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

[strebe at aol.com]

Mr. Ossipoff:

You are working on what is technically known as a Hard Problem. It has been solved for conformal maps, but not for equal-area maps, and even less so for unconstrained maps. There isn't much theory to appeal to. Most of the approaches I've seen in the literature to date have been iterative least-squares solutions (by simulated annealing or other means) for some set of constraints.

In your case one constraint, which you note, would be to minimize the greatest scale factor within the region of interest. That constraint alone will not yield a unique solution because it says nothing about what happens to all the scale factors which are less than the greatest scale factor. (The conformal property, on the other hand, strictly controls what happens once you specify the region of minimal variation in scale factor, and does yield a unique solution.) It is the intractability of even a definition for optimal in non-conformal maps that has stymied research.

I note that you do not require a unique solution; you merely wish to constrain the maximum error across a region. However, I am not aware of any studies of that, either, except statistical and empirical. It is not at all clear to me that meridian multiplying takes you anywhere toward a solution, though it is also not clear to me that it does not. Mostly you need to be careful that what you're doing does not lead to improvement in one kind of distortion only to give up the gains to the other kind.

Regards,
-- daan Strebe







Thanks for the constructive and helpful comments about meridian multiplying. 

For me, the adjusted-width Aitoff or Hammer-Aitoff would be for continents, 
countries and states, and my goal would be to minimize the greatest factor by 
which the scale varies on the map, so that a certan accuracy can be guaranteed 
to someone measuring distances.

Mike Ossipoff




 


 


 

-----Original Message-----
From: Michael Ossipoff <mikeo2106 at msn.com>
To: proj at lists.maptools.org
Sent: Fri, 18 Jul 2008 12:39 pm
Subject: [Proj] Re: Transverse or oblique adjusted-width Aitoff or Hammer-Aitoff for oblong regions













Thanks for the constructive and helpful comments about meridian multiplying. 

For me, the adjusted-width Aitoff or Hammer-Aitoff would be for continents, 
countries and states, and my goal would be to minimize the greatest factor by 
which the scale varies on the map, so that a certan accuracy can be guaranteed 
to someone measuring distances.

Mike Ossipoff




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