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

Thu Aug 2 17:25:01 PDT 2007

Daan--

You wrote:

It's very uncommon for a map projection to satisfy two metric criteria 
simultaneously.

I reply:

Yes, and that’s why the sinusoidal projection is so remarkable. It’s 
equidistant and it’s equal-area.
And not just equidistant. Not only are the parallels truly-spaced and 
uniformly divided, but they’re all truly divided.

We’ve discussed two desiderata for a data map: I’ve mentioned the 
desirability of feasibly, hopefully easily,  finding out if a particular 
place is within a certain zone on the map. What good is a data map if you 
can’t determine that? You mentioned that people expect a data map to be 
equal-area.  Ok, the sinusoidal has both of those desired properties.

As the price for those properties, the sinusoidal doesn’t do well by scale 
variation, distances, and directions (unless it’s a world map, because 
interruption is accepted for world maps).

But how fair is it to judge a data map according to how well it works as 
navigational map? It isn’t a navigational map. It’s a special purpose map 
for showing _where_ certain spatial distribution zones are.

For example, the gnomonic projection too is a special purpose projection, 
for showing great circles as straight lines. No one criticizes the gnomonic 
for not being a good source of accurate distances, directions or areas. They 
don’t, because they know that that is not what it’s for. Likewise for a 
spatial distribution map.

The hiker isn’t going to depend on an atlas rainfall-distribution map or a 
species range map in a nature guidebook to determine the distances and 
directions that s/he needs. How far is the rare salamander habitat from the 
trailhead, and what heading should the compass be set for? S/he’ll find that 
out from another map, probably a USGS topographic map.

Does a botanist or wildlife biologist want to know how wide a habitat range 
is, or its distance from the nearest river or lake? For one thing, I 
question whether s/he’ll need that more often than s/he’ll need to know the 
position of the habitat range’s boundaries. And, when s/he needs that, and 
has gotten some lat/long co-ordinates from an equidistant projection, then 
s/he can calculate the distance, or get it from a map better suited for that 
than is a little atlas or guidebook range map.

And does a botanist or wildlife biologist really need to get that 
information from a little distribution map in a public-consumption atlas or 
nature guidebook purchased in a bookstore?

It’s difficult to find a scenario in which someone needs accurate distances 
from an atlas spatial distribution map or a nature guidebook species range 
map. But what makes those maps unique is that they tell where those zones 
are, and I’m merely asking that they do so _usably_.

You wrote:

You certainly can't get the best distance measurements AND the equidistant 
property simultaneously.

I reply:

Well, for the most accurate distances, you can calculate them as accurately 
as you want to  from a conformal map. That’s more feasible with a conformal 
because, at any particular point, the scale is the same in ever direction.

But if you’re referring to distances measured directly from the map, without 
being corrected for scale in its part of the map, I don’t know that 
equidistant maps do worse. The sinusoidal, Bonne, and Stabius-Werner have 
more scale variation because they concentrate their distortion in the 
corners. And the sinusoidal more so, because it’s an equatorially-centered 
map, rather than a locally-centered map.

But, considering for instance the conic projections, the equidistant doesn’t 
do badly in comparison to the other conics. The equal area conic has more 
scale variation than the equidistant or the conformal. If the parallel with 
the largest east-west scale has 1.1 times the east-west scale of the 
parallel with the smallest east-west scale, then, to make up for that and 
maintain equal area, that first parallel must have 1.1 times _less_ 
north-south scale, in comparison to the second parallel. Resulting in a 
maximum scale variation factor of 1.21 instead of just 1.1   Equal area maps 
square the maximum scale variation factor. They do _worst_ by the standard 
of scale-variation. And equal area maps are the ones that cartographers like 
for data maps.

Equidistant conic has pretty much the same scale variation as conformal 
conic. But if the equidistant conic spaces its parallels truly, then its 
typical, average distances, measured according to its official scale, will 
be more accurate than on the conformal, because typically the distances on 
the equidistant conic will have some north-south component, and that’s the 
direction in which scale is true.

So, in that regard, equidistant looks best, and equal area looks worst.

And, as I said, after getting two points’ lat and long co-ordinates from an 
equidistant projection, you can calculate accurate great circle distances 
and directions. More accurate than you could measure directly from any map. 
On no map are distances always accurate. But they are when you calculate 
them from the accurate lat/long co-ordinates that you get from an 
equidistant map.

Of course, if distances, including route distances, were the important 
thing, then the conformal maps would, because route distances can be 
calculate on a conformal as accurately as you want to.

Anyway, it seems to me that the sinusoidal is what satisfies the person who 
is interested in the areas of the zones, and also the person who is 
interested in _where_ the zone boundaries are.

Sometimes it seems as if cartographers regard data maps as just general 
purpose maps on which zones have been drawn, and therefore want them to have 
the same properties that they look for in general purpose maps. Atlases 
often use, for data maps, the same projection that they use for their 
general purpose maps.

It seems to me that the sinusoidal is the projection that would please 
everyone, accurately and conveniently giving the two kinds of information 
that people actually need, want, and use on a data map.

I understand that, for instance, course-instructors might want spatial 
distribution maps that show their students how the total area of the Earth’s 
boreal forest compares to that of the tropical rainforest, etc.

You wrote:

You don't see more maps on the projections you describe because not many 
people share your priorities.

I reply:

I certainly can’t argue with that.

I guess sinusoidal isn’t going to become widely used for data maps, because 
cartographers apply general-purpose requirements to those special purpose 
maps.

I’m the first to admit that I’m the only person I’m aware of who has 
expressed the preferences that I’ve been expressing here.

So then what would be a more realistic request for me to make to the 
publishers of atlases and nature guidebooks? I’d ask them to at least avoid 
the azimuthal equal-area projection. Or, if they must use it, then at least 
state, somewhere, the X,Y map co-ordinates, and the lat/long co-ordinates, 
of the projection’s center. If they don’t want to write it on the map page, 
the could have it on part of one page of an appendix. So that someone could 
find the position of zone boundaries without having to first solve a system 
of five simultaneous nonlinear equations to find the projection’s center and 
orientation.

I’d ask them to avoid the polyconic too. It isn’t that the polyconic is more 
difficult than other projections. It’s that it has no properties, has 
nothing to justify its use for a data map.

The polyconic is often used to map the U.S. But it’s the wrong projection 
for a country of east-west extent.

Better yet, to make the request more complete, I’d recommend that publishers 
only use the following kinds of projections for data maps:

Cylindrical, conic, polar azimuthal (and maybe oblique azimuthal if they 
tell the map’s center position [in map co-ordinates and lat/long 
co-ordinates] and the map’s orientation about that center), and 
pseudocylindrical.

Well, usually the only major offender is the azimuthal equal area map, 
without the information that should go with it. If it weren’t for that one, 
I might not have posted these messages.

But this list is proof that I’m not the only person who is interested in 
co-ordinate transformations and considers them important, and wants ways of 
making them more feasible. In fact just the other day, on this list, there 
was discussion about software for transforming from the azimuthal equal 
area’s map co-ordinates to lat/long co-ordinates. Exactly the transformation 
that motivated me to post here, on the very projection that creates the most 
difficulty for the person who wants positions of zone boundaries on a data 
map. But how would you like it if you had to solve a system of five 
simultaneous nonlinear equations, to find the position of the projection’s 
center (in both co-ordinate systems) and its orientation about that center, 
before you could begin the actual co-ordinate transformation task?

Michael Ossipoff

I have a better idea of where you're coming from now. I see your need, more 
or less, though you seem more interested in precise coordinate inversion 
than I really see the need for. You don't see more maps on the projections 
you describe because not many people share your priorities. Be careful not 
to get too cavalier about the other traits. It's very uncommon for a map 
projection to satisfy two metric criteria simultaneously. You may mean that 
you can get good "enough" distance measurements even wit your "graduated" 
equidistant, but good "enough" for you might not be good enough for someone 
else. You certainly can't get the best distance measurements AND the 
equidistant property simultaneously

Daan--