<HTML><FONT FACE=arial,helvetica><HTML><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2"><BR>
Michael,<BR>
<BR>
Your "equidistant elliptical" is Apianus II. Your "graduated equidistant" doesn't really have a name; it's just the equirectangular projection with standard parallel set to need.<BR>
<BR>
You wrote:<BR>
<BR>
<BLOCKQUOTE CITE STYLE="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px" TYPE="CITE"></FONT><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2">Though I’ve already replied to “Your analysis of the issues surrounding your<BR>
thesis seems to evolve as we talk”, and “You’ve now evolved to an<BR>
interrupted sinusoidal”, I want to mention those false statements again, as<BR>
examples of your overall tendency toward falsity in these postings. You said<BR>
that I’d “evolved to” the sinusoidal, though I’d been suggesting it from my<BR>
first posting here, for when equal-area is desired. My position has been<BR>
consistent in my postings here.<BR>
</BLOCKQUOTE></FONT><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2"><BR>
Balderdash. You made no mention of the utility of equal-area in your original posting, and this is your only mention of sinusoidal:<BR>
<BR>
<BLOCKQUOTE CITE STYLE="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px" TYPE="CITE"></FONT><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2">So, for that person, how about an equidistant elliptical projection. It<BR>
would be a compromise between two extreme equidistant projections--the<BR>
cylindrical equidistant and the sinusoidal. </FONT><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2"><BR>
</BLOCKQUOTE></FONT><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2"><BR>
This mention comes completely without any recommendation for the sinusoidal, and of course there is no sign at all of interruptions. I've copied your original posting below in its entirety, since you seem to have lost it and forgotten its content. Hence I'm going to ignore your aspersions of "falsities"; they carry no credibility. Of course I am quite done with trying to engage you in any conversation.<BR>
<BR>
</FONT><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2">Regards,<BR>
-- daan Strebe<BR>
<BR>
</FONT><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2"><BR>
<BLOCKQUOTE CITE STYLE="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px" TYPE="CITE"></FONT><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2">This list’s main topic is solutions for co-ordinate transformations. But<BR>
this posting isn’t off-topic, because it’s about co-ordinate transformations<BR>
too: I claim that, for data maps in atlases, the transformation from the<BR>
map’s X & Y co-ordinates to the latitude/longitude co-ordinates should be<BR>
easy and convenient for everyone, and I suggest a solution that I haven’t<BR>
read mentioned anywhere.<BR>
<BR>
But I’d also appreciate it if someone could tell me of a general map<BR>
projections discussion list, or any Internet mailing list, newsgroup or<BR>
other forum where this posting would be more appropriate.<BR>
<BR>
When atlases have data maps showing such things as climate, vegetation and<BR>
population, it could be of some interest to the reader to find out exactly<BR>
where the map is saying the boundaries of the zones are. With nearly all<BR>
data maps, determining the latitude and longitude corresponding to a point<BR>
on the map involves calculation. Regrettably that can be an inconvenient<BR>
amount of calculation, as is the case with, for instance, the Lambert<BR>
azimuthal equal area map--a common projection for data maps.<BR>
<BR>
For some projections, such as Mercator, Miller, and the pseudocylilndricals,<BR>
the calculation is more feasible, because it's only necessary to calculate<BR>
the latitude from the point's Y co-ordinate, the point’s distance from the<BR>
map's equator. But most atlas users would like to get an accurate<BR>
measurement without even having to do that.<BR>
.<BR>
So it's obvious that a data map should use an equidistant projection, a<BR>
projection in which the latitude and longitude vary linearly with distance<BR>
north or east on the map. Then anyone could easily determine the<BR>
geographical co-ordinates of a point on the map.<BR>
<BR>
Some would answer that the data map zone boundaries are approximate anyway.<BR>
But it would still be better to not add position-guessing error.<BR>
<BR>
Of course the measurements would be even easier if the projection is<BR>
cylindrical, with meridians and parallels straight and perpendicular to each<BR>
other.<BR>
<BR>
So I suggest that the best projection for data maps would be the cylindrical<BR>
equidistant. Below I’ll suggest an improvement on that projection.<BR>
<BR>
Because a cylindrical projection is centered on the equator, while a conic<BR>
is centered on a parallel in the mapped region, the use of a cylindrical<BR>
projection would mean more distortion, but that's an acceptable price for a<BR>
data map on which positions are more easily measured.<BR>
<BR>
Of course if one needed the low-distortion advantages of a conic, such as<BR>
easy and relatively accurate distance measurements, then one might want to<BR>
use a conic equidistant, as a compromise between easy position-measurement<BR>
and low-distortion advantages. But I suggest that the cylindrical is usually<BR>
better, because surely the positions of the zone boundaries are the most<BR>
important information in a data map, and their ease of measurement is<BR>
all-important. With a conic, longitude measurement isn't as easy and<BR>
accurate as with a cylindrical.<BR>
<BR>
What I'm saying is intended to apply to all data maps, whether showing the<BR>
world, a continent, a country, or a state or province.<BR>
<BR>
But of course a cylindrical equidistant world map shape-distorts at some<BR>
latitudes, usually the near-polar latitudes. And the radically different<BR>
scales in the two dimensions can complicate distance-measurement. But<BR>
there's no reason why the north-south scale has to be uniform over the whole<BR>
range of latitudes: Why not specify “dividing parallels” (say, every 10<BR>
degrees, for instance), and have the distance from the map's equator vary<BR>
linearly with latitude, but at a different scale, between each pair of<BR>
dividing parallels. So, using the 10 degree example, the north-south scale<BR>
between 0 and 10 degrees latitude would be equal to the geometric mean of<BR>
the east-west scales at 0 and 10 degrees latitude. And the north-south<BR>
scale between 10 and 20 degrees latitude would be equal to the geometric<BR>
mean of the east-west scales at 10 and 20 degrees latitude…and so on for<BR>
each 10 degrees of latitude.<BR>
<BR>
That's what I'm calling a “graduated equidistant projection“. Determining<BR>
geographic co-ordinates from map position would be as easy as with an<BR>
ordinary equidistant projection, but shapes and directions wouldn't be<BR>
visibly distorted, and scales would be more nearly the same in all<BR>
directions. The map would have, to some degree, the advantages of a<BR>
conformal projection, while retaining the easy position measurement of an<BR>
equidistant projection.<BR>
<BR>
I propose the use of the graduated equidistant cylindrical map for all data<BR>
maps. But, if one desires the low-distortion advantages of a conic, then I’d<BR>
propose the graduated equidistant conic for data maps of ontinents,<BR>
countries, states and provinces, with the cylindrical used only for world<BR>
maps. As I said, I consider easy position measurement to be the most<BR>
important property of a data map, which is why I’d use the cylindrical for<BR>
all data maps.<BR>
<BR>
Why not use recommend Bonne for continents? With curved parallels and<BR>
meridians, accurate determination of longitude would be especially<BR>
inconvenient. And Bonne has more scale variation than conic--and distances<BR>
are probably the most often-measured quantity on maps.<BR>
<BR>
I don't necessarily claim to be the first advocate of graduated equidistant<BR>
projections, but I've never found one in an atlas, or anywhere else. And<BR>
I've never read any mention of them.<BR>
<BR>
When the only calculation needed for position-measurement is that of<BR>
calculating the latitude based on the distance from the map’s equator, as in<BR>
the case of the Mercator or a pseudocylindrical map, some would be willing<BR>
to do that calculation.. For many purposes the advantages of conformality<BR>
would be desirable--uniform scale in every direction at any particular<BR>
point, and more accurate shapes and directions. So, for someone who doesn’t<BR>
mind calculating latitude from the Y co-ordinate, the Mercator might be a<BR>
better choice than the graduated cylindrical equidistant. Of course the<BR>
latter projection can be regarded as a very rough beginning of an<BR>
approximation to the Mercator.<BR>
<BR>
Likewise, someone who is willing to calculate latitude from Y co-ordinate<BR>
might prefer the conformal conic to the graduated equidistant conic, for<BR>
the same reason. Or maybe not, because that calculation involves more work<BR>
with the conformal conic than with the Mercator.<BR>
<BR>
Of course, even someone willing to calculate latitude from Y co-ordinate<BR>
might often prefer the convenience of a graduated equidistant projection,<BR>
with which very little calculation is needed.<BR>
<BR>
To save space on the page, one could use a somewhat smaller north-south<BR>
scale in one or more dividing-parallel sections in the extreme north and<BR>
south parts of the map. Those regions aren’t where most people would<BR>
usually need to measure distance and direction anyway. Of course one could<BR>
do that with the Mercator too, using the Mercator for all latitudes except<BR>
those where the expansion seriously uses up page-space. There, the Mercator<BR>
would be replaced by cylindrical equidistant or graduated cylindrical<BR>
equidistant, or maybe a grafting of the Miller there. I’d prefer those<BR>
combinations to the ordinary Miller projection because it would keep<BR>
Mercator’s properties in the most important parts of the map.<BR>
<BR>
Someone might want to combine the easy position measurements of an<BR>
equidistant with the beauty and round appearance of an world elliptical map.<BR>
So, for that person, how about an equidistant elliptical projection. It<BR>
would be a compromise between two extreme equidistant projections--the<BR>
cylindrical equidistant and the sinusoidal. Parallels are spaced equally and<BR>
each is divided uniformly. And the parallels’ lengths are determined by the<BR>
map’s elliptical shape. It would resemble Mollweide, but with equidistant<BR>
parallels. Longitude measurement, with the curved meridians, wouldn’t be as<BR>
easy as it would be with a cylindrical projection. For that reason, speaking<BR>
for myself, I’d prefer the graduated equidistant cylindrical for data maps.<BR>
<BR>
Michael Ossipoff</FONT><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2"><BR>
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