Projection-Proposal: Behrmann-Eckert IV graft.

[RogerOwens]

Because Behrmann looks fine in the tropics, and because its min/max scale is equally low at lat 41.4 and at the equator, I suggest grafting, to Behrmnn, at lat 42, the Eckert IV projection.

But, instead of the Eckert IV section just being the original Eckert IV, starting at its own original lat 42, I’d start the quarter-circles vertical at the graft-latitude, lat 42. …so that the meridians will be smooth, and the vertical scale won’t abruptly change with Y position.

So, Behrmann’s east and west outer meridians, at lat 42, become Eckert IV’s quarter circles. The radius of those two circles would be determined so as to give the above-graft Eckert IV part of the map the correct area, in comparison to the Behrman below-graft area.

As with original Eckert IV, each of the inner meridians would be determined by the uniform division of each parallel between the outer circles.

Of course each above-graft parallel would be Y-positioned for equal-area.

Behrmann’s only problem is its high latitudes, where, in Northern Europe, at lat 60, its ShAcc is only 1/3. (…not so much different from Lambert CEA’s ShAcc of 1/4 at that latitude.)

So then why not have all the advantages of a cylindrical, with Behrman’s good shapes from the equator to lat 41.4, and then have Eckert IV (the changed form of it that I described) grafted at lat 42, to ease the distortion of the upper latitudes.

Because the map’s conformal latitude (lat 30) is in the cylindrical section, the overall map, with the Eckert IV graft, has the same area, for a given equator-width, as the map would have if it were Behrmann all the way up.

I suggest Eckert IV instead of Sinusoidal (though I aesthetically prefer Sinusoidal’s topologically-realistic point-pole, and its better portrayal of the polar region) because the Eckert IV line-pole approach gives better ShAcc in much of the above-graft latitudes, probably including much or most of the Arctic.

Advantages:

1. Large area for a given equator-width.

2. Cylindrical properties up to lat 42.

3. Good realism and aesthetic appearance below lat 42, and not so bad above that latitude, due to the curved meridians there.

4. The construction of CEA and Eckert IV are easily explainable and demonstrate-able to anyone.

(Admittedly Eckert IV’s construction would require that the hearer be willing to listen to a bit more talk. But though not as brief, the construction-explanation is still completely explainable to someone who is curious enough to be willing to listen to all of it.)

Michael Ossipoff

[daan]

Given the configurable equal-area pseudocylindric projections available these days—Hufnagel, Tobler hyperelliptic, my sinucyli—I don’t know why someone would spend time gafting together projections anymore. Just seems tedious, and then you’re left with discontinuities.

Tobias has already mentioned his Hufnagel explorer several times. I have one too.

• https://www.mapthematics.com/interactive/Hufnagel.html

It is Very Easy™ to go online and try things out.

— daan

[RogerOwens]

Hyperelliptic's meridians are horizontal when they reach the pole, resulting, for one thing, in very large NS scale-exaggeration, which was one reason why I preferred PF8.32 to it.

Of course, with any line-pole map, you're writing-off the polar regions, to improve the non-polar high latitudes.

How easy is it to explain Hyperelliptic's or Huffnagel's construction to people?

Do those projections have the cylindrical advantages over low & medium latitudes?

Michael Ossipoff

[daan]

Hyperelliptic's meridians are horizontal when they reach the pole, resulting, for one thing, in very large NS scale-exaggeration, which was one reason why I preferred PF8.32 to it.

Nonsense. It’s configurable, ranging from cylindric to pointed pole.

How easy is it to explain Hyperelliptic's or Huffnagel's construction to people?

Not the faintest concern of mine. In all my years of being The Projection Guy, I could count on one hand the number of times anyone has asked “how” a projection is constructed if they didn’t just want the formulas. Those who did ask weren’t looking for a description complete enough to draft the projection; they were looking for generalities, and they got them.

Do those projections have the cylindrical advantages over low & medium latitudes?

We give you the fishing pole and a barrel stuffed with fish… and you starve. I’m not going to hand you the fish, though. Sorry.

— daan

[RogerOwens]

How easy is it to explain Hyperelliptic's or Huffnagel's construction to people?

Not the faintest concern of mine. In all my years of being The Projection Guy, I could count on one hand the number of times anyone has asked “how” a projection is constructed if they didn’t just want the formulas. Those who did ask weren’t looking for a description complete enough to draft the projection; they were looking for generalities, and they got them.

True, most people don't ask. I'm just saying that, speaking for myself, I'm more interested in projections for people who do, however few they might be.

Speaking for myself, I wouldn't make, advocate or put-up an equal-area map whose construction couldn't be explained to everyone.

I didn't say that people who differ on that matter are wrong.

Because there's only one conformal cylindroid map, a weaker kind of explainability has to be accepted: Starting at the equator, directly spacing each next parallel for a NS scale equal to the EW scale at that particular latitude. Mercator must have done something like that, and that makes it an acceptable explanation.

Do those projections have the cylindrical advantages over low & medium latitudes?

We give you the fishing pole and a barrel stuffed with fish… and you starve. I’m not going to hand you the fish, though. Sorry.

I guess that can be taken as "No they don't."

Yes, you and Tobias have told me about all sorts of map-making software, and I appreciate that. I don't want to seem ungrateful. But the software doesn't answer the importance (to me) of easy explainability to everyone. ...and you above-quoted answer certainly has nothing to do with the question that you quoted before your answer.

Your above-quoted passage seems to be referring to my earlier requests for maps to be posted, from their formulas. But I've made no such request during this particular visit to this forum.

By the way, the Behrmann-Eckert IV graft that I proposed doesn't have discontinuity at the graft.

My proposal specified one particular map with that graft.

Michael Ossipoff

[daan]

We give you the fishing pole and a barrel stuffed with fish… and you starve. I’m not going to hand you the fish, though. Sorry.

I guess that can be taken as "No they don't."

Are you being deliberately obtuse? That’s not what I said or meant, and I’m pretty sure the rest of the world would have no trouble understanding that.

Yes, you and Tobias have told me about all sorts of map-making software, and I appreciate that. I don't want to seem ungrateful. But the software doesn't answer the importance (to me) of easy explainability to everyone. ...and you above-quoted answer certainly has nothing to do with the question that you quoted before your answer.

Now you’re making me angry. Stop setting up straw men. Your question was not about easy explainability; that was a prior question. Your question was, as everyone can see:

Do those projections have the cylindrical advantages over low & medium latitudes?

That’s not a question about “explainability”. At all. And you know it, and everyone else knows it, so stop pretending otherwise.

Your above-quoted passage seems to be referring to my earlier requests for maps to be posted, from their formulas.

NO. It refers to your question directly prior about how the projection behaves. I gave you the tools to answer that question and many more with the greatest of ease: without having to leave that armchair that you prognosticate so liberally from, without doing anything but clicking a few buttons. But no; you want to be spoon fed anything that doesn’t fall out of your own brain. I’m not interested in actionless pontification. Pontificating is easy. Most people don’t bother because they know it amounts to nothing. Where are the products of all this yacking? Why wouldn’t you just go •do• something to answer questions, confirm what you think you remember, generate maps… •anything•.

By the way, the Behrmann-Eckert IV graft that I proposed doesn't have discontinuity at the graft.

By the way, yes it does: if not a the first derivative, then at the second. And your description of how to graft them does not include any instructions about how to preserve areas in the transition.

— daan

[RogerOwens]

And your description of how to graft them does not include any instructions about how to preserve areas in the transition.

I said that the radius of the two circles in the upper, Eckert IV, section would be chosen to make the area of the upper section in the correct proportion to that of the lower, Behrmann, section.

And I said that the above-graft parallels would be Y-positioned for equal-area.

Obviously that's going to be very similar to Eckert IV.

It's true that I didn't post the formulas for the proposed projection. I've always posted a verbal specification of a projection-proposal, before posting its formulas.

And, also, I've always posted projection-proposals before checking the amount of their distortion. The min/max scale (which I abbreviate as 'ShAcc", for "Shape-Accuracy") isn't bad in Behermann. It's equal at the equator, and at lat 41.4
At those latitudes, it's 3/4.

And, up to lat 41.4, there's no absolute-compression in Behrmann. I use the term "absolute-compression", at a point, to refer to the factor by which the shortest scale at that point is less than the equatorial scale.

It's absolute as opposed to the relative compression consisting of a low value for ShAcc at a point.

But, above the graft, I don't yet know how low ShAcc will get, or how much absolute compression there'll be.

(But no, I'm not asking you to tell me.)

(But I don't suppose it can be too bad, because plain Behrmann has, as the shortest scale at a point divided by the equatorial scale, 2/3 at lat 60, and .43 at the Europe's northernmost arctic coast.)

On a CEA map, the latitude up to which there's no absolute compression is the same latitude at which the ShAcc is the same as it is at the equator.

If absolute or relative compression is problematically large in the above-graft section of B.E., that would spoil its value. No doubt this idea is familiar to cartographers: The length of the pole-line and the curvature of the meridians could be adjusted to minimize the absolute &/or relative compression above the graft, if they'd otherwise be unacceptable.

Of course a rightly-designed projection would be at least somewhat optimized according to some notion of optimization. But it seems to me that optimization of equal-area maps has a long way to go before it means anything.

As you'd said, there are many different things that different people could want, and it's arbitrary how different distortion-problems are compared and combined in a rating, and so rating and optimization are problematic anyway--aside from the fact that current optimization methods (including the one that I've suggested) leave important things out and are nowhere near ready to be usefully and meaningfully used.

What my optimization suggestions have left out is absolute compression, which is a completely different problem from low ShAcc. That occurred to me when Piotr said that north-south compression seems to him, worse than east-west compression. I'd answered that they're both equally-bad. But that was before it occurred to me that he has a point, in the sense that absolute compression is worse than the (mostly cosmetically objectionable) relative compression (low ShAcc).

There's the problem of how to compare and combine absolute and relative compression, in a rating or optimization. ...not to mention the problems of how to compare and combine with eachother, those, and all the other considerations that I've mentioned. ...many of which cartographers don't seem interested in.

When describing how I'd rate projections, I also named various other considerations, such as specific-area, space-efficiency, aspect-ratio, or reciprocal aspect-ratio, (or some combination of them, &/or others too) as possible things to multiply a merit-rating by. It would depend on which of the map's dimensions is the limiting factor, for fitting it into some available space.

So, lacking a rating or optimization standard, I'd just look at B.E.'s above-graft distortion to find out if it's unacceptably large.

By the way, one thing about absolute-compression is that, being potentially more of a usefulness problem than the more cosmetic relative-compression, then that means that I'd be wrong to assign a zero-rating for low ShAcc, when absolute compression can be much more detrimental to usefulness (which I consider important).

Anyway, I withdraw my rating suggestion, only because it isn't complete, and because the necessity of including and weighting absolute-compression makes even worse the arbitrariness that you mentioned.

Here's a partial quote of what I said in my initial post, relevant to how I'd make B.E. equal-area:

But, instead of the Eckert IV section just being the original Eckert IV, starting at its own original lat 42, I’d start the quarter-circles vertical at the graft-latitude, lat 42. …so that the meridians will be smooth, and the vertical scale (and its rate of change with respect to y-value, won’t abruptly change with Y position.

So, Behrmann’s east and west outer meridians, at lat 42, become Eckert IV’s quarter circles. The radius of those two circles would be determined so as to give the above-graft Eckert IV part of the map the correct area, in comparison to the Behrman below-graft area.

As with original Eckert IV, each of the inner meridians would be determined by the uniform division of each parallel between the outer circles.

Of course each above-graft parallel would be Y-positioned for equal-area.

Michael Ossipoff

[daan]

It's true that I didn't post the formulas for the proposed projection. I've always posted a verbal specification of a projection-proposal, before posting its formulas.

Maybe reconsider that habit. I cannot understand your description. It still makes no sense to me, but I admit I’m not willing to spend hours grappling with it.

To start with, you keep talking about “Eckert IV’s quarter circles”, but Eckert’s meridians don’t consist of circular arcs; they consist of elliptical arcs. Hence I presumed the circular arcs were a new bit of geometry that acts as a transition between Behrmann and Eckert, but apparently that isn’t what you mean. I cannot tell what portion of Eckert IV you propose to use. I cannot tell how you transition. If you propose to use Eckert IV all the way from its equator, which is the only way you can trivially avoid discontinuities, then I cannot understand what you do with the rest of the projection such that you only have the whole world, once.

— daan

[RogerOwens]

It's true that I didn't post the formulas for the proposed projection. I've always posted a verbal specification of a projection-proposal, before posting its formulas.

Maybe reconsider that habit. I cannot understand your description. It still makes no sense to me...

I intend to post the formulas too. It's just that I'm always so busy that it isn't possible to do everything as soon as I'd like to. The formulas for my proposed projection will be posted within a day or two.

, but I admit I’m not willing to spend hours grappling with it.

I hope that the formulas will be clearer than my words were.

To start with, you keep talking about “Eckert IV’s quarter circles”, but Eckert’s meridians don’t consist of circular arcs; they consist of elliptical arcs.

Eckert IV has quarter-circles as the north and south sections of its peripheral meridians, and so those were the Eckert IV quarter-circles to which I referred.

Note: When referring to Eckert IV's quarter-circles, I didn't say that all of the meridians were quarter-circles.

Hence I presumed the circular arcs were a new bit of geometry that acts as a transition between Behrmann and Eckert

Yes, they are. The vertical bottom end of each of the two quarter-circle Eckert outer-meridians grafts smoothly to the end of the corresponding Behrmann outer meridian. Likewise does each inner meridian start out vertical at the graft-latitude (as a consequence of the definition of "pseudocylindrical", and the form of the outer meridians). ...and thereby graft smoothly to the corresponding inner meridian of Behrmann.

, but apparently that isn’t what you mean.

It is. The verticality of the the quarter-circle outer-meridians, and of all the the inner-meridians (resulting from the outer meridians and the pseudocylindricity) are what make the transition from Behrman's vertical meridians, as you suggested.

I cannot tell what portion of Eckert IV you propose to use. I cannot tell how you transition.

I said that, from lat 42 on up, the outer meridians would be Eckert's quarter-circles. I said that the other meridians would be determined by the uniform division of each parallel, as with any pseudocylindrical.

That results in all of the meridians above the graft being vertical at the graft, where they meet their corresponding Behrmann meridians.

I said that, above the graft, the parallels would be spaced for equal-area, as is the case for other equal-area cylindroid maps, including the below-graft Behrmann part of the map that I propose.

If you propose to use Eckert IV all the way from its equator

I propose to use Eckert IV's quarter-circle outer-meridians, starting at the lat 42 graft-latitude. Yes, in Eckert IV, those quarter-circles go all the way down to Eckert IV's equator.

But I don't use those quarter-circle outer-meridians from B.E.'s equator. Only from lat 42, on up, as I said.

And, as I mentioned, the form of those outer meridians, and the map's pseudocylindricality, determine the inner meridians, and, in particular, make them vertical where they meet their corresponding meridians of Behrman.

, which is the only way you can trivially avoid discontinuities

As you said, with increasing Y, there are no abrupt changes in the vertical scale, or its y rate of change.

, then I cannot understand what you do with the rest of the projection such that you only have the whole world, once.

As I said, from the equator to lat 42, the map is Behrmann.

Admittedly, my proposed projection, which I call "B.E.", short for "Behrmann-Eckert IV", accepts some extra distortion above lat 42, in return for having a cylindrical from the equator to lat 42.

When I examine the above-graft part, for distortion, I'll find out if that price is acceptable.

Michael Ossipoff

[daan]

I understand you think you were plain. You were not.

The Eckert IV and the Behrmann do not even have the same width, so, starting with that, anyone reading your description would be confused because your very launching point isn’t understandable.

Then, you keep talking about Eckert IV, but as it has become clear in your later attempts to clarify, what you’re proposing isn’t an Eckert IV by any useful explanation. You’re not “grafting” the top 48° of an Eckert; you’re not “grafting” a full Eckert; you’re not grafting any portion of an Eckert. You’re not transforming an Eckert in any identifiable way for your graft. You’re not preserving the parallel spacing, the pole-line width, or anything. You’ve simply concocted an ad hoc equal-area (partial) projection that happens to have elliptic arcs as (partial) meridians, end-capped with quarter circles at the outer meridians.

So, Behrmann’s east and west outer meridians, at lat 42, become Eckert IV’s quarter circles.

They do not “become”. Your full projection’s (not Behrmann’s) outer meridians (abruptly) switch to your ad hoc equal-area caps at 42°. More source of confusion for the reader.

— daan