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

[Strebe at aol.com]

Michael,

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.

Regards,
-- daan Strebe


In a message dated 7/30/07 22:29:36, mikeo2106 at msn.com writes:

> 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
> 
> 
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