Which projections you favor?

[RogerOwens]

I like maps that are as easy as possible to explain

Then you should like the American Polyconic: One look at Jason Davis’ beautiful visualization says it all.
And of course, any true projection is very easy to explain, too. Though I really wouldn’t use the Central Cylindric just because it’s easy to explain… well, maybe to demonstrate why mathematical projections are needed.

What are the properties of the American Polyconic? What information would one get from it?

I don't think that position-finding is as easy on a Polyconic, as it is on a Pseudocylindrical.

(,,,or, for local small-region maps, on an conic or pseudo-conic)

Central Cylindric--is that a Cylindrical-Equidistant? Of course it's great for position-finding, and has all the cylindric advantages, but, if you have a cylindroid's lat/Y formula (and formulas for magnification and scales, for a cylindic) or a latitude-ruler, then you could get the same information on a conformal or equal-area cylindroid map. So I'd prefer a conformal or equal-area version, because it packs more information into the map. As I said, i guess I'm greedy for usefulness.

Michael Ossipoff

[Atarimaster]

Well, I merely referred to that »easy to explain« bit. Nonetheless:

What are the properties of the American Polyconic?

Meridians are equally spaced along the equator, parallels are equally space along the central meridian.

I don't think that position-finding is as easy on a Polyconic, as it is on a Pseudocylindrical.

… but surely, position-finding on the American Polyconic isn’t any harder (or easier) as on the August (speaking of cusped projections).

Central Cylindric--is that a Cylindrical-Equidistant?

No.
This is the Central Cylindric.
In my opinion, not good for anything, but easy to explain.

[Piotr]

Also, American Polyconic has low distortion around central meridian (useful for gores) and it has equidistant parallels.

[RogerOwens]

Well, I merely referred to that »easy to explain« bit. Nonetheless:
… but surely, position-finding on the American Polyconic isn’t any harder (or easier) as on the August (speaking of cusped projections).

Sure, and the lesser ease of position-finding is,for me, a disadvantage of August.

Which is going to be more helpful and useful for some particular map-user (such as you or me)?: Showing the poles and having less magnification, or easier position-finding, and (to a lesser extent) somewhat easier determination of scale and magnification.

(On Mercator, the scale with respect to the equatorial scale is secant lat. And the magnification with respect to the equatorial region is the square of that. ...easier than measuring along a curved line to measure the scale.)

Two requests:

1. Equal-Area Cylindrical has been described, and is available at your comparison website, in versions conformal at lat 45, and at lat 30. But Cylindrical Equidistant is described, and available at your website, conformal at lat 45, but not at lat 30. So could you add, to your comparison website, a version of Cylindrical Equidisant that's conformal at lat 30?

...achievable by vertically compressing the conformal-at-lat-45 (Gall Isographic) version by a factor of .8165

...multipllying the vertical distances, coordinates in the map by a factor of .8165

2. It seems to me that you said that you now have GeoCart. Daan said that his Oblated-Lagrange is easily implemented on GeoCart. That suggests that GeoCart can make an image of a map, from its formula. ...that it can make custom-made, user-specified maps, from inputted formulas.

Could you find out if that GeoCart facility can accept backwards formulas, and, if it can, could you use GeoCart to make an image of Equal-Area PF8.32, conformal at lat 30 (using the formula for it, in the thread about that projection, using the last K value that I posted, the one that was correct)

It would be good to, additionally, have an image for the version with the K value that gives equal and opposite NS/EW scale-disproportion at (lat 45, lon 0), and at the equator. But, mostly, I request the one with the K value that makes the map conformal at lat 30.

Could you make that image via GeoCart?

Michael Ossipoff

[Atarimaster]

1. Equal-Area Cylindrical has been described, and is available at your comparison website, in versions conformal at lat 45, and at lat 30. But Cylindrical Equal-Area is described, and available at your website, conformal at lat 45, but not at lat 30. So could you add, to your comparison website, a version of Cylindrical Equidisant that's conformal at lat 30?

Well, there is a Cylindrical Equidisant with standard parallels at 36.5° (for a reason).
I’ll have to think if adding a quite close variant is going to offer some benefit.

...achievable by vertically compressing the conformal-at-lat-45 (Gall Isographic) version by a factor of .8165
...multipllying the vertical distances, coordinates in the map by a factor of .8165

I prefer to achieve this by setting a standard parallel, which is a single option in Geocart (and other projection applications).

2. It seems to me that you said that you now have GeoCart. Daan said that his Oblated-Lagrange is easily implemented on GeoCart. That suggests that GeoCart can make an image of a map, from its formula. ...that it can make custom-made, user-specified maps, from inputted formulas.

Yes, I purchased Geocart a few months ago.
But you must have misunderstood daan. I guess he was saying that he could easily add Oblated-Lagrange to Geocart; but there’s no way for me, as a user, to input formulae.

And by the way, earlier in the thread you said:

We don't have a posted image of Equal-Area PF8.32, because it uses backwards formulas, and the image-making method that was being used doesn't accept backwards formulas for custom maps.

As far as I remember, that issue has never been resolved.
I said that I don’t know whether the d3 scripts accept backwards formulae. I wouldn’t recognize a backwards formula if it’d jump right into my face.
Here, again, is the list of projections that are supported by d3. If only one of them requires a backwards formula, then the scripts support them obviously.
(That still doesn’t mean that I’ll be able to translate your formula adequately to JavaScript syntax.

Kind regards,
Tobias

[daan]

Two requests:

1. Equal-Area Cylindrical has been described, and is available at your comparison website, in versions conformal at lat 45, and at lat 30. But Cylindrical Equal-Area is described, and available at your website, conformal at lat 45, but not at lat 30. So could you add, to your comparison website, a version of Cylindrical Equidisant that's conformal at lat 30?

...achievable by vertically compressing the conformal-at-lat-45 (Gall Isographic) version by a factor of .8165

...multipllying the vertical distances, coordinates in the map by a factor of .8165

2. It seems to me that you said that you now have GeoCart. Daan said that his Oblated-Lagrange is easily implemented on GeoCart. That suggests that GeoCart can make an image of a map, from its formula. ...that it can make custom-made, user-specified maps, from inputted formulas.

Could you find out if that GeoCart facility can accept backwards formulas, and, if it can, could you use GeoCart to make an image of Equal-Area PF8.32, conformal at lat 30 (using the formula for it, in the thread about that projection, using the last K value that I posted, the one that was correct)

It would be good to, additionally, have an image for the version with the K value that gives equal and opposite NS/EW scale-disproportion at (lat 45, lon 0), and at the equator. But, mostly, I request the one with the K value that makes the map conformal at lat 30.

Could you make that image via GeoCart?

Let me disabuse you of a few notions.

Geocart is not extensible by users. The many projections it supports are highly configurable, but as an end-user, you cannot just add in a new projection. When I say I can “easily” add a projection, I literally mean “I”. Many of the novelties you have seen me post are a consequence of my owning the source code and being able to make changes at will. My purpose in creating Geocart was for my own research. The fact that it is a commercial product is incidental.

Second, “easily” does not mean trivial; it just means in comparison to some projections that I have spent literally hundreds of hours refining. Setting up a new projection is not just a matter of plugging in a formula. The simplest projection would take me on the order an an hour of work if you include getting the development environment set up, the source code for the experiment sufficiently isolated, programming the formulas (at the very least; more on that later), integration into the menuing system and UI so that the projection is available, and cursory testing.

Third, as for the formulas themselves, if Geocart were like any other map projection program, formulas alone might suffice. They do not suffice for Geocart. The raster image processing and distortion analysis facilities require correct results even in numerically difficult spaces. Some projections are numerically stable everywhere, but many are not. They require series expansions around difficult points, and sometimes even more difficult, special case coding. Some projections that look simple on paper require thousands of lines of intricate programming.

Fourth, projections have outer boundaries. These outer boundaries have to be expressed precisely. For an equatorial aspect of a cylindric or pseudocylindric projection, or those that are topologically similar, this is easy. Still, the fact of the boundary has to be declared, along with a lot of other ancillary information about the projection so that it has context and relationship to the other projections and for purposes of analysis. When the outer boundary is elaborate, good luck. See my recent posting on equal-area transitions. Can you even come up with the outer boundary?

Fifth, rendering flawless raster projection images of the sort Geocart is capable of, as well as distortion maps, requires massive computational power. If Geocart accepted arbitrary projection formulas from users, the formulas would have to be interpreted, rather than executed natively in the machine language of the computer. That would slow down rendering by thousands of times, making Geocart’s facilities all but useless. Furthermore, there is no practical way to give the user a means to express an arbitrary projection’s outer boundaries. Nor would there any way to deal with regions of numerical difficulty.

And lastly, this business of “backwards” formulas is nothing like trivial. (They’re not “backward”; they just don’t have a closed form solution.) Solving them is generalized root-finding. People write papers on this, and I’ve recently refereed papers on it specifically for map projections. Geocart has had a generalized root finder for far longer than anything that has come out in public research, but even then, you need to set up each instance individually, also analyzed for numerically treacherous conditions. There may not be just one root; the root-finder may not converge on the one root even if there is just one for any number of mathematical reasons; and you almost always have to have some notion of where that root is in order to even start the search.

The amount of work involved is why I have ignored your continued wheedling to post this or that projection image. Tobias put a massive amount of work into setting up a system for you to experiment with and generate images. As far as I can tell, you have completely ignored what he has done and instead have continued to wheedle him to do your work for you. Now you go on about how easy it must be for other people to do your work for you. Well. It’s not, and I consider you lucky for no one exploding over your requests already. I can’t blame you, exactly, for not grasping the amount of work involved, but on the other hand, asking people to push buttons for you that they set up for you to push to do what you want, as Tobias has done, really strikes me as the extreme of rudeness.

— daan

[RogerOwens]

Let me disabuse you of a few notions.

Geocart is not extensible by users.

No, I didn't say that I thought Geocart was extensible by users. When you said that making an image of Oblated-Lagrange was something that could be easily done via Geocart, that sounded as if you were saying that making a map image from user-supplied formulas was a feature of Geocart. Evidently that wasn't what you meant.
No offense intended :^)

[...]

And lastly, this business of “backwards” formulas is nothing like trivial. (They’re not “backward”; they just don’t have a closed form solution.)

Oh, excuse me--I thought that "backwards formulas" meant formulas that give latitude and longitude as functions of X and Y. :^) But evidently I must have been mistaken, if you say that such formulas aren't "backwards formulas". :^)

Yes, at least one of those backwards formulas (the one for latitude) doesn't have a closed form solution (to get a "forwards formula" for Y). Yes, and that's why I posted the backwards formula instead of the forwards formula, because there wasn't a closed solution that would yield a forwards formula.

Solving them is generalized root-finding.

Oh, ok. I've never made a map that didn't have forwards formulas, and I was hoping that there might be some way to just directly use backwards formulas in the automated drawing of a map, instead of solving them numerically for X land Y in terms of lat and lon. But I didn't claim to know how it would be done, or how difficult it would be. Note that my request was in the form of a question, using the word "could".

So there's no need to get all upset.

People write papers on this, and I’ve recently refereed papers on it specifically for map projections. Geocart has had a generalized root finder for far longer than anything that has come out in public research, but even then, you need to set up each instance individually, also analyzed for numerically treacherous conditions. There may not be just one root; the root-finder may not converge on the one root even if there is just one for any number of mathematical reasons; and you almost always have to have some notion of where that root is in order to even start the search.

...and I didn't say or imply anything about the easiness or difficulty of that root-finding task, or even about the need for it. As I said, my request merely consisted of a question using the word "could".

The amount of work involved is why I have ignored your continued wheedling to post this or that projection image.

Linear PF8.32 doesn't need backwards formula. But you're the one who knows how difficult it would nevertheless be in Geocart, and I have no interest in debating that.

(and of course images of Linear PF8.32 have already been made available, thanks to Tobias.)

Tobias put a massive amount of work into setting up a system for you to experiment with and generate images. As far as I can tell, you have completely ignored what he has done and instead have continued to wheedle him to do your work for you.

As I mentioned at the time, it would involve finding and acquiring the software tools needed to use it (something that I have no skill or experience with), and accomplishing various computer-file tasks that I've never done. I didn't know the meaning of most of the words specifying what I'd need to do in order to achieve the imaages. l clarified that that wasn't a capability of mine. Tobias knows how to do that, and he made the images. I thanked him for it. What's your problem with that?? (rhetorical question)

And could it be that making map-images wasn't something that was outside Tobias's interests (in addition to being something that he, and not I, was qualified for?

Now you go on about how easy it must be for other people to do your work for you.

Actually no, I didn't "go on" about it. I quoted what you said, when you said that it would be easy to use Geocart to make an image of Oblated-Lagrange, and, once, I said (based on what you'd said about Oblated Lagrange) that it would be easy for you do do, using the software that you have. Mistaken? Sure. Big angry grievance for you? :^) No. I said it once.

And at no time did I say (much less "go on about") that it would be easy for Tobias to make the images that I requested. It was nice of him to do so, and I thanked him for it. And, as I said, my requests were always in the form of a question using the word "could". You spend too much time going on about, obsessing about, alleged wrongdoing.

asking people to push buttons for you that they set up for you to push to do what you want

As I clarified, i wasn't qualified for it, and didn't even know the meanings of the terms for the things that would need to be done, involving the finding and acquisition of file-tools, and file-operations.

, as Tobias has done, really strikes me as the extreme of rudeness.

...and we wouldn't want any rudeness, would we. :^)

Michael Ossipoff

— daan[/quote]

[daan]

And lastly, this business of “backwards” formulas is nothing like trivial. (They’re not “backward”; they just don’t have a closed form solution.)

Oh, excuse me--I thought that "backwards formulas" meant formulas that give latitude and longitude as functions of X and Y. :^) But evidently I must have been mistaken, if you say that such formulas aren't "backwards formulas". :^)

This is the hazard of using non-standard terminology. What you describe here is the inverse formula. In the context of PF8.32, what you have is either the forward formula that has no closed form solution, or you have the inverse formula in closed form. Now that I understand what you are saying, yes, programs need the forward form, and so yes, this is an exercise in root-finding.

You spend too much time going on about, obsessing about, alleged wrongdoing.

I gave a list of problems that need to be solved when imaging map projections. Your interjections in the body of that list dealt only with perceived slights. Perhaps my announcing my intent to disabuse you of this or that put you on guard, but the purpose of the list was to inform you about what it would mean to fulfill your requests, as well as to give an overview of how map projections get imaged in a raster environment. It was not to insinuate that everything on the list indicated some failing of yours. It is unfortunate if that information doesn’t interest you. Some people who care about projections would find the list illuminating.

As for alleged wrongdoing, I will simply repeat that (a), I do not expect that you would have known in advance the magnitude of what you keep requesting; but also (b) I found your response to Tobias’s work offensive, as well as your repeated requests for images. If you think the map projections you dream up have enough sufficient merit to be shown to the world, then you should draft them yourself the way people did fifty years ago; pay someone else to produce the images for you; or at least demonstrate some willingness to learn how to do the automated work yourself. The non-equal-area versions of your PF series that Tobias coded up need nothing more than for you to push a few buttons, and yet you seemingly could not be bothered to learn to do even that. Your protestations of poor qualifications ring hollow when anyone with the ability to post on this forum could easily learn to do what it takes. (I do not intend to speak for Tobias here; heʼs a good bit more tolerant than I am.)

For my own part, on the occasions I have been willing to spend time coding up a new projection in Geocart in order to render images for a discussion here, it is because I took an interest in the underlying problem, not in the appearance of the results. Endless slight variations with labored commentary do not interest me; the world does not need Yet Another Pseudocylindric Projection,™ let alone another cylindric.

— daan

[RogerOwens]

the world does not need Yet Another Pseudocylindric Projection,™ let alone another cylindric.

I didn't say that the world needs PF8.32.

But PF8.32, in its linear and equal-area versions, combines some arguably good attributes that aren't often found together:

1. High space-efficiency
2. Equal-area or linearity
3. Point-Pole, for topological accuracy, and a more accurate and better-looking Arctic
4. Meridians that never become horizontal
5. Meridians and parallels that never have infinite scale anywhere.
6. Some close approach to the cylindrical advantage of vertical meridians and equal treatment of all longitudes, in non-Arctic regions

In Behrmann, Africa, though horizontally-compressed, isn't terribly or really objectionably compressed (especially considering what is gained), And, Behrmann's U.S. looks fine.

Equal-Area PF8.32 is quite similar to Behrmann in non-Arctic (and non-Antarctic) regions. So Equal-Area PF8.32 shares some of Behrmann's acceptable appearance in the -30 to +30 latitude-band, and in mid-latitudes.

...while still having a point-pole and a resulting better Arctic.

The demands and problems of a linear map are less than those of an equal-area map, and so the Linear version of PF8.32 look even better.

As for a new Cylindrical, there's no reason why a Cylindrical Equidistant shouldn't have the compromise of having its standard parallel at lat 30, which is the boundary between the more equatorial half of the world and the less equatorial half of the world, I suggest that that makes lat 30 a good compromise place for the standard parallel of a cylindrical or pseudocylindrical map.

I'm not saying that the world needs that, but it would be a good compromise.

The 45/0 lat compromise that I've suggested, as an alternative is an obvious and natural compromise between the equator and the most mid-latitude place. Whether or not the world needs it, it's well-justified as a possible choice.

But the U.S. looks so un-distorted in Behrmann that there seems no need to move the standard parallel any farther north than lat 30.

So conformality at lat 30 is my main version for equal-area and linear PF8.32 and Cylindrical maps.

Michael Ossipoff

[Piotr]

Which standard parallels and central meridians you favor?

I think a 10 degree East central meridian is good because it shows Russia undivided. 20 degree West split is good for hemispheric projections.

About standard parallels, I think 37 degrees for cylindrical equal-area is a good idea, as well as 45 degrees for equirectangular.

What is your opinion on hemispheric maps?
A. non-circular (Sinusoidal, Quartic, Van der Grinten, Lagrange, Hammer)
B. circular, but not fully symmetrical (Mollweide, Bacon, Fournier's and Apian's globulars)
C. azimuthal (equidistant, Lambert equal-area, stereographic, orthographic, gnomonic)
Also American Polyconic, which is nearly circular. Nicolosi globular is nearly equidistant.

I'm not into A. because circular shape is much better for a hemisphere. Hemispheric American Polyconic stretches outer meridians a lot.

Hemispheric Mollweide, Fournier I and Apian's are good but I really prefer one of the azimuthal ones. Bacon and Fournier II have too much distortion.

I think hemispheric equidistant, Lambert equal-area and stereographic are great. Hemispheric orthographic has large distortion far from center point, and Hemispheric gnomonic cannot show the entire world.

EDIT: I actually think 45 is a better idea for cylindrical equal-area (Gall-Peters) but a higher or lower value may be used depending on context.