Request for feedback on my map projection paper

[quadibloc]

Offhand, I do not have advice to give you on how to obtain a map which approaches the very beautiful one you have created which also has a mathematically well-defined projection. But after some thought...

My suggestion would be to attempt to do what was done to create the GS50 projection. Start from a conformal projection that is not interrupted in a way that conflicts with your map. So if you can find one meridian along which you are interrupting the map, you could use August's conformal. Of course, the Stereographic of the whole world is only interrupted at one point, but the large scale of such a map in some places could be a problem. Then have a computer find a complex polynomial that minimizes scale error everywhere, since any differentiable function is a conformal mapping. If Snyder had published his source code, this would be "easy".

Upon further thought, I remember that, to reduce distortion, I used a mapping of a Stereographic projection to another Stereographic projection on a different scale to produce this derived from August's conformal projection:

This reduced distortion by interrupting the map along 270 degrees of a meridian instead of 180 degrees. If you do the same in reverse, then you could fit the interruption into the Atlantic Ocean, and you would have a starting point for a projection very similar to your illustration to which to apply Snyder's technique.

It reminds me a little of an equal-area map I did,

but that map had severe distortion in parts of Africa and North America, unlike yours, which has very low distortion everywhere. Making odd-shaped equal-area maps in a patchwork fashion was easy enough for me, but for conformal maps, my mathematical expertise being limited, I could only try the most obvious options.

[justinkunimune]

That's beautiful! I've never seen an interrupted map that goes so out of its way to avoid splitting up archipelagos, and the interruption is really imperceptible to me.

I would go for the 5° grid myself. I know Jacob Rus made a conformal map based on an octohedron by specifying a Cahill-Keyes-like boundary and then having MATLAB solve for the inside, but he said that it took over a day to solve, and he ended up just saving a grid of points anyway.

[quadibloc]

On further thought, it may not be easy to introduce new interruptions in the projection using a polynomial. While repeated mappings to and from the plane on different projections can introduce interruptions, at least one in the Atlantic ocean can't easily be duplicated by any conformal projection I know of.

Oh, wait, that's not necessarily a problem. Introduce the interruption along a meridian in the Atlantic ocean with a projection that can be continued beyond the interruption. So the fact of an interruption area bounded on three sides is not an issue.

I'm surprised not seeing a comment from Daniel Strebe. Perhaps he is busy. On the other hand, he may just come back in a day or so, and surprise us all by supplying the mapping you're looking for.

[daan]

On further thought, it may not be easy to introduce new interruptions in the projection using a polynomial.

Right. The polynomial method can yield a projection that approximates continuous characteristics (such as distortion) that you want for a limited region, but would not be useful for creating or controlling interruptions. The problem is that the polynomial only gives you what you want at a specific number of points, with no control over what happens between those points. Therefore you could sprinkle the interruption with points along it in a suggestion of the path of the interruption, but you would not be guaranteed it would stay path-like between those points. If the one line itself were your only concern, you would be able to find a polynomial that worked if the number of points sufficed for the length of the path, but would have no control elsewhere. This is why GS50 does not give a proper path of constant flation around the US; it only approximates one, with unlimited deviations away from that path between the controlling points.

Justin mentioned Jacob Rus’s projection and its construction. The Schwarz-Christoffel mechanism is completely general; it can even be used for continuously changing boundaries. MATLAB is basically what everyone uses for this sort of thing; it’s the only publicly known out-of-the-box machinery for illustrating conformal maps having arbitrary boundaries. It implements the Schwarz-Christoffel mapping, along with tools to support that, such as solving the “side-length parameter problem”, which is something you have to do before you can apply the S-C integral to get your mapping. These MATLAB developments, along with much of the research that went into S-C numerical computation, were largely due to Tobin Driscoll and Lloyd Trefethen. MATLAB uses Gaussian quadrature for its S-C calculation. This is why Jacob Rus’s solution took so long.

So, to answer Luca’s question, yes, you can use Schwarz-Christoffel to compute Luca’s map-within-a-complicated boundary. MATLAB is the only machinery available to do that, and when you have finished, you will as many projected points as you want, but still no way to compute the map in any other environment except as interpolation.

Why have I been silent? <rant>Basically, I’ve bashing my face against my keyboard trying to satisfy Apple’s “software notarization” process for Geocart. While I agree with the need for its security benefits, I curse Apple for being the richest company in the world and yet still feeling no particular obligation to put out sufficient document for independent developers to comply with their mandates. Thousands of technical writers and editors could use those jobs, and hundreds of thousands of developers could produce more, better software for the platform. But apparently Apple doesn’t think their next $1 trillion will come from that business, so it’s left to barely lurch along.</rant> Oh, I was also enjoying the Northwest Chocolate Festival, as a counterpoint to vexation.

— daan

[quadibloc]

trying to satisfy Apple’s “software notarization” process for Geocart.

I am glad to hear that soon Macintosh owners will be able to obtain Geocart for their computers through the App Store for the Apple Macintosh.

It's unfortunate the process is difficult. I had only heard of it recently, due to news stories about one particular development environment (Electron) that generated code containing disallowed API calls - leading to a large number of programs being booted from the App Store and general consternation.

I was not thinking in terms of using Schwarz-Christoffel. That kind of machinery would end up creating a new elliptic integral for the map. Instead, I envisaged something like this:

The interruption around Antarctica is perhaps the toughest one, so do that first. Map the globe to the Stereographic projection, centered on the South Pole, then superimpose a Stereographic projection on it at a smaller scale, so that the area around Antarctica becomes a hemisphere. Put that in one of the lobes of August's conformal projection.

Do a number of mappings involving taking one's existing map, sending it to the sphere with an inverse Stereographic, and to the plane with an August's conformal, to create the cusp shaped beginnings of the various cuts. Use the fact that August's conformal can be extended to a repetition of the sphere, plus a large-scale Stereographic in the first step, to avoid wrecking what one has constructed of the projection so far.

Finally, when one has a projection interrupted in the right places, search for a polynomial mapping with minimal areal distortion relative to the sphere. So the mess from multiple repeated composed projections is the starting point, instead of the Stereographic, for the same optimization process as Snyder used for GS50.

Very long, very messy, but it's all relatively simple math, staying away from advanced stuff like Schwarz-Christoffel.

There is another reason that Schwarz-Christoffel did not occur to me. As I understand what Luca Concialdi is seeking to accomplish, that technique would be overspecified.

Schwarz-Christoffel allows an integral to be constructed for a conformal mapping given the boundaries of the source region and the target region. As I understand the proposed map projection, while the source region on the globe is defined - one can see exactly where the interruptions should go - the shape of the resulting map, including that of its boundary, is variable, capable of adjustment to achieve the goal of minimizing distortion.

[Atarimaster]

trying to satisfy Apple’s “software notarization” process for Geocart.

I am glad to hear that soon Macintosh owners will be able to obtain Geocart for their computers through the App Store for the Apple Macintosh.

Ummm, I’m no developer so I don’t know much about these things, but the way I get it, the fact that daan is satisfying the software notarization process does not mean that Geocart will be available on the Mac App Store – it’s required (in macOS Catalina) for all applications that are distributed the “traditional“ way, i.e. directly at the developer’s website. Correct me if I’m wrong.
By the way, I’m not even sure if being available at the Mac App Store would boost Geocart’s sales, but that’s completely off-topic…

daan, I wish you good luck with the process!

[quadibloc]

Correct me if I’m wrong.

You are right, as I had confused the App Store review process, where things like illegal API calls are checked for, and simply application signing, which indeed is an absolute requirement on the Macintosh (instututed with MacOS version 10.15, known as Catalina) to install programs obtained from anywhere. I did not read the articles carefully enough or remember their contents well enough.

[daan]

You are right, as I had confused the App Store review process, where things like illegal API calls are checked for, and simply application signing, which indeed is an absolute requirement on the Macintosh (instututed with MacOS version 10.15, known as Catalina) to install programs obtained from anywhere. I did not read the articles carefully enough or remember their contents well enough.

It’s yet more complicated. Code-signing has been a requirement for several OS releases, albeit able to be relaxed on the user side if the user chooses to and knows how. For Catalina (OS 10.15), yet another level of security called “notarization” is being enforced (albeit, again, able to be relaxed). The major problem is the alarm to the user, and having to give instructions to defeat notarization, if you do not distribute your software as notarized.

Notarization is similar to the App Store review process. It sucks up whatever information it wants from the application or package, sends it off to Apple’s servers, analyzes it, and then notifies you sometime later whether it has passed or not. If it has passed, a watchdog process on your computer receives notification and adds the notarization signature to the binary.

— daan

[daan]

I was not thinking in terms of using Schwarz-Christoffel. That kind of machinery would end up creating a new elliptic integral for the map.

Oh, it would be vastly more complicated than an elliptic integral. Grotesque, really.

Instead, I envisaged something like this:

Difficult to visualize, but I think I get the gist.

I am still skeptical of using complex polynomials against a world map. Those polynomials practically always break out of control when the expanse is too broad—and it’s impossible to guess how broad “too broad” is in a given circumstance without just trying things because their behavior ends up largely chaotic. Basically, the tighter you try to constrain them, the more likely they are to squirt out around the edges you want. Besides puncturing your desired boundary, they will generally causes folding and overlapping, sometimes very soon after their breakout.

— daan

[quadibloc]

Not that it is particularly important, but after adding additional modified forms of the original Van der Grinten projection to my site, as noted above in this thread, I finally got around to also drawing the Van der Grinten III projection, and giving it a page on my site, given that it has been utilized with some frequency.