Which projections you favor?

[quadibloc]

Adams does not address the octahedral case in Elliptic Functions Applied to Conformal World Maps (1925). I know Cahill credits Adams for his conformal butterfly’s mathematical development, but I don’t find any publication that obviously contains that. Adams published Conformal Projection of a Sphere Within a Square in 1929. That’s one I don’t have that, but the title doesn’t suggest it would be in there, and Snyder does not note a publication for Adams’s octahedral development, either.

So… I’m skeptical. But, perhaps you can enlighten us.

I think I can, although I admit I'm a bit intimidated by your expertise in the field, as it's far greater than mine.

Basically, pull the same trick that turns the Stereographic projection into this:



but instead of using the Stereographic, start from this projection in Elliptic Functions Applied to Conformal World Maps (1925):



and, bingo, you're just using four of the six points of the star and that's your octahedron. If I didn't have the polynomial for the six-pointed star from Adams' book, and if the transform to the octahedral case weren't trivial like this, I would never have harbored hopes of someday implementing it.

[daan]

I see. You conformally map the sphere to ⅔ of the sphere and then project the “live” parts to the Star of David.

Lee elaborates on Adams’ development, including a series expansion, but curiously, does not relate the octahedron to that. Instead, the “tantalizingly close” refers to the world in a triangle.

— daan

[daan]

And I think interrupted projections are good, since distortion is bad.

But, interruptions are just another kind of distortion. I call that proximity distortion, as opposed to what I call differential distortion.

It would be nice if you could keep distortion "low enough" on an uninterrupted map of the whole world...

All world maps are interrupted.

Not that the Mercator, Winkel's Tripel, or the Briesemeister, or the Eckert IV are terrible, mind you, but trying to look for something that is much better than any of them and presents the entire world without any interruptions is likely a vain quest.

Quite so—that is, after amending without any interruptions to without any more interruptions than an entire meridian.

Someday we will have a mathematical theory that puts robust bounds on how much a given amount of interruption can improve differential distortion. We know that unlimited interruptions can reduce differential distortion to zero. We know that interrupting at only a single point yields unlimited differential distortion. But what happens in between is unexplored territory, as far as I know, beyond simply observing that the more interruption, the less inevitable the differential distortion.

— daan

[quadibloc]

But, interruptions are just another kind of distortion. I call that proximity distortion, as opposed to what I call differential distortion.

It is true that interruptions are... a fault of a flat map that a globe does not share. I think the normal meaning of the word "distortion" implies that it's "differential distortion", but I do see what you are aiming to make your terminology more precise about.

Quite so—that is, after amending without any interruptions to without any more interruptions than an entire meridian.

Yes, all world maps are interrupted, even if only at a single point. I like what the August conformal and the Eisenlohr manage to do - interrupting the world on an entire meridian, but mapping it to a shape that, because of its two cusps, matches the sphere, so that it can occupy a finite area and yet be conformal at every point - someone who had only seen the Mercator, the Stereographic, and the Lagrange, might not have believed that possible.

The usual use of the term, of course, reserves "interrupted" for what gets done to the Mollweide and the Sinusoidal.

I approve of precise terminology, as it leads to clear thinking. And we need more of that in a lot of places, many more badly than in map projections.

[daan]

I like what the August conformal and the Eisenlohr manage to do - interrupting the world on an entire meridian.

Eisenlohr and August are a fascinating pair for perhaps a couple of reasons.

The usual use of the term, of course, reserves "interrupted" for what gets done to the Mollweide and the Sinusoidal.

I agree people often casually mean meridional splits on pseudocylindric maps, but I don't think they reserve the term for that. If we were to ask people if they thought Fuller's dymaxion was interrupted, or if globe rosettes were interrupted, I presume they would agree, and I also presume many of them would spontaneously describe such maps as interrupted.

Interruption’s value is in reducing differential distortion. All interruptions participate in that reduction, so when we talk about interruptions, I feel like we should acknowledge all of them, rather than privilege the common configuration of an interrupted outer meridian. Privileging that configuration makes it hard to then compare against projections that are interrupted much less, and also distorts the perception of the effect of adding another interruption. Given what you have had to say in the forums so far, I trust that you understand what you meant. Unfortunately, I have conducted conversations with many cartographers about interruptions, only to discover that their conclusions were weak or false because they had not noticed that they were always working with interruptions anyway.

I rail against useless pedantry, by the way: precision beyond a contribution to understanding. I don’t think this is a case of that. Too many people misunderstand what is going on with interruptions because of (?) the way they talk about them.

Cheers.
— daan

[quadibloc]

If we were to ask people if they thought Fuller's dymaxion was interrupted, or if globe rosettes were interrupted, I presume they would agree, and I also presume many of them would spontaneously describe such maps as interrupted.

Interruption’s value is in reducing differential distortion. All interruptions participate in that reduction, so when we talk about interruptions, I feel like we should acknowledge all of them, rather than privilege the common configuration of an interrupted outer meridian.

Oops! Yes, of course Fuller's Dymaxion was interrupted by any definition.

You are right that the... division... of the globe by an outer meridian does the same thing as an interruption. Privileging it creates confusion; after all, the outer meridian is just as capable as any... internal interruption... of a projection of cutting Antarctica in half, or cutting off the eastern tip of Siberia from the rest.

I would say that the casual use of the term "interruption", though, normally means something in addition to the usual scheme of a projection, which is why the edge isn't counted. (This definition, though, is also problematic, because then Fuller's Dymaxion was only interrupted because one or more of the triangles had been cut in half.) But inventing a new word, instead of using the old one more consistently and precisely has the consequence that people have to learn the new word first in order to understand you. How to improve the precision of the language of the subject, without winding up being perceived as like Humpty-Dumpty (of Alice in Wonderland, of course) is not a trivial question. But specifically because any choice will confuse people in a subject where the existing language is imprecise, your approach of being bold in the matter certainly is a highly valid one.

[Piotr]

Daan Strebe's favorite map projection seems to be Sinusoidal, as he/she uses it in the Geocart icon, and as default map projection.

I have no favorite. People should be exposed to myriad projections, aspects, and configurations so that they don’t canonize any particular warped view of the world.

— daan

I found the button that exposes people to myriad maps!!!

[mapnerd2022]

For Valentine's day, either the Stabius-Werner or Mr.Strebe's Heart projection. For a pointed-pole projection that has an Globe-like shape, if equal area,then either the Mollweide or the Hammer. If not, then either the Apian II or the Aitoff. As for Halloween... The Wegtreue Ortskurskarte or, as It's more known as since late Mr.Tobler independently presented it in 1966, the Loximuthal projection.

[PeteD]

Which standard parallels and central meridians you favor?

...

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

...

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.

I know I'm replying to a post that's nearly six years old, but I was going through this thread and it occurred to me that it should be possible to calculate the optimal standard parallel. If my calculations are correct, the Kavrayskiy-type (i.e. log squared) angular distortion is minimized for phi0 = arccos(2/e) = 42.63° for both the cylindrical equal-area and equirectangular projections.

[Atarimaster]

I know I'm replying to a post that's nearly six years old

Arrghh, that was six years ago?

If my calculations are correct, the Kavrayskiy-type (i.e. log squared) angular distortion is minimized for phi0 = arccos(2/e) = 42.63° for both the cylindrical equal-area and equirectangular projections.

Interesting, because Canters arrived at almost the same value (43°)* for the equirectangular projection using a different metric, namely the mean linear scale distortion measured only across the continental areas. For the entire globe, his result was 37.5°. Regarding what I said in 2017, it’s not surprising that personally I prefer the latter.


*) see Small-scale Map Projection Design, chapter 2.3