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

strebe at aol.com strebe at aol.com
Sat Aug 4 10:59:26 EDT 2007

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Michael,

The sinusoidal is not equidistant by any standard definition. Equally spaced parallels are a necessary condition for equidistance, but not sufficient. Constant scale along parallels is not related to equidistance because parallels are not great circles. The shortest path between two points on a parallel is never the parallel itself unless the parallel is the equator. It is constant scale along meridians, combined with straight meridians, that defines equidistance. The sinusoidal fulfills neither criterion.

Your post is quite long. I can't really get into a whack-a-mole game of responding to each of your points only to have several more spring up in their places. I will just note that I don't seem to have the trouble you have in determining geographic coordinates on a map as long as the map comes with a graticule. On a medium-scale map (a whole state, for instance) it's easy enough to arrive a lat/long coordinate accurate to a few seconds' accuracy in a minute or two. It just takes two measurements and a short calculation. That's FAR easier than trying to correct for the projection's vagaries in assessing distances, whether the map is conformal or not.

Regards,
-- daan Strebe

-----Original Message-----
From: Michael Ossipoff <mikeo2106 at msn.com>
To: Strebe at aol.com; proj at lists.maptools.org
Sent: Thu, 2 Aug 2007 5:25 pm
Subject: Re: Graduated equidistant projections for convenient co-ordinate transformations

Daan--

You wrote:

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

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:

property simultaneously.

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

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

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