Request for feedback on my map projection paper

[Luca_bat_map]

That's it. It's nothing more than an experiment... The outer shape is ugly but the overall look of lands is very nice to my eyes. I played with the conformal cube of Geocart (demo version, you can see the watermarks) and I found the parameter to have all land a little away from the cube vertexes. Still I think you can see the potential of conformality, don't you agree?

[Luca_bat_map]

I tried to make a fully conformal projection with fairly low distortion of area and general shape using just the Lambert conic conformal:
Nowadays, though, of course the biggest demand is for projections that are fully equal-area.

Not new to me, I really like your website Check here https://www.mapthematics.com/forums/vie ... 3&start=10, your map inspired this one (another experiment).

[justinkunimune]

I don't know if I personally like the way eastern South America and western North America look in that particular one, but I certainly see the potential. Maybe I'll try something like what you suggested. Or at the very least, I'll mention conformality in the future work part of my conclusion and hope that someone else does it if not future me.

[quadibloc]

Not new to me, I really like your website

Glad to hear it. I suppose it's the Bird Projection you were referring to, although I agree that it's more inspired by the Cahill.

Somehow, viewing that discussion inspired me to update my page on the Van der Grinten projection. I added my true feelings about it: on the one hand, it's a projection that only someone warped by the Mercator could think is a good idea... and yet I admire very much the ingenuity that led to such a good approximation to conformality with a simple conventional construction.

I also added some additional forms of a modified Van der Grinten, including one



that seems to resemble a map being sold for classroom use that is advertised as being on the "Modified Van der Grinten Projection". The parallels are still circular arcs; their intersections with the central meridian are unchanged, those with the edges are the average in height of the center (or those of the Van der Grinten III) and those of the original Van der Grinten I. I had originally proposed a modified Van der Grinten with the center heights being added only as a 1/4 part; that seemed to be enough to eliminate some excess stretching in Alaska, but now I've also tried 1/3, and that is attractive as well, although I tend to feel the 1/2 proportion goes slightly too far: now shear has become noticeable, so now it is no longer as nearly conformal as it had been.

[daan]

The other reason I didn't feel compelled to pursue conformality is that I think there are more efficient ways to create conformal projections. If I'm not mistaken, a conformal projection is completely defined by its boundary (i.e. Adams's is the only elliptical conformal projection, and August's is the only epicycloidal one). So I felt like if I was going to optimize conformal maps, I should optimize the boundary and then solve for the conformal map projection that fills it. Except I don't know how to do that.

Your description is correct. It’s possible your mesh method would be more efficient. The Schwarz-Christoffel mapping in the general case is not efficient, requiring numerical integration of a potentially huge function of a complex variable. Simple cases with symmetries are much easier, but that’s not the “island world” case.

Hammond put out an atlas in the mid 90s that used optimal conformal maps for continents and regions. That was done in collaboration with Mitch Feigenbaum of chaos theory note. The atlas did not attempt such a thing for a world map. As far as I know, the details of their method were never revealed. MATLAB has an extensive library that could be used for this purpose based on earlier FORTRAN source code that is still openly available. Possibly that’s what was used in the Hammond atlas.

— daan

[daan]

The outer shape is ugly…

Guy doing a push-up…

You have a knack for piecing together conformal maps!

— daan

[quadibloc]

The other reason I didn't feel compelled to pursue conformality is that I think there are more efficient ways to create conformal projections. If I'm not mistaken, a conformal projection is completely defined by its boundary (i.e. Adams's is the only elliptical conformal projection, and August's is the only epicycloidal one).

Your statement is almost correct. A conformal projection is completely defined by its boundary, if you include in its "boundary" where points are mapped to on that boundry.

I had thought that the Eisenlohr versus August's conformal projection was an example, but I see that the overall shape of its boundary is also different.

Also, I might mention that on my page about the August conformal projection, I note that the GS50 projection is somewhat flawed - since the area between North America and Hawaii differs in scale from the main areas of the map around it that are being depicted, the relationship between the three main areas isn't quite shown correctly - which would seem to be the object of putting them all on one map on one projection. Given that I'm that picky about this point, it's not surprising I didn't warm to the last two of the Danseiji projections.

[Luca_bat_map]

I don't know if I personally like the way eastern South America and western North America look in that particular one, but I certainly see the potential. Maybe I'll try something like what you suggested. Or at the very least, I'll mention conformality in the future work part of my conclusion and hope that someone else does it if not future me.

Yeah I don't like that parts too but... looking at area error all parts near the cube vertexes are quite the same, I think it's not possible to do better with a cube. However that's my dream: a generic conformal engine to put the globe on a generic shape (for example the shape of your Danseiji IV or something like that) to get almost zero area error on all lands and exactly zero angle error. The perfect map...

However, today we have a lot of nice world maps that almost reach that goal...

[quadibloc]

However, today we have a lot of nice world maps that almost reach that goal...

I am glad you smiley'd when you said that. We do have a lot of different world maps. Some are very nice. And some do... approach... the goal of having low error. But as there is a fundamental limit to how low you can get error in shape if you avoid error in area, and error in area if you avoid error in shape... for a given level of interruption, even if we don't have a formula for what that minimum is, although we may be approaching that limit, it doesn't follow that our maps are nearly perfect. Instead, they're doomed to be forever far enough away that globes will still be worth having, bulky and expensive though they are.

[Luca_bat_map]

Hi all,

I know that somehow we are gone quite off-topic here but after the "guy doing push-ups map" I wanted to do the same but better. So... call me the crazy photoshop guy but I wanted to waste some time to get this. This map derives from the cube with some big manual work... grid is totally hand made...

Now... how to translate it into a fully defined map? Should I make a 5° grid and write all vertex coordinates? Or should I just define to boundary and then say "hey, computer, do the conformal map within this boundary? Such a shame I guess no computer will be able to do that without further instruction (and I have no clue about what instruction to give...), am I right?

However, have a nice weekend!

Ciao
Luca