[Proj] Graduated equidistant projections for convenient co-ordinate transformations

[Michael Ossipoff]

This list’s main topic is solutions for co-ordinate transformations. But 
this posting isn’t off-topic, because it’s about co-ordinate transformations 
too: I claim that, for data maps in atlases, the transformation from the 
map’s X & Y co-ordinates to the latitude/longitude co-ordinates should be 
easy and convenient for everyone, and I suggest a solution that I haven’t 
read mentioned anywhere.

But I’d also appreciate it if someone could tell me of a general map 
projections discussion list, or any Internet mailing list, newsgroup or 
other forum where this posting would be more appropriate.

When atlases have data maps showing such things as climate, vegetation and 
population, it could be of some interest to the reader to find out exactly 
where the map is saying the boundaries of the zones are. With nearly all 
data maps, determining the latitude and longitude corresponding to a point 
on the map involves calculation. Regrettably that can be an inconvenient 
amount of calculation, as is the case with, for instance, the Lambert 
azimuthal equal area map--a common projection for data maps.

For some projections, such as Mercator, Miller, and the pseudocylilndricals, 
  the calculation is more feasible, because it's only necessary to calculate 
the latitude from the point's Y co-ordinate, the point’s distance from the 
map's equator. But most atlas users would like to get an accurate 
measurement without even having to do that.
.
So it's obvious that a data map should use an equidistant projection, a 
projection in which the latitude and longitude vary linearly with distance 
north or east on the map. Then anyone could easily determine the 
geographical co-ordinates of a point on the map.

Some would answer that the data map zone boundaries are approximate anyway. 
But it would still be better to not add position-guessing error.

Of course the measurements would be even easier if the projection is 
cylindrical, with meridians and parallels straight and perpendicular to each 
other.

So I suggest that the best projection for data maps would be the cylindrical 
equidistant.  Below I’ll suggest an improvement on that projection.

Because a cylindrical projection is centered on the equator, while  a conic 
is centered on a parallel in the mapped region, the use of a cylindrical 
projection would mean more distortion, but that's an acceptable price for a 
data map on which positions are more easily measured.

Of course if one needed the low-distortion advantages of a conic, such as 
easy and relatively accurate distance measurements, then one might want to 
use a conic equidistant, as a compromise between easy position-measurement 
and low-distortion advantages. But I suggest that the cylindrical is usually 
better, because surely the positions of the zone boundaries are the most 
important information in a data map, and their ease of measurement is 
all-important. With a conic, longitude measurement isn't as easy and 
accurate as with a cylindrical.

What I'm saying is intended to apply to all data maps, whether showing the 
world, a continent, a country, or a state or province.

But of course a cylindrical equidistant world map shape-distorts at some 
latitudes, usually the near-polar latitudes. And the radically different 
scales in the two dimensions can complicate distance-measurement. But 
there's no reason why the north-south scale has to be uniform over the whole 
range of latitudes: Why not specify “dividing parallels” (say, every 10 
degrees, for instance), and have the distance from the map's equator vary 
linearly with latitude, but at a different scale, between each pair of 
dividing parallels. So, using the 10 degree example, the north-south scale 
between 0 and 10 degrees latitude would be equal to the geometric mean of 
the east-west scales at 0 and 10 degrees latitude.  And the north-south 
scale between 10 and 20 degrees latitude would be equal to the geometric 
mean of the east-west scales at  10 and 20 degrees latitude…and so on for 
each 10 degrees of latitude.

That's what I'm calling a “graduated equidistant projection“. Determining 
geographic co-ordinates from map position would be as easy as with an 
ordinary equidistant projection, but shapes and directions wouldn't be 
visibly distorted, and scales would be more nearly the same in all 
directions. The map would have, to some degree, the advantages of a 
conformal projection, while retaining the easy position measurement of an 
equidistant projection.

I propose the use of the graduated equidistant cylindrical map for all data 
maps. But, if one desires the low-distortion advantages of a conic, then I’d 
propose the graduated equidistant conic for data maps of ontinents, 
countries, states and provinces, with the cylindrical used only for world 
maps. As I said, I consider easy position measurement to be the most 
important property of a data map, which is why I’d use the cylindrical for 
all data maps.

Why not use recommend Bonne for continents? With curved parallels and 
meridians, accurate determination of longitude would be especially 
inconvenient. And Bonne has more scale variation than conic--and distances 
are probably the most often-measured quantity on maps.

I don't necessarily claim to be the first advocate of graduated equidistant 
projections, but I've never found one in an atlas, or anywhere else. And 
I've never read any mention of them.

When the only calculation needed for position-measurement is that of 
calculating the latitude based on the distance from the map’s equator, as in 
the case of the Mercator or a pseudocylindrical map, some would be willing 
to do that calculation.. For many purposes the advantages of conformality 
would be desirable--uniform scale in every direction at any particular 
point, and more accurate shapes and directions. So, for someone who doesn’t 
mind calculating latitude from the Y co-ordinate, the Mercator might be a 
better choice than the graduated cylindrical equidistant. Of course the 
latter projection can be regarded as a very rough beginning of an 
approximation to the Mercator.

Likewise, someone who is willing to calculate latitude from Y co-ordinate 
might prefer the conformal conic to  the graduated equidistant conic, for 
the same reason. Or maybe not, because that calculation involves more work 
with the conformal conic than with the Mercator.

Of course, even someone willing to calculate latitude from Y co-ordinate 
might often prefer the convenience of a graduated equidistant projection, 
with which very little calculation is needed.

To save space on the page, one could use a somewhat smaller north-south 
scale in one or more dividing-parallel sections in the extreme north and 
south parts of the map.  Those regions aren’t where most people would 
usually need to measure distance and direction anyway. Of course one could 
do that with the Mercator too, using the Mercator for all latitudes except 
those where the expansion seriously uses up page-space. There, the Mercator 
would be replaced by cylindrical equidistant or graduated cylindrical 
equidistant, or maybe a grafting of the Miller there. I’d prefer those 
combinations to the ordinary Miller projection because it would keep 
Mercator’s properties in the most important parts of the map.

Someone might want to combine the easy position measurements of  an 
equidistant with the beauty and round appearance of an world elliptical map. 
So, for that person, how about an equidistant elliptical projection. It 
would be a compromise between two extreme equidistant projections--the 
cylindrical equidistant and the sinusoidal. Parallels are spaced equally and 
each is divided uniformly. And  the parallels’ lengths are determined by the 
map’s elliptical shape. It would resemble Mollweide, but with equidistant 
parallels. Longitude measurement, with the curved meridians, wouldn’t be as 
easy as it would be with a cylindrical projection. For that reason, speaking 
for myself, I’d prefer the graduated equidistant cylindrical  for data maps.

Michael Ossipoff