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

[strebe at aol.com]

 Mr. Ossipoff:

The technique of "meridian multiplying" has been exploited in several equal-area projections, such as Hammer, Eckert-Greiffendorf, and Strebe's heart. The meridian multiplication coupled with the x-axis compression retains the the projection's equivalence and tends to stretch the isocols (level curves of angular distortion) east-west with respect to the original graticule at the expense of their north-south extent. By original graticule, I mean before applying any coordinate transformations to obtain an oblique or transverse aspect.

Meridian multiplying has also been used in compromise projections such as Aitoff, as you note. In the case of compromise projections the utility is less clear. Distortion is characterized by both the areal inflation and angular distortion; typically they do not coincide if they are not circular, and in those cases they are often perpendicular to each other. That is, if the isocols of angular distortion are ovals flattened horizontally, then isocols of areal inflation will tend to be ovals flattened vertically. Hence meridian multiplying to favor a region whose extent is 3:1 east-west may improve its angular distortion but degrade its areal inflation throughout that region. There is no natural or commonly recognized method of comparing angular distortion to areal inflation, so there is no natural way of deciding whether the operation resulted in any improvement. One could study the results to see if, for example, distance measurements throughout the region improved statistically. Methods to do that come with some controvery over how distance measurements should be distributed, but it at least it can all be kept scientific as long as all assumptions are noted explicitly.

Regards,
-- daan Strebe


 


 

-----Original Message-----
From: Michael Ossipoff <mikeo2106 at msn.com>
To: proj at lists.maptools.org
Sent: Thu, 17 Jul 2008 5:17 am
Subject: [Proj] Transverse or oblique width-aduusted Aitoff or Hammer-Aitoff for oblong regions.











When mapping an oblong or oval region, a region with unequal dimensions, a 
region that isn't close to being circular or square, and when linearity and 
conformality aren't needed, then why not use a transverse or oblique Aitoff or 
Aitoff-Hammer projection, centered in the mapped regiion?

Where Aitoff and Aitoff-Hammer multiply the azimuthal's X dimension by 2 (and 
change the grid the accordingly), here we multiply by a factor appropriate for 
the region's shape. If the region has one dimension 3.5 times its other 
dimension, then multiply by 3.5 instead of 2. Of course the projection's 
"equator" is along the region's long dimension, and the projection's center is 
at the center of the region.

The choice between Aitoff and Hammer-Aitoff would depend on whether or not 
equal-area is needed.

As I've said before, I suggest that, for general purpose maps of countries or 
states, equal area is a poor choice.

As I said when I proposed Apianus II, this obvious idea has probably been 
proposed before. If so, I'm proposing it again.

Mike Ossipoff




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