Eisenlohr’s optimal conformal map of the world

[Atarimaster]

I think part of the problem is that the Mercator projection has a reputation of being "the map used by professionals" (…)

Hmmm. My theory is that it’s often used because it’s often used.
People don’t think at all who uses this projection, and why. They just take the map that looks familiar to them.

[Milo]

Sure, that's part of it, but the cycle must have gotten started somewhere, and I find it unlikely that the Mercator projection ending up in this role rather than some other projection was completely random chance.

It's certainly not the fault of Gerardus Mercator, who was well aware of the need for multiple map projections, and drew plenty of maps in projections other than the one that now bears his name.

[daan]

I wish people would rely more on experts—regardless of domain.

I respect people who seek to understand something on their own more than those who blindly follow the advice of someone who claims to be an expert.

So do I… but most people do not have the time or necessarily the capacity to understand that “something” when they have to execute on a goal that includes elements outside their own expertise. People grossly underestimate the amount of expertise that goes into even the most mundane-seeming things, and so they go into a project with only the most naïve grasp of the less familiar elements: so naïve that they don’t even know that they are doing anything wrong, and no way to know without becoming at least an informal expert.

And it is not about “blindly” following experts. My point isn’t about that at all. It’s that experts need to be included in the enterprise. Consulted with. Expertise implies nuance. The expert cannot offer nuanced advice without being involved.

I don't think I count as an "expert" on the field of map projections. I have no formal education in the subject and have not held any professional employment in the field.

I have no formal credentials in map projections, mathematics, or even computer science. And yet the last is my vocation; the first is my avocation; and the middle is needed for both. I do not consider myself to be an expert in mathematics, but I do consider myself to be an expert in map projections. I have expertise in some domains of computer science. Map projections do not earn me a living; computer science does. I have no opinion on your “expertise” in map projections, but I do have an opinion on your ability to think about them. That opinion is high regard.

If you can't understand basic concepts like "some map projections preserve angles but distort areas while other map projections preserve areas but distort angles" on your own, you have no business publishing any maps at all, even in a projection that someone else recommended to you.

I don’t think it boils down that simply. Most mapmakers understand those concepts but still don’t deeply understand map projections, and never will. Map projection is the ugly red-headed stepchild* of mapmaking for the reason that it is unrelated to the remaining concepts and expertise needed to create a fine map. It is also highly mathematical, which most mapmakers are not. Conversely, most experts on map projections are not not experts in mapmaking, myself included. Therefore, mapmakers should be educated enough in map projections to know how to choose an appropriate one for routine maps and should understand when they need to consult an expert for the remaining cases. If I should ever have to make a commercial map (heaven help the world), then I hope I am humble enough to consult with an expert.

*(Apologies to the redheads of the world, whom I really don’t agree are ugly; to the stepchildren, whose lives are sometimes harder because of it; and to ugly people, who are judged and treated cruelly for an arbitrary construct.)

I don’t think this dynamic is different in other fields. Specialists are critical.

The newspaper part doesn't surprise me much. I think the quality of journalism and journalists is decaying, at least here in Portugal.

I believe that’s because the readership isn’t willing to pay for expertise anymore. Why? Because they can get their news for free now. News outlets are not hiring expert reporters and expert editors because they can’t afford them.

I think part of the problem is that the Mercator projection has a reputation of being "the map used by professionals", due to having properties that are valuable for navigators, the one profession where being able to accurately read maps is an important part of the job (or was…

…the cycle must have gotten started somewhere, and I find it unlikely that the Mercator projection ending up in this role rather than some other projection was completely random chance.

I think I can shed some light on this. The 16th century was a period of great experimentation in map projections. A lot of one-off and novel projections showed up, were used, and mostly discarded quickly. Mercator’s projection, which Mercator himself was quite proud of, showed up in 1569. It did not change the world of mapping for the simple reason that navigation wasn’t advanced enough yet to make good use of the projection, mostly because longitude wasn’t measurable onboard. The 17th century was dominated by double-hemisphere maps, typically on the stereographic projection.

By the late 18th century, the longitude problem had been solved, and the value of the Mercator projection bloomed. The British Empire published innumerable maps on the projection for navigation. Meanwhile navigation was so esteemed and critical to the wealth and power of the empire that its culture and artifacts bled over into other aspects of British life. This included maps. It helped a lot that the Mercator enlarged Europe compared to much of the world because Europe was so cramped with countries and place names; spreading them out helped with the display and labeling.

We have to suppose that Europeans were also satisfied with how important it made Europe look. Arno Peters was wrong about a lot of things, but not entirely wrong. I find in A Little Book on Map Projections (Mary Adams, 1914) this quote:

Most atlases contain a World Map on Mercator’s projection. The maps as printed are only part of the World Map, for on this system the poles are projected to infinity and latitudes above 83½° are not usually represented. This is not of much importance as land above this latitude is of little value, and few persons desire to travel there. In Mercator’s projection countries in high latitudes are shown on a much larger scale than those near the equator, and Canada compares very favourably as regards area with the United States. Mercator’s is therefore a favourite projection for British Empire maps. It has two other merits…

Meanwhile the plate carree projection is actually a pretty good projection, as cylindrical projections go, but its extreme simplicity gives it a reputation as a naive map projection used by amateurs who don't know what they're doing, because people just can't believe that something that obvious could be "professional".

Having spent a lot of time around people in GIS and cartographic domains, I find that most of the disparagement about plate carrée is about its misuse, not its simplicity. ArcGIS made it the default, and so hundreds of thousands non-professionals have miscreated millions of maps that ought to have been on equal-area or conformal projections (depending on context) but instead were just in “unprojected coordinates”, as these naïfs call it. Since GIS packages normally do their computations in projected space, rather than in spheroidal space, millions of maps are simply wrong in their display of population densities, territorial and real estate area metrics, distances—you name it. These have real consequences: governmental allocations and decisions are often made on grossly wrong area measures; this was a topic in a paper by Nick Chrisman 2016, “Calculating on a round planet”. So, the disparagement has been about its use as a default, resulting in misuse, rather than about its value as a projection.

Finding myself finally untangling from a lot of snarls, I’ll migrate back to threads I’ve left hanging for way too long. I still don’t have a lot of time, but at least I have some.

Cheers,
— daan

[mapnerd2022]

Sure, that's part of it, but the cycle must have gotten started somewhere, and I find it unlikely that the Mercator projection ending up in this role rather than some other projection was completely random chance.

It's certainly not the fault of Gerardus Mercator, who was well aware of the need for multiple map projections, and drew plenty of maps in projections other than the one that now bears his name.

Well, yes, he used the Sinusoidal (the classic one which has points for the poles) for anything to do with areas, so presumably he invented it as well.

[quadibloc]

But maybe I’m missing what you’re getting at.

What I was getting at was, I thought, rather simple. In one post, you gave the definite integral which I reproduced graphically, because cutting and pasting the text from your post didn't work. You said evaluating that integral gave a mess.
Then, in a later post, you said "the mess was", but in the illustration, it showed that a different integral, which was an indefinite integral, equalled a long and messy closed-form solution.
Was that integral equivalent to the other one? It's never stated.
The indefinite integral shown as equal to the mess was the integral of phi sin phi * sqrt( cos phi ) / sqrt( 1 + cos phi ), d phi.

Which is not the same integral as 2π – 8∫₀^π/2 φ sin(φ)/√(1 + sec(φ)) dφ which is all I was getting at.

[Milo]

The indefinite integral shown as equal to the mess was the integral of phi sin phi * sqrt( cos phi ) / sqrt( 1 + cos phi ), d phi.

Which is not the same integral as 2π – 8∫₀^π/2 φ sin(φ)/√(1 + sec(φ)) dφ

Yes it is.

Divide both the numerator and the denominator of your first integral by sqrt(cos(phi)), and you arrive at just the integral clause of your second formula, without the "two pi minus eight times..." part, which however is trivial.

My understanding was that the indefinite integral produces an awful mess that we haven't been able to simplify yet, but then plugging the value φ = π/2 into this indefinite integral (in order to obtain the definite integral we're interested in) allows the formula to be greatly simplified, as the functions comprising it evaluate to convenient values at this point.

[quadibloc]

You won't believe me when I say I saw a Mercator projection having been used to show the worldwide distribution of covid19 cases and that I also saw it on TV being used in one Portuguese school(an elementary school, no less!) ( Particularly when children believe anything they see, even a nightmare-ish, unbalanced and greatly distorted representation of the world and especially of the polar regions) that has taken in/integrated some chilidren from Ukraine.

It is true that an equal-area projection should be used for a map that shows distributions and densities. I'm so used to the Mercator projection being used in schools from my own childhood, though, that I find it hard to be shocked at its being used in an elementary school, even nowadays.
If the classroom also has a globe, and children are shown that the map makes some places larger...
An interrupted Sinusoidal, in my opinion, is the best equal-area substitute for the Mercator, as it minimizes shape distortion. Things are sheared away from the central meridians, as one might expect, but the projection is accurate on the equator, and accurate along each central meridian.
Compare that with the Mollweide, say, where someone wouid have to look at another map, or a distortion chart, to know what the amount of stretch is for each different latitude.
So I am definitely of the opinion that whatever the faults of the Mercator, the Peters projection is not the answer.
In a poor country, where each child can't be provided with his own atlas, the Mercator projection has the advantage that several children can go to the map at once, and see reasonable maps of different areas of the world - because it is conformal. And it has North at the top everywhere, another bonus for avoiding confusion!
This is the same reason why it was a sensible choice for Google Maps.

[quadibloc]

Yes it is.

Well, it looked different, and I was too lazy to try to figure out how to change the formula. So my point was, I guess, that I was asking Daniel Strebe to show more of his work to make this easier to follow.
Now, if I remember correctly, a lot of integrals containing trigonometric functions can be converted, through a change of variables, into a form that doesn't have them.
From the Internet Archive, quite some time back, I had found an old book with oodles of definite integrals in it...
Tables d'intégrales définies
by D. Bierens de Haan, and it seemed to me I would be likelier to find it there in that form.

[Atarimaster]

We have to suppose that Europeans were also satisfied with how important it made Europe look.

On the other hand, I remember Eckert dismissing the Mercator projection because it was showing the german colonial territories too small. So he used a colonialistic argument to dismiss a projection that has often been accused as being colonialistic…
I’ll have to search for the exact quote, which will take some time because I don’t have Eckert’s works in searchable digital versions.

btw, in political terms, Eckert was dreadful anyway, but that is an entirely different subject.

[mapnerd2022]

We have to suppose that Europeans were also satisfied with how important it made Europe look.

On the other hand, I remember Eckert dismissing the Mercator projection because it was showing the german colonial territories too small. So he used a colonialistic argument to dismiss a projection that has often been accused as being colonialistic…
I’ll have to search for the exact quote, which will take some time because I don’t have Eckert’s works in searchable digital versions.

btw, in political terms, Eckert was dreadful anyway, but that is an entirely different subject.

I wonder why he made the 1st and 2nd projections useless/novelties when from his 3rd projection onwards we have actually useful world maps...