[Proj] Revised definitions & suggestions. Peters proj has value.

[Michael Ossipoff]

Revised definitions and suggestions. Peters projection has value.

Linearity definitions:

I’d like to tighten my definition of linearity:

A projection has the linearity property if a map precisely constructed on 
that projection can be divided into regions such that, within every such 
regiion, Y distance varies linearly with latitude, for a particular X value, 
and can also be divided into regions such that, within every such regioin,   
X distance varies linearly with longitude, for a particular Y value.

A projection possessing that property is a “linear projection”.

[end of definition of linearity property]

A projection has the linearity2 property if a map precisely constructed on 
that projection can be divided into regions such that,  within every such 
region, Y distance varies linearly with latitude, for a particular X value, 
and X varies linearly with longitude, for a particular Y value.

[end of definition of linearity2 property]

As I said, linearity, the first property defined above is what I officially 
mean by “linearity”. But linearity2 is probably possessed by all the linear 
projections of interest. So, as a practical matter, I’ll often use the 
simpler linearity2 definition.

Obviously any linear function is linearly interpolable, and so any linear 
projection possesses the linearly interposable positions property (LIPP). 
Sometimes I’ll speak of linearity, sometimes I’ll speak of LIPP.

Projection definitions:

I’d like to revise my terminology for the projections that I’d previously 
named “graduated equidistant projections”

Those projections were linear projections that approximate co formality. 
But, by a similar construction, linear projections can approximate 
equal-area too.

A “linear approximate conformal projection” is a linear projection that 
approximates conformality. One that approaches conformality, as an 
unattainable limit, as the latitude bands become smaller and more numerous.

Within each latitude band, the north-south scale is equal to the geometric 
mean of the east-west scales along the two parallels that bound that 
latitude band.

[end of definition of a linear approximate conformal projection]

A linearly approximate equal-area projection is a linear projection that 
approximates equal-area. One that approaches equal-area as an unattainable 
limit, as the latitude bands become smaller and more numerous.

Within each latitude band, the north-south scale varies inversely with  the 
geometric mean of the east-west scales along the two parallels that bound 
that latitude band.

[end of definition of a linear approximate equal-area projection]

“linear approximate conformal projection” will be abbreviated “LA conformal 
projection”.
“linearly approximate equal-area projection” will be abbreviated “LA 
equal-area projection”.

A projection is an LA projection if it is an LA conformal projection or an 
LA equal-area projection.

The projections that I formerly called “graduated equidistant projections” 
are LA conformal projections.

LA[projection-name] is the LA projection that approaches the 
[projection-name] projection as an unattainable  limit, as the latitude 
bands become smaller and more numerous.

For instance, the projection that I’d formerly called the graduated 
equidistant cylindrical is the LA Mercator projection.

Of course there is an LA Mollweide projection, and LA Eckert IV projection, 
an LA Gall’s Orthographic projection, and LA Peters projection, etc.

I admit that I advocated LA cylindrical equidistant out of personal 
prejudice. I like conformal projections. Conformality is, of course a very 
useful property. The fact that the scale, at any particular point, is the 
same in every direction greatly facilitates distance measurement, for 
instance. And directions can be more accurately judged from the map.

I felt that a data map might as well have an additional property that I 
like, in addition to linearity, and so I chose near-conformality as that 
property.

But I must admit that conformality, as useful as it is, is unlikely to be 
needed on a data map.

As Daan pointed out, for the purpose that I expressed (accurate 
determination and measurement of position)  the cylindrical equidistant is 
the best. It is the best projection for data maps when equal-area isn’t 
needed.

And, as Daan pointed out, and as I immediately agreed, equal-area is 
desirable on data maps, for many (maybe most) kinds of data, such as species 
ranges, vegetation-regions, land-use, etc.

There is exactly one linear equal-area map: The sinusoidal. And so the 
sinusoidal was my first recommendation (and still probably my best 
recommendation) for a projection for a data map when equal area matters. For 
one thing, as I said, it’s the only truly equal-area linear projection.

But, if equal-area doesn’t have to be  _exactly_ attained, then the LA 
equal-area maps become possibilities.

Linear maps for people who are willing to calculate the transformation 
between lat/lon and X,Y:

I mentioned in my first posting here that linear data maps are only needed 
for those who don’t want to calculate conversions between lat/lon and X,Y, 
other than by linear interpolation.

For anyone else, linearity isn’t needed. It could still be convenient, but 
other properties could be more desirable.

First, let me say tha the equal-area cylindrical has such a brief lat/lon  
X,Y conversion formula that that calculation might be acceptable to people 
who ordinarily would only be willing to use linear interpolation. So 
projections such as Peters, Gall Orthographic, Balthazart, etc., might be 
acceptable for such people.

But maybe not. I still claim that, for the general public, for atlases and 
nature-guides for the general public, data maps should be linear.

I’m just saying that maps such as Peters are near the margin of becoming 
acceptable to those who don’t want to do calculation.

For the person who is willing to do more calculation to convert between 
lat,lon and X,Y , linearity isn’t essential. For such persons, when 
equal-area is desired for a data map, other equatorial-aspect equal-area 
maps could be considered. Such as Mollweide, Eckert VI, Eckert IV, or such 
cylindrical equal-area maps as Peters, Gall Orthographic, Balthazart, or 
Tristan Edwards.

Eckert IV moderates the less-desirable qualities of the cylindrical 
equal-area.  Moderating the cylindrical equal-area just enough to achieve a 
more civilized appearance.. Therefore, being as close as possible to 
cylindrical, Eckert IV seems more justified than Eckert VI.

Mollweide is justified by its great realism. I consider Mollweide to be the 
most realistic-looking  world map projection (Orthographic elliptical is 
more realistic looking, but I consider it a picture rather than a map). Yes, 
even though Africa is portrayed a bit skinny on Mollweide.

LA Mollweide is much more realistic-looking than Apianus II, and of course 
has more accurate areas. Both are linear projections.

Such cylindrical equal-area versions as Peters, Gall Orthographic,  
Balthazart and Tristan Edwards (actual) are the extreme (Tristan Edwards is 
the most extreme that I’ve heard of).

So, sinusoidal, Mollweide, Eckert IV, and cylindrical equal area (Peters, 
Gall Orthographic, Balthazart, Tristan Edwards, etc.) seem the projections 
most of interest if someone willing to do calculation wants an equal-area 
data map.

As we all agree, sinusoidal should be interrupted when used as a world map. 
I like the USGS’s interrupted sinusoidal best.

Good shapes can be helpful in using a data map. Not necessary if measurement 
is the important thing, but otherwise helpful for clarity of direct 
perception. Among the aforementioned projections, interrupted sinusoidal 
wins by that standard. All the pseudocylindricals, including Mollweide and 
Eckert, can be interrupted. It would then be a choice between interrupted 
sinusoidal’s better shapes and natural linearity, vs. the better 
space-filling efficiency of the other projections, with resulting 
improvement in minimum scale for certain latitudes.

But sometimes getting the most accurate position measurements possible might 
be the important thing. Then, the other equal-area equatorial aspect maps 
mentioned above could be helpful. Especially the more extreme north-south 
expanded cylindrical equal area projections such as Peters.

Of special interest, then, is the minimum scale on the map. That will be a 
north-south scale at extreme latitudes. Here I don’t mean “scale” in the 
usual sense. When I say here “north-south scale” I mean the scale for the 
purpose of measuring latitude--the Y-direction scale between parallels. (as 
opposed to true north-south scale, measured along meridians).

Obviously, as we go from sinusoidal to Mollweide to Eckert IV to cylindrical 
equal-area, the east-west scale increases for extreme latitudes. Therefore  
the north-south scale decreases.

If one looked at that alone, one would expect the minimum scale to be less 
for a cylindrical equidistant map than for the sinusoidal. But the 
cylindrical equal area more efficiently uses its available rectangular 
page-space. Therefore it has greater total map area, and therefore greater 
area in every part.

For a given page width and map width, a cylindrical equal-area (CEA) map 
with a 2:1 aspect ratio (the aspect ratio of sinusoidal, Mollweide, and the 
Eckert pseudocylindricals), will have greater minimum scale (for 
position-measurement purposes) than the sinusoidal, up to latitude 50.

Peters and Gall Orthographic have greater minimum scale than sinusoidal up 
to latitude 60.

At the extreme, Tristan Edwards (actual) has greater minimum scale than the 
sinusoidal right up to the arctic circle.

Where a CEA map has greater minimum scale than the sinusoidal, it allows 
more precise measurement of positions.

All the CEA projections have greater east-west scale than the sinusoidal.

Mollweide and Eckert IV are of course intermediate between sinusoidal and 
cylindrical equal-area.

Similar comparisons could of course be done between any pair of projections 
among the set consisting of sinusoidal, Mollweide, Eckert IV, and CEA.

Obviously, the more a cylindrical equal-area map is expanded north-south, 
the greater its minimum scale is. And a book-page allows lots of room for 
north-south expansion. So that expansion is limited only by what it does to 
the map aesthetically. (or a desire to uses the rest of the page for text or 
other maps).

Alright, Peters looks awful. But not when you consider that it serves a 
valuable purpose--improvement in minimum scale up to some minimum latitude, 
as a result of more efficiently filling the available rectangular 
page-space.

Behrmann treats Africa and South America a lot better than the other CEA 
versions I’ve mentioned here, but it only has greater minimum scale than the 
sinusoidal up to latitude 41.

Of course, then, for the person who wants to convert between lat,lon and X,Y 
without any calculation other than linear interpolation,  the LA Mollweide, 
LA Eckert IV and LA cylindrical equal area would be possibilities. The 
considerations for choosing between those and the interrupted sinusoidal 
would be the same as those described above for the person who is willing to 
do calculation.

Of course when regions smaller than the entire Earth are being mapped, it 
makes less difference which projection is used. All of the projections 
mentioned above would be good equal-area data maps for smaller regions.

What if the map information is such that equal-area isn’t important? Maps 
with equally-spaced parallels tend to have better shapes than equal-area 
maps. (But LA Mollweide will still look better than Apianus II, because 
Apianus II’s nearly upright continents look artificial and out of place in 
the ellipse).

Equally-spaced parallels suggest a sequence that includes sinusoidal, 
Apianus II, Eckert V, Eckert III, and cylindrical equidistant.

But, when equal-area isn’t needed,  the choice seems simpler than when 
equal-area is desired. There’s less reason to not use the cylindrical 
projection.

Michael Ossipoff