<html><body name="Mail Message Editor"><div><br></div><div>In other words, a "linear" projection is a pseudocylindric projection with equally spaced parallels.</div><div><br></div><div>There is no limit to how many such projections could be created. They follow the form:</div><div><br></div><div>x = λ • f (φ)</div><div>y = k • φ</div><div><br></div><div>where λ is longitude and φ is latitude and k is some arbitrary constant and f is some arbitrary function of (only) latitude. On the sphere, then, scale along parallels (also the horizontal scale factor) would be</div><div><br></div><div>f (φ) / cos (φ)</div><div><br></div><div>and scale along meridians would be</div><div><br></div><div>√(λ² • [∂f/∂φ]² + k²)</div><div><br></div><div>with a vertical scale factor of k.</div><div><br></div><div>Regards,</div><div>-- daan Strebe<br></div><div><br></div><br>On Jun 27, 2008, at 12:48:08 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] (no subject)</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>June 27, 2008 12:48:08 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>List members--<span class="Apple-converted-space"> </span><br><br>Last year I defined linearity, as a map property, here, and argued for the desirability of that property on data maps, maps that show d<br>distributions of such things as temperature, species habitat range, etc. I'd like to improve a little on the definitions that I posted at that time. (by X and Y position, I mean the values of the X and Y map co-ordinates)<span class="Apple-converted-space"> </span><br><br>Linearity:<span class="Apple-converted-space"> </span><br><br>A map projection is linear if, on maps using that projection, Y position on the map varies linearly with latitude, and X position on the map varies linearly with longitude.<span class="Apple-converted-space"> </span><br><br>[end of linearity definition]<span class="Apple-converted-space"> </span><br><br>Zonal linearity:<span class="Apple-converted-space"> </span><br><br>A map projection is zonally linear if maps using that projection can be divided into regions such that, within each region, Y position varies linearly with latitude, and X position varies linearly with longitude.<br><br>[end of zonal linearlity definition]<span class="Apple-converted-space"> </span><br><br>I've given linearity the simpler and briefer definition above because all the zonally linear projections, other than my own linear approxmiate (LA) projections, have that briefly defined property that I've defined above as "linearity". So, the only zonally linear maps used anywhere are linear as defined above. My LA projections are the only projections I've heard of that are zonally linear without being linear, by the above definitions.<span class="Apple-converted-space"> </span><br><br>In previous postings, I defined an even wordier kind of linearlity, which I'll give a name here, though I don't think this property really needs naming, because I know of no projection that has it without having the zonal linearity that I've defined above. It's alright for its name to be awkward, since I don't expect to say it often:<span class="Apple-converted-space"> </span><br><br>Especially-broadly-defined linearity:<span class="Apple-converted-space"> </span><br><br>A map is especially-broadly-defined-linear if it can be divided into regions such that within each region, Y position varies linearly with latitude, and if it can also be divided into regions such that X position varies linearly with longitude.<span class="Apple-converted-space"> </span><br><br>[end of especially-broadly-defined linearity definition]<span class="Apple-converted-space"> </span><br><br>As I said, that property has little value since I know of no projecion possessing it without also possessing zonal linearity as defined above.<span class="Apple-converted-space"> </span><br><br>Well, one reason to speak of especially-broadly-defined linearity is to mention that it aparently amounts to the same thing as the linearly interpolable positions property (LIPP) that I defined last year. Since I'm not changing my definition of LIPP, there's no need to repeat it here.<br><br>Though not linear, the equidistant conic, Bonne, and Stabius-Werner projections obviously are more convenient for determining and finding lat/lon co-ordinates than are other non-linear projections. They deserve credit for their position-conveniene, and it deserves a name.<br><br>Polar linearity:<br><br>A map projection is polar-linear if the parallels are concentric circles, and if distance along the each parallel varies linearly with longitude, and if distance along lines passing through the center of the parallels' common center varies linearly with latitude.<span class="Apple-converted-space"> </span><br><br>[end of polar linearity definition]<span class="Apple-converted-space"> </span><br><br>Of course the equidistant conic, Bonne, and Stabius-Werner are polar-linear.<span class="Apple-converted-space"> </span><br><br>Obviously polar linearity isn't as good as linearity, but it's better than what most non-linear prjecions offer for convenience in finding or determining lat/lon co-ordinates.<br><br>Some linear projections:<span class="Apple-converted-space"> </span><br><br>Cylindrical Equidistant<span class="Apple-converted-space"> </span><br>Sinusoidal<span class="Apple-converted-space"> </span><br>Trapezoidal<span class="Apple-converted-space"> </span><br>Eckhert III (if I remember it correctly)<span class="Apple-converted-space"> </span><br>Apianus II<span class="Apple-converted-space"> </span><br><br>My LA projections, while not being linear as defined here, are zonally linear.<span class="Apple-converted-space"> </span><br><br>Mike Ossipoff<span class="Apple-converted-space"> </span><br><br><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="u2EDA023BA114495DADF6E894E8D04E40"><FONT style="color: black; font: normal 10pt ARIAL, SAN-SERIF;"><HR style="MARGIN-TOP: 10px"><A title="http://toolbar.aol.com/moviefone/download.html?ncid=aolcmp00050000000011" href="http://toolbar.aol.com/moviefone/download.html?ncid=aolcmp00050000000011" target="_blank">Get the Moviefone Toolbar</A>. 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