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

strebe strebe at aol.com
Mon Jul 21 21:49:22 EDT 2008

Mr. Ossipoff:

>if scale variation is reduced when those curves are figures whose dimensions
>are in a ratio resembling that of the the region's dimensions,

For conformal maps the same was conjectured in 1856 by Tshebyshev [1] and proved by Gravé [2] in 1896. Snyder hypothesized the same for equal-area maps [3], but I have proved by counter-example (Wiechel compared to Lambert azimuthal) that the condition is insufficient for equal-area maps, and therefore, by inference, all other maps that are not conformal. It is possible the condition is necessary but insufficient; however, I also have evidence that even that is not true, at least not in all cases. (E.g., a band straddling the equator from edge to edge on a cylindric equal-area projection has considerably better distortion characteristics than any equal-area projection with a closed isocol approximating the same region.)

Where does this leave you? You could take measurements of any improvements in some specific metric of distortion across a broad range of examples. If the technique does, in fact, result in quantifiable improvement in all cases, then you could credibly recommend the technique.

-- daan Strebe

[*1] Chebyshev, P.L. 1856. “Sur la Construction des Carte Géographiques.” Bulletin, Académie Impériale des Sciences, Classe Physico-Mathématique, vol. 14, pp. 257–261. St. Petersburg. Reprinted in Oeuvres des P.L. Tchebychef, vol. 1. New York: Chelsea Publishing Co., 1962.

[*2] Gravé, D.A. 1896. Ob Osnovnykh Zadachakh Matematicheskoyh Teorii Postroyeniya Geograficheskikh Kart, pp. 177–183. St. Petersburg.

[*3] Snyder, J.P. 1988. “New Equal-Area Map Projections For Noncircular Regions.” The American Cartographer, vol. 15, no. 4, pp. 341–355.

On Jul 19, 2008, at 1:43:22 PM, "Michael Ossipoff" <mikeo2106 at msn.com> wrote:
From:   "Michael Ossipoff" <mikeo2106 at msn.com>
Subject:    [Proj] RE: Proj Digest, Vol 50, Issue 19
Date:   July 19, 2008 1:43:22 PM PDT
To: proj at lists.maptools.org

I was being a bit loose when I used the word "minimize". I should have just said "reduce". True, actually minimizing the greatest scale variation in a region would be a much more complicated problem than I intended to tackle.

My reasoning was merely this: If Aitoff and Hammer-Aitoff have an oval pattern for their lines of equal scale variation, and if scale variation is reduced when those curves are figures whose dimensions are in a ratio resembling that of the the region's dimensions, then Aitoff could reduce the scale variation compared to what it would be with an unmodified azimuthal equidistant.

Scale variation of meridian multiplying maps is a kind of scale variation problem that I haven't looked at yet, and so I admit that I don't know how well it would do.

Mike Ossipoff

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