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

[Michael Ossipoff]

Daan--

You wrote:

Your jargon is idiosyncratic

I reply:

I’ve unsuccessfully tried to find, on the Internet, an ica glossary of map 
projection terms. I try to use terms consistently with the usage in various 
books that I’ve seen. Apparently I guessed wrong about “equidistant”. If so, 
then I don’t know what I should call the projection that I’ve been calling 
“equidistant elliptical”. Maybe “equally-spaced elliptical”?

As for “graduated equidistant” projections, I’m the first to admit that I 
don’t know what ica would call it, or whether or not it was named before I 
called it “graduated equidistant”.

You continued:

and your analysis of the issues surrounding your thesis seem to evolve as we 
talk.

I reply:

Not sure how that’s so. From the start, graduated equidistant cylindrical 
was my favorite, and all along I’ve been suggesting sinusoidal as a 
compromise between those wanting easy positions and those wanting 
equal-area. I mentioned a few other compromises too, but the above two are 
the ones I’ve consistently emphasized. Nor have I changed my justifications 
for those suggestions. All along I’ve claimed that one of those two 
projections would be the best for range maps in nature guidebooks and 
spatial distribution maps in atlases.

Later I added that, if a publisher must use azimuthal equal area as a data 
map, it would be nice if they’d at least fully specify the projection (its 
center and orientation). That wasn’t a change in my suggestions, but merely 
an addition.

You continued:

These make it hard for me to engage you in conversation. When a person 
wishes to announce that the world ought to be doing things differently, then 
it behooves the person to exercise due diligence. That includes a rigorous 
analysis and vocabulary of the domain. Without both, conversation gets 
distracted by the parties involved just trying to figure out what the other 
is saying.

I reply:

I admit that guessed wrong the official meaning of “equidistant”. As I said, 
I’ve tried to find an ica glossary of projection terms. If there are other 
incorrect usages of mine that are making me difficult to understand,  they 
aren’t intentional.

You wrote:

Equal-area is a rigorous concept. "Giving accurate and easy 
directly-measured lat/long coordinates" is not, and it is not a "metric 
criterion".

I reply:

Ok then, let me say it more precisely:

The “linearly-interpolable positions” property is possessed by a projection 
if, using a map precisely constructed on that projection, it is possible to 
determine, with accuracy limited only by measuring error, the latitude and 
longitude of a point on the map by using only measurement, with a ruler, of 
distance on the map,  and linear interpolation (and no other calculation)

[end of linearly-interpolable positions property definition]
.
The equidistant cylindrical, the graduated equidistant cylindrical, and the 
sinusoidal all have that property.

Not knowing exactly what is meant by a metric criterion, I’d better not call 
that a metric criterion, but I do call it a precisely-expressible 
measurement property.

You continued:

You have performed no rigorous studies of this property of the sinusoidal

I reply:

Now that I’ve precisely defined the property, it’s obvious that it is 
possessed by the sinusoidal, the equidistant cylindrical, and the graduated 
equidistant cylindrical.

You wrote:

I could easily claim some other projection is better than sinusoidal for 
"accurate and easy directly-measured lat/long coordinates"

I reply:

So could I: The equidistant cylindrical and the graduated equidistant 
cylindrical. The measurement is even easier on those projections because 
they have straight meridians. As I said, I suggest the sinusoidal as a 
compromise to those who want equal area in a data map.

But I welcome any other suggestions for projections with that property. Or, 
if you know of a projection on which latitude and longitude are reasonably 
easy, even if the projection doesn’t have the property I defined above, then 
I’m sure I wouldn’t object to it in nature guidebook range maps and spatial 
distribution maps in atlases.

You continued:

So. How ought I engage your assertion in (2)? This assertion is peripheral 
to your thesis, so, to me, it's whack-a-mole digression.

I reply:

I don’t consider it peripheral or a digression, because I claim that the 
sinusoidal is the best choice for maps whose purpose is to show information 
(such as forest type and land use) for which many people want equal area, 
but for which many others will want accurate linearly-interpolable lat/long 
co-ordinates. And I emphasize that, in everything I’ve been saying, I’m only 
talking about nature guidebook range maps and atlas spatial distribution 
maps. The data maps in those two kinds of books sold to the general public 
in bookstores.

You wrote:

It's not clear to me how often the cartographer shares your priorities.

I reply:

Not very often :^)

You continued:

You may want to think of a map as one that fits your notion of a "data map", 
but people may be using it for many other purposes as well.

I reply:

But isn’t it the natural presumption that the purpose of a forest-type 
spatial distribution map in an atlas is to show where the forest-regions 
are, and maybe the relative areas of those regions?

You wrote:

If the only purpose for the map truly is to easily determine lat/long 
positions, you're best off with a plate carrée, regardless of how distorted 
it gets as you move away from the equator. As soon as you start making 
concessions to other uses, well, suddenly the waters muddy a lot. There 
aren't any easy answers when you compromise because suddenly everyone 
interested in the map wants a different balance of compromises.

I reply:

Quite so. I’d like to say that linearly interpolable position is all that 
matters, but I realize that, for some kinds of mapped data, many people want 
equal area. But the sinusoidal clearly offers both, and so the choice is 
clear.

You wrote:

You started out claiming a "graduated equidistant" is the answer.

I reply:

And I’ve never stopped saying that it’s the answer that I like best. But, 
from my first posting here, I acknowledged that some might want equal area 
for some kinds of mapped data, and that that would make the sinusoidal the 
best choice.

If data maps were made only for me, then sure, I’d specify graduated 
equidistant cylindrical. I  haven’t wavered from that position during these 
postings.

You wrote:

That's already a compromise away from the "easiest" determination of 
lat/long.

I reply:

True. If you’re interpolating latitudes in several latitude zones of a 
graduated equidistant cylindrical, you’d have to measure the north-south 
width of each zone you’re interpolating in, whereas, with the ordinary 
equidistant cylindrical, fewer measurements would be needed. But I felt that 
that compromise is justified to give better shapes on a world map, and near 
conformality on maps of smaller areas.

But I certainly have no objection to the ordinary equidistant cylindrical as 
a data map.

I reply:

You've now evolved to an "interrupted sinusoidal".

I reply:

I didn’t “evolve to” the sinusoidal. I suggested it from the start, in my 
first posting here, as the best choice for data maps if people want equal 
area. And of course I have been acknowledging that many people do want equal 
area for certain kinds of mapped data.

As for “interrupted sinusoidal”, as I said from the start, it greatly 
reduces the sinusoidal’s faults, for a world map.

There was no evolution over time of what I was saying about graduated 
equidistant cylindrical, sinusoidal, and interrupted sinusoidal.

You continued:

Many projections could be argued to provide similar compromises, or a 
"better" balance of those compromises [better for whom?].

I reply:

For guidebook and atlas data maps, in addition to my favorite, the graduated 
equidistant cylindrical, I’ve suggested a few compromises:

1. The sinusoidal, when many people want equal area (which is almost surely 
the case for such things as forest-type and land-use).

2. The graduated equidistant conic, for those who want the low  distortion 
of a locally-centered map, but with reasonably easy lat/long determinations. 
Conic projections don’t possess the linearly interpolable positions 
property, so I don’t like them as much for data maps.

I like the sinusoidal better, because I want linearly interpolable position 
more than I want low distortion, for data maps.

I’d be interested in other compromises that other people might propose.
.
You wrote:

And, most glaringly, how many interruptions are there, and where are they?

I reply:

My favorite interrupted sinusoidal projection is that of the USGS. Shapes on 
it look perfectly good enough.

You continued:

The more interruptions, the greater the accuracy... but the less usable for 
any purpose that spans an interruption.

I reply:

Well, you’d have to pick up a route on the other side of the interruption, 
but it wouldn’t be difficult to find it, due to the sinusoidal’s 
equally-spaced parallels.



You wrote:

What, then, is your thesis?

A) There is a class of maps whose primary function is to allow people to 
easily determine geographic coordinate. Therefore the projection must be 
suited to that function alone.

or

B) The primary use of ground cover maps is an easy determination of 
geographic coordinates of the extents of the ground cover. Therefore these 
maps should be drawn on the projection that makes that determination 
easiest.

or

C) Ground cover maps serve several functions. The most important is an easy 
determination of geographic coordinate, but allowances must be made for 
other uses.

or

D) Something else.

I reply:

I’d say D, though A, B, and C are close. I’d say that A is a true statement, 
if we replace “must” with “should”, but it doesn’t completely say what I 
want to say. So D is my answer.

Here is what I claim:

[The below refers only to species range maps in nature guide-books, and to 
spatial distribution maps in atlases. These are two kinds of books sold to 
the general public in bookstores and found in public libraries. These maps 
are what I mean whenever, below, I say “data maps”. None of this applies to 
maps used only by or intended only for professionals.]

1. All data maps should make it as easy as possible to determine lat/long 
co-ordinates of points on the map. Specifically what I mean by that is that 
data maps should have the linearly-interpolable positions property, except 
where a demand for low distortion justifies using the equidistant conic or 
graduated equidistant conic instead, even though it doesn’t have that 
property.

2. But I claim that the linearly-interpolable positions property and equal 
area are the only properties that can be needed on a data map. So, because 
(with the sinusoidal) it’s possible to have the linearly-interpolable 
positions property, and equal area too, claim that actually all data maps 
should have the linearly-interpolable positions property.

3. For some kinds of mapped data, such  forest-type and land-use, where many 
people want equal-area, the sinusoidal projection is the best choice, 
because it has the linearly-interpolable position property and is an 
equal-area map.

4. For other kinds of mapped data, where equal-area isn’t demanded, the 
cylindrical equidistant or graduated cylindrical equidistant is the best 
choice, because of its maximally easy lat/long determinations.

5. Obviously, when the data map is a world map, and the sinusoidal is used 
(due to desire for equal area) the map should be interrupted, for lower 
distortion. The USGS interrupted sinusoidal is a good projection of that 
type.

[end of data map claims]

If you know anyone who makes decisions about the data maps I’m talking 
about, would you pass these claims on to them?

Or, at least, if they are using the azimuthal equal area map for a data map, 
ask them to at least state (somewhere in the book containing that map) the 
position of the projection’s center (in map co-ordinates and lat/long 
coordinates) and the map’s orientation about that center.

Commenting on theses A, B, and C.

A is true because equal area probably isn’t demanded for all kinds of mapped 
data.

I don’t know about B, because some people might want equal area. And if B 
includes maps used only by professionals, and not just the ones that I’ve 
here called “data maps”, then it is outside the scope of my claims (though 
I’d expect that easy positions, or easy positions and equal area, would be 
the important thing even for professional users of groundcover maps).

As for C, maybe, for some people, equal area is more important than easy 
positions. Certainly allowance must be made for those who want equal area. 
Or, if necessary, even for those who want the low distortion of a 
locally-centered map, though I don’t think that is necessary for any data 
map. All of that is covered in my statement of my claims, above.




Michael Ossipoff