Low scale-variation conformal world map?

[daan]

Is the conflict between average and worst distortion such that, when scale-variation is measured by least-squares, or by area for a given minimum scale, the Lagrange has less scale-variation than the elliptical conformal world map?

I didn’t quite follow that, and it’s not “scale variation” that’s less in the Lagrange. Scale variation is less in the conformal world in an ellipse. Think of it this way. Assuming the lowest scale factor in the map is 1.0 whether Lagrange or the conformal world in a 3:1 ellipse, the Lagrange has much less total area, which implies much lower least squares scale factor. I actually did a posting on this topic awhile ago.

The difference between the min/max scale variation of the Eisenlohr and the Lagrange is astonishing. I guess the Eisenlohr's interruption, the split, has something to do with it.

I think the split itself has rather less to do with it than the details of the boundary shape. Eisenlohr is no more interrupted than Lagrange. Either way, the interruption is 180° degrees of arc. Chebyshev’s min/max theorem states that the optimal conformal map of a region (the one with the least scale variation) is the one with constant scale along its boundary for the region. Eisenlohr’s shape forces the boundary scale factor to a constant. The closer the map’s shape is to that, the closer it is to optimal by the Chebyshev criterion. However, that is not the best by least squares.

If interruptions are used, would more and deeper interruptions be better?

Certainly more interruptions means you can drive the scale variation lower, but interruptions themselves are a kind of distortion. You’re just trading one kind for another. In some circumstances that can be preferable. Interrupted conformal maps, by the way, are hardly studied.

For instance:

I might try other interruption methods too. Maybe an oval conformal world map, interrupted and re-centered on three or four meridians, from both poles, all the way to the equator.

I wonder how much that would help the scale-variation.

In this case, it wouldn’t. Not without some other sort of manipulation. That is because the maximum distortion is at the poles, and you’ll still have the poles in your simple interruption scheme.

Good luck!
— daan

[RogerOwens]

Thanks again for the answers, surely helpful to many other forum-visitors too, on that interesting subject.

[RogerOwens]

I looked on the Internet for Miller's oblation function, which, when applied to the Stereographic (interpreted as a set of complex numberss) gives the Oblated Stereographic.

What I've been able to find so far wasn't a formula for X+iY as a function of x+iy. Instead, what I found was two separate functions for X and for Y, as functions of the real numbers x and y.

A few questions about Miller's oblation function present themselves:

I understand that, if it's a polynomial function of a complex variable, then, if the original map was conformal, then the resulting map will be conformal--with the grid-lines being the same ones in the original map, and the region thats mapped being the same region that's mapped by the original map (as is the case, for instance, when the August maps the same Earth as the Lagrange).

But the Miller Oblated Stereographic maps a _differently_ shaped piece of land than does the Stereographic map from which the Oblated Stereographic map is made. So, how can the technique that makes August from Lagrange transform a Stereographic map to a conformal map of a differently shaped region?

Maybe there's a more complicated procedure, a trick to achieve that? That would explain why the Miller Oblated Stereographic was introduced so much later than was the August projection.

Could you explain how the Miller Oblated Stereographic is made, the principle, and what paricular function of a complex variable can be used?

...or a link or URL to a website that explains it?

And is/are there any other particular function(s) of a complex variable that you could suggest for oblating the Lagrange?

...or a link or URL to something about that?

I found, on the Internet, an image of the elliptical conformal world map. As you'd mentioned, it quite drastically magnifies the regions at low latitude near the outer meridians. That's probably why it hasn't been used as a published map. But I still intend to make an oval conformal world map (Oblated Lagrange) for the wall.

But, seeing the elliptical conformal world map gives one appreciation for the August and the Eisenlohr.

Between those two, I like it that the Eisenlohr doesn't magnify the arctic and antarctic as much. It magnifies the low-latitude regions near the outer meridians more, but magnifcation of the arctic and antarctic, regions usually of less interest, seems like more of a waste of space.

So, can you tell me how the Eisenlohr is made? Not just the formula(s) for X and Y, but the principle and the procedure? If not in detail, then in outline? ...and/or a link or URL to a website that explains that?

Michael Ossipoff, posting as Roger Owens

[daan]

What I've been able to find so far wasn't a formula for X+iY as a function of x+iy. Instead, what I found was two separate functions for X and for Y, as functions of the real numbers x and y.

Yes, when dealing with conformal projections Snyder commonly gave the real-valued X,Y developments in his utilitarian works rather than the single-valued complex functions, and that is true of most books. You can find the analytic version in his 1985 “Computer-assisted Map Projection Research”. I don’t have most of my library on hand, unfortunately, so my comments here are going to be from memory, and I can’t give page numbers or quotes.

I understand that, if it's a polynomial function of a complex variable, then, if the original map was conformal, then the resulting map will be conformal--with the grid-lines being the same ones in the original map, and the region thats mapped being the same region that's mapped by the original map (as is the case, for instance, when the August maps the same Earth as the Lagrange).

Yes. However, the “region” is irrelevant. Once you have projected the globe conformally to the plane, you can then apply whatever analytic function you want, and the result will be a conformal map. How that results relates to the concerns of the globe, including the graticule, is completely incidental; the analytic function operates on the plane, not the globe.

But the Miller Oblated Stereographic maps a _differently_ shaped piece of land than does the Stereographic map from which the Oblated Stereographic map is made. So, how can the technique that makes August from Lagrange transform a Stereographic map to a conformal map of a differently shaped region?

I don’t quite follow your question, but I think the answer is as I already gave. The shape of the region is irrelevant. The analytic function maps all points on the plane to other points on the plane and does not care about regions as you think of them on the globe. Now, possibly the analytic function has a restricted domain that it can operate on, but that’s not going to have anything to do with geographical regions; it has only to do with the function and the domain. Also, depending on the domain and the function, the resulting map might overlap itself. Obviously you would choose functions and domains within them that do not come with these kinds of troubles.

Maybe there's a more complicated procedure, a trick to achieve that? That would explain why the Miller Oblated Stereographic was introduced so much later than was the August projection.

The technique Miller used is not sophisticated; as you mention above, it is just a simple polynomial acting on the complex plane. The reason it came so much later has more to do with need and interest. Miller’s technique was developed first by Laborde and a collaborator whose name I don’t recall, and the reason they were interested in it is that it is “customizable” because both the degree of the polynomial and its coefficients can be chosen.

Could you explain how the Miller Oblated Stereographic is made, the principle, and what paricular function of a complex variable can be used?

w = a ∙ z + b ∙ z³

where

a = 0.9245
b = 0.01943
z = x + i y (that is, the stereographic coordinate)
w = X + i Y

However, the stereographic is centered on 18°N, 20°E, rather than the pole, so you would first perform a spherical coordinate rotation.

And is/are there any other particular function(s) of a complex variable that you could suggest for oblating the Lagrange?

Not particularly. There must be many, but the polynomial is simple, flexible, and easily controlled, so I have never looked into alternatives.

So, can you tell me how the Eisenlohr is made? Not just the formula(s) for X and Y, but the principle and the procedure? If not in detail, then in outline?

I haven’t read Eisenlohr’s paper (it’s in German), but I assume he arrived at his solution by solving the Dirichlet problem fairly directly. Surprisingly the solution has a closed form in this case. Dirichlet had made a study of boundary conditions for differential equations already some decades before that, so presumably Eisenlohr knew the Dirichlet problem and also knew about Gauss’s conjecture on the optimal world map. Therefore he was motivated to combine them to arrive at the optimal conformal world map with a single interruption along 180° of arc and bilaterally symmetrical.

In the case of an arbitrary differential equation, the Dirichlet boundary condition does not suffice to yield a unique solution. For that you would resort to the Cauchy boundary condition. However, in the special case of conformal maps the boundary suffices to completely determine the solution (and the interior).

Michael Ossipoff, posting as Roger Owens

Ah, hello again, Michael.

Regards,
— daan Strebe

[RogerOwens]

Oops--I was thinking that the Oblated Stereographic maps a different-shaped region, but, as you pointed out, it isn't about the shape of the region shown.

Thanks for these useful and helpful answers regarding conformal maps.

I'll experiment with the oblation of the Lagrange via Az+Bz^3, for my oval conformal world wall-map project. Even if it isn't as practical, I like the oval because it looks planet-like. People can regard it as a decorative novelty. I also intend to put up Lagrange and August. The conformal world maps' different magnification-patterns just makes them more interesting.

First I must find out more about the details of how graphics can be done with Visual Basic for Applications (VBA). I guess I'll use VBA, because it's part of Microsoft Office, and so it's already available (and usable when I find out how) on my computer. Someone told me that some (maybe easier) programming langauges are available on every Windows computer, but didn't give details about how to gain access to them.

Mike

[daan]

It’s trivial to code this up in Geocart, so here’s what it looks like.

Oblated Lagrange world.jpg (50.76 KiB) Viewed 2611 times

Oblated Lagrange distortion.jpg (43.43 KiB) Viewed 2611 times

with Lagrange for comparison:

Lagrange distortion.jpg (42.33 KiB) Viewed 2611 times

There’s no reason to use a linear coefficient a other than 1.0; what happens depends on its ratio to b, and meanwhile leaving it at 1.0 keeps the scale at the center at 1.0. But the assemblage is remarkably insensitive to the cubic coefficient b as well, as long as your goal is to maximize oblateness while preserving convexity. In this example I have a at 1.0, b at 0.1, and the scaling s before transformation at 0.86. In other words, if you properly adjust s, it hardly matters what b is as long as it is non-zero and rather less than 1.0. The effective formulæ I used are,

z′ = z ∙ s
w = (a ∙ z′ + b ∙ z′³)/s

Adding in a quintic term does nothing useful for your enterprise; it creates a dimple on the east-west and at the poles or else a hat at the poles, depending on whether negative or positive. And of course a z² term would introduce east-west asymmetry, so that’s not productive. And meanwhile a negative b gives you a north-south stretch instead of east-west, so what you see here is all there is to see with this method.

Regards,
— daan Strebe

[RogerOwens]

Ok, thanks for the image and the information. From the images, the Oblated Lagrange has more scale max/min than the Lagrange. I guess there's no point making a map that would only increase Lagrange's magnification in the arctic and antarctic, and at low latitudes near the outer meridians.

I'd hoped to make a 2:1 oval Oblated Lagrange, the oval shape of Mollweide, Aitoff and Aitoff-Hammer, because that's the ratio of the equator's length to that of a meridian. ...partly in hopes that it would move some of the greatest magnificaton from the arctic and antarctic, to the low latitudes near the outer meridians, and have a less magnified Antarctica than Lagrange does.

Mike

[daan]

Ok, thanks for the image and the information. From the images, the Oblated Lagrange has more scale max/min than the Lagrange. I guess there's no point making a map that would only increase Lagrange's magnification in the arctic and antarctic, and at low latitudes near the outer meridians.
Mike

No, it does what you want. Lagrange’s scale factor at the poles is about 280, compared to “only” 190 for the oblated version. At 89° along the central meridian, it’s 19.0 vs. 13.5. The oblated version achieves this by shoving more of the distortion into the out edges of the map near the equator, where it’s 2.7 vs. 2.0 for the Lagrange.

— daan

[RogerOwens]

Thanks for the custom map. I'll either use it directly copied from the posted image, or construct and print it out by applying Az+Bz^3 to Lagrange, with the parameter-values that you posted, using geographical data from a map that I scan (and reverse-map) or from a map database.

Of course, I realize that, though copying the posted image is faster, the construction and full-size printing would give better resolution.

My wall-posted conformal world maps will be Lagrange, that Oblated Lagrange, and August.

That's my goal with Oblated Lagrange--moving some of Lagrange's arctic and antarctic magnification to the more-of-interest lower latitudes. Lagrange's main fault seems to be that it devotes so much of its space to the arctic and antarctic.

Thanks again for the map.

Mike

[RogerOwens]

When the image of the Oblatedf Lagrange was posted, I mis-read the distorition pictures. But, after you pointed out that it achieved my goal, I compared that image to a Lagrange map that have, comparing the parallel-spacings in the antarctic, and at low latitudes near the outer meridians, and noticed that, as you said, the oblation moves some magnification from the arctic and antarctic, to the low latitudes near the outer meridians--which is what my goal was, for Oblated Lagrange.

So, thanks for making and posting the Oblated Lagrange map.

Mike