<HTML><FONT FACE=arial,helvetica><HTML><FONT COLOR="#000000" FACE="Palatino" FAMILY="SERIF" SIZE="2"><BR>
Michael,<BR>
<BR>
Easily converting between map coordinates and spherical coordinates is one reason to choose a projection. When a mapmaker decides that's the most important reason, the mapmaker does just what you suggest. Generally the mapmaker decides other factors are more important.<BR>
<BR>
Regards,<BR>
-- daan Strebe<BR>
<BR>
<BR>
In a message dated 7/30/07 22:29:36, mikeo2106@msn.com writes:<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">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<BR>
<BR>
<BR>
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