[Proj] Re: Graduated equidistant projections for convenient
mikeo2106 at msn.com
Mon Aug 6 16:33:01 EDT 2007
Your jargon is idiosyncratic
Ive 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 Ive seen. Apparently I guessed wrong about equidistant. If so,
then I dont know what I should call the projection that Ive been calling
equidistant elliptical. Maybe equally-spaced elliptical?
As for graduated equidistant projections, Im the first to admit that I
dont know what ica would call it, or whether or not it was named before I
called it graduated equidistant.
and your analysis of the issues surrounding your thesis seem to evolve as we
Not sure how thats so. From the start, graduated equidistant cylindrical
was my favorite, and all along Ive 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 Ive consistently emphasized. Nor have I changed my justifications
for those suggestions. All along Ive 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 theyd at least fully specify the projection (its
center and orientation). That wasnt a change in my suggestions, but merely
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
I admit that guessed wrong the official meaning of equidistant. As I said,
Ive 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
Equal-area is a rigorous concept. "Giving accurate and easy
directly-measured lat/long coordinates" is not, and it is not a "metric
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, Id better not call
that a metric criterion, but I do call it a precisely-expressible
You have performed no rigorous studies of this property of the sinusoidal
Now that Ive precisely defined the property, its obvious that it is
possessed by the sinusoidal, the equidistant cylindrical, and the graduated
I could easily claim some other projection is better than sinusoidal for
"accurate and easy directly-measured lat/long coordinates"
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 doesnt have the property I defined above, then
Im sure I wouldnt object to it in nature guidebook range maps and spatial
distribution maps in atlases.
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 dont 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 Ive been saying, Im 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
It's not clear to me how often the cartographer shares your priorities.
Not very often :^)
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.
But isnt 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?
If the only purpose for the map truly is to easily determine lat/long
positions, you're best off with a plate carrÃ©e, 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.
Quite so. Id 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
You started out claiming a "graduated equidistant" is the answer.
And Ive never stopped saying that its 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
If data maps were made only for me, then sure, Id specify graduated
equidistant cylindrical. I havent wavered from that position during these
That's already a compromise away from the "easiest" determination of
True. If youre interpolating latitudes in several latitude zones of a
graduated equidistant cylindrical, youd have to measure the north-south
width of each zone youre 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.
You've now evolved to an "interrupted sinusoidal".
I didnt 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 sinusoidals faults, for a world map.
There was no evolution over time of what I was saying about graduated
equidistant cylindrical, sinusoidal, and interrupted sinusoidal.
Many projections could be argued to provide similar compromises, or a
"better" balance of those compromises [better for whom?].
For guidebook and atlas data maps, in addition to my favorite, the graduated
equidistant cylindrical, Ive 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 dont possess the linearly interpolable positions
property, so I dont 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.
Id be interested in other compromises that other people might propose.
And, most glaringly, how many interruptions are there, and where are they?
My favorite interrupted sinusoidal projection is that of the USGS. Shapes on
it look perfectly good enough.
The more interruptions, the greater the accuracy... but the less usable for
any purpose that spans an interruption.
Well, youd have to pick up a route on the other side of the interruption,
but it wouldnt be difficult to find it, due to the sinusoidals
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.
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
C) Ground cover maps serve several functions. The most important is an easy
determination of geographic coordinate, but allowances must be made for
D) Something else.
Id say D, though A, B, and C are close. Id say that A is a true statement,
if we replace must with should, but it doesnt 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 doesnt have that
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) its 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
4. For other kinds of mapped data, where equal-area isnt 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
[end of data map claims]
If you know anyone who makes decisions about the data maps Im 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 projections center (in map co-ordinates and lat/long
coordinates) and the maps orientation about that center.
Commenting on theses A, B, and C.
A is true because equal area probably isnt demanded for all kinds of mapped
I dont 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 Ive
here called data maps, then it is outside the scope of my claims (though
Id 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 dont think that is necessary for any data
map. All of that is covered in my statement of my claims, above.
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