On distortion and optimal projections

[dummy_index]

I still can’t find an equivalent term in mathematics. The third derivative of the position vector is often called “jerk”, and the fourth derivative is sometimes called “jounce”. Those have similar everyday meanings to “twitch”. Maybe there are engineering fields that use “twitch”, but I was not able to find any. What is the mathematical principle that 引き攣れ expresses?

Sorry, it is not a technical term. 2点を引き離したら（大圏コースが断裂をまたいでいない限り）どうやっても2点の間が引き伸ばされることになります。一方で2点を近づけたとしてもその間のコースが短縮されるとは限らない（e. g. curved meridians in Wiechel projection, or your https://www.mapthematics.com/forums/vie ... p?f=8&t=88）。なので図形の問題を考えるときの手がかりとして（tensed ropeなど）引張に着目することがよくあるのですが、つまり『強制的に引き伸ばされる』と言いたかったのです。

If you represent the sphere on the plane as a nearly double hemisphere, with only 100km of connection between the two near-hemispheres, then you will end up with points that were separated by a mere 100km on the sphere being separated by 4R = 25,484km equivalent on the plane. This means there must be large angular deformation within the map. Even if you only interrupt along the 180° meridian, as per the projection under discussion, the original πR distance between the poles will get extended to 4R.

Is this an argument for how to minimize the angular deformation of the boundary, or how to minimize the average angular deformation across the map? It is not clear to me that the argument necessarily achieves either one. For example, while it is possible that the circular Hammer has lower angular deformation along its boundary than my modified Eisenlohr, it definitely has higher average angular deformation.

This is an argument for "which is bigger, angular deformation of the boundary or angular deformation of anywhere in central meridian." As you know, circular Hammer has non-conformal central point. How on the other interruption form?

Oh, なるほど、Eisenlohrのisocolを利用して…isocolとantisocolを表す複素関数が欲しくなって…

Regards,

[dummy_index]

(i am interested in Snyder's Oblated Equal Area projection, but the publication is not free. Now i'm reading your "An efficient ...")

[quadibloc]

Despite not knowing Japanese, I tried to use Google to sort this out. but I had no luck.
From context, I now suspect that the "strong twitch", since the word is defined as "convulsion", could refer to what happens at the north and south poles: if one projects the sphere to a circle or ellipse, instead of an epicycloid, one can't have equal angular deformation on the entire boundary, since these two points on the boundary are surrounded by 360 degrees on the globe, and 180 degrees on the map - this makes them non-conformal points on an otherwise conformal projection, and it also complicates optimizing distortion on the periphery in other types of projection.

[daan]

Certainly the boundary on the modified Eisenlohr is longer because the circular Hammer’s boundary is a circle, and of all shapes with the same area, a circle yields the shortest perimeter. Your assertion that this implies lower distortion along the boundary is novel to me. I have to think about it. Superficially, it sounds correct.

It needs more qualification to be correct. If you permit a hole in the interior, then it’s possible for the outer boundary to have no distortion at all.

The absence of a hole is fundamentally topologically different, though. So far, I can’t think of a reason that your assertion would be false if holes are not permitted.

— daan

[daan]

Sorry, it is not a technical term. 2点を引き離したら（大圏コースが断裂をまたいでいない限り）どうやっても2点の間が引き伸ばされることになります。一方で2点を近づけたとしてもその間のコースが短縮されるとは限らない（e. g. curved meridians in Wiechel projection, or your https://www.mapthematics.com/forums/vie ... p?f=8&t=88）。なので図形の問題を考えるときの手がかりとして（tensed ropeなど）引張に着目することがよくあるのですが、つまり『強制的に引き伸ばされる』と言いたかったのです。

Sorry, “twitch” is not a technical term. What I mean is, if you increase the separation of two points from the globe when projecting them to the plane, then you have inevitably stretched [and therefore distorted] the distance. However, if you decrease the distance between them, you have not necessarily distorted the distance because the path between them could be curved, such as in the Wiechel or in your [Masque projection] https://www.mapthematics.com/forums/vie ... p?f=8&t=88. Therefore I often look at “tension” as a clue (such as in a stretched rope) when thinking about a shape problem. What I wanted to say by “twitch” is that is was ‘forced to stretch’.

I would say “strain” is a better translation as you are using the word.

Oh, なるほど、Eisenlohrのisocolを利用して…isocolとantisocolを表す複素関数が欲しくなって…

Ah, I see: you use the Eisenlohr isocols. That makes me want to know the complex function that expresses the isocols and antisocols.

The antisocol is the continuous gradient from boundary to the point of interest.

I haven’t been able to distill the infinitesimal geometric conditions into differential conditions yet, but expect to be able to eventually.

— daan

[dummy_index]

Despite not knowing Japanese, I tried to use Google to sort this out. but I had no luck.
From context, I now suspect that the "strong twitch", since the word is defined as "convulsion", could refer to what happens at the north and south poles: if one projects the sphere to a circle or ellipse, instead of an epicycloid, one can't have equal angular deformation on the entire boundary, since these two points on the boundary are surrounded by 360 degrees on the globe, and 180 degrees on the map - this makes them non-conformal points on an otherwise conformal projection, and it also complicates optimizing distortion on the periphery in other types of projection.

(To be sure, I don't consider the deformation just at the both pole (the "non-conformal points") - it can't be treated as Tissot ellipse)
I get ahead concerning about various interruption (as you know, interruption length affects scale distortion http://www.quadibloc.com/maps/mcf0702.htm) (however I don't know precisely about Chebychev's study...)
Non-conformal points wasn't my point at that post, but I can discuss. In the case of interruption as Pierce Quincuncial, cutting four quarter-circle makes 4πR periphery in natural length. If we make it a circle with 4πR² area, it will be able to free from angle distortion on entire of periphery. But around the "non-conformal point" (including the inside of the map), 360 degrees to 180 degrees mapping makes a/b ≧ 2 (Tissot ellipse major/minor) inevitably. (If mapping uniformly, all points near the pole will be a/b = 2. if mapping un-uniformly, narrowly mapped region will be a/b > 2.) Originally, don't we want minimizing the range of distortion in entire map? In this case, we should reduce the incompatibility of angle on the "non-conformal point," in return for elongate the periphery.
Simple formula is also our interest...

[daan]

It needs more qualification to be correct. If you permit a hole in the interior, then it’s possible for the outer boundary to have no distortion at all.

Sorry for this hasty, completely wrong posting.

The reality is the opposite: a closed path having no distortion on an equal-area projection can’t exist because the bounded area will always be too small. I was thinking of a different situation here and misapplying it.

— daan

[daan]

In summary, by dummy_index’s observations:

• Concerning a region of interest on the ellipsoid, the equal-area projection that gives the shortest possible closed path of constant angular deformation (isocol) bounding that region is the projection whose bounding isocol has the smallest possible measure for a bounding isocol.
• The shortest closed path is circular. Therefore the map with the smallest min/max angular deformation will be a circle with the same surface area as the region from the ellipsoid as long as some interior point has no distortion.

Odd observation: These observations hold regardless of the region’s shape on the globe.
Odd corollary: In order to construct this map of some arbitrary region such as to have circular boundary, the interior would, in the general case, be arbitrarily jumbled and distorted, with many interior points possibly having much greater distortion than the boundary.

In order to see that this must be true, consider a narrow strip along the entire equator. For the boundary of this strip to have minimal distortion, that strip would have to be bent into a circle. By bending it into a circle, the equator itself (or some other, even more arbitrary path) will get collapsed to a point, and so will have infinite distortion inside.

The circularized Hammer projection dummy_index presented has a boundary of constant angular deformation except at the poles. Interior to the projection, the minimum angular deformation is 0. Nowhere in the interior does the angular deformation reach the value along the bounding isocol. We know that this modified Hammer has considerably greater “average” distortion than my proposed optimal projection, but we also know it has a smaller distortion measure along its bounding isocol. We do not know whether its interior could be improved, and we have no theory that could inform whether it could be improved. Even if it could be improved, it seems fairly clear (to me) that it could not be improved enough to drive its average distortion lower than my proposed optimal projection.

We already know that the boundary of an equal-area map says nothing about its interior, but this set of observations shows the error I committed when I posited that the min/max criterion would hold when constructing the “optimal” equal-area map. Thus, even if the requirement of an isocol bounding the region of optimality holds by some theory of optimality, that isocol will not satisfy min/max optimality semantics.

I am not sure I have any “theory” left.

— daan

[quadibloc]

Obviously, taking an odd-shaped area on the globe, and sistorting it into a circle, won't produce less angular error in an equal-area map than producing a map that is close to the same shape would be able to. Either the theory is wrong, or there has been an error in expressing it - or in translation and so on. Since there is a good chance of the latter, it's probably not worth the effort to attempt a serious critique of that theory.

[daan]

Obviously, taking an odd-shaped area on the globe, and sistorting it into a circle, won't produce less angular error in an equal-area map than producing a map that is close to the same shape would be able to. Either the theory is wrong, or there has been an error in expressing it - or in translation and so on.

I don’t interpret dummy_index’s comments that way. “He” has given a counterexample to my hypothesis, disproving it. It would not matter if I somehow misinterpreted his comments; my observations about those purported comments still disprove my hypothesis and make it hard to see how to modify my hypothesis for this new reality.

It is still possible that my method gives minimum-error equal-area maps, but I don’t know by what definition of minimum-error that would be.

— daan