the region, or even to show that it is better than a wide selection of possible alternatives. The latter, at least, would be important in establishing a robust claim of improvement.

Your musing over how atlases choose projections might be a little kind. 

Regards,
-- daan Strebe


On Jul 22, 2008, at 12:15:26 PM, "Michael Ossipoff" <mikeo2106 at msn.com> wrote:
Again, thanks for the information. So then, there's no particular reason to expect a locally-centered and suitably oriented Aitoff map to have less scale variation than a similarly centered and oriented cylindrical equidistant map, for an extended oblong region. Actually that's a great relief, to hear that my belief about Aitoff and Hammer-Aitoff for oblong regions isn't true, or that the matter would be difficult to determine, because it means that I can stop wondering why I couldn't prove it to be true. Because I don't understand the determination of Aitoff's maximum scale variation, in comparison to other projections for such regions, it's a relief to hear that it isn't an elementary problem.

So maybe there's a good reason why Aitoff and Hammer-Aitoff aren't found in atlases, mapping oblong continents, countries and states.

Mike Ossipoff
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