<html><body name="Mail Message Editor"><div><br></div><div>Mr. Ossipoff:</div><div><br></div><div><span class="Apple-style-span" style="font-family: -webkit-monospace; font-size: 11px; ">>if scale variation is reduced when those curves are figures whose dimensions</span></div><div><span class="Apple-style-span" style="font-family: -webkit-monospace; font-size: 11px; ">>are in a ratio resembling that of the the region's dimensions,</span><br></div><div><br></div><div>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.)</div><div><br></div><div>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.</div><div><br></div><div>Regards,</div><div>-- daan Strebe</div><div><br></div><div><div>[*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.</div><div><br></div><div>[*2] Gravé, D.A. 1896. Ob Osnovnykh Zadachakh Matematicheskoyh Teorii Postroyeniya Geograficheskikh Kart, pp. 177–183. St. Petersburg.</div><div><br></div><div>[*3] Snyder, J.P. 1988. “New Equal-Area Map Projections For Noncircular Regions.” The American Cartographer, vol. 15, no. 4, pp. 341–355.</div><div><br></div></div><br>On Jul 19, 2008, at 1:43:22 PM, "Michael Ossipoff" <mikeo2106@msn.com> wrote:<br><blockquote style="padding-left: 5px; margin-left: 5px; border-left-width: 2px; border-left-style: solid; border-left-color: blue; color: blue; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0; "><div style="width: 100%; "><div id="felix-mail-header-block" style="color: black; background-color: white; border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: silver; padding-bottom: 1em; margin-bottom: 1em; width: 100%; "><table border="0" cellpadding="1" cellspacing="1" width="100%"><tbody><tr><td width="70px" style="font-family: 'Lucida Grande'; font-size: 8pt; color: gray; text-align: right; vertical-align: top; font-weight: bold; "><span>From:</span></td><td style="font-family: 'Lucida Grande'; font-size: 8pt; color: black; text-align: left; vertical-align: top; padding-left: 5px; "><span title=""Michael Ossipoff" <mikeo2106@msn.com>">"Michael Ossipoff" <mikeo2106@msn.com></span></td></tr><tr><td width="70px" style="font-family: 'Lucida Grande'; font-size: 8pt; color: gray; text-align: right; vertical-align: top; font-weight: bold; "><span>Subject:</span></td><td style="font-family: 'Lucida Grande'; font-size: 8pt; color: black; text-align: left; vertical-align: top; padding-left: 5px; "><span style="font-weight: bold; ">[Proj] RE: Proj Digest, Vol 50, Issue 19</span></td></tr><tr><td width="70px" style="font-family: 'Lucida Grande'; font-size: 8pt; color: gray; text-align: right; vertical-align: top; font-weight: bold; "><span>Date:</span></td><td style="font-family: 'Lucida Grande'; font-size: 8pt; color: black; text-align: left; vertical-align: top; padding-left: 5px; "><span>July 19, 2008 1:43:22 PM PDT</span></td></tr><tr><td width="70px" style="font-family: 'Lucida Grande'; font-size: 8pt; color: gray; text-align: right; vertical-align: top; font-weight: bold; "><span>To:</span></td><td style="font-family: 'Lucida Grande'; font-size: 8pt; color: black; text-align: left; vertical-align: top; padding-left: 5px; "><span title="proj@lists.maptools.org">proj@lists.maptools.org</span></td></tr></tbody></table></div><div id="felix-mail-content-block" style="color: black; background-color: white; width: 100%; "><div style="font-family: monospace; color: black; background-color: white; font-size: 8pt; "><br><br><br>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.<br><br>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.<br><br>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.<br><br>Mike Ossipoff<br><br>_______________________________________________<br>Proj mailing list<br>Proj@lists.maptools.org<br>http://lists.maptools.org/mailman/listinfo/proj<br><br></div></div></div></span></blockquote><br><div><br></div><div class="aol_ad_footer" id="u0595F6D6BC5746D18524DCC2063A5389"><FONT style="color: black; font: normal 10pt ARIAL, SAN-SERIF;"><HR style="MARGIN-TOP: 10px">The Famous, the Infamous, the Lame - in your browser. <A title="http://toolbar.aol.com/tmz/download.html?NCID=aolcmp00050000000014" href="http://toolbar.aol.com/tmz/download.html?NCID=aolcmp00050000000014" target="_blank">Get the TMZ Toolbar Now</A>!</FONT></div></body></html>