Eckert IV vs Eckert VI. Other related topics.

[RogerOwens]

Yes, that's the general boundary-shape that I expected. That's interesting how it treats lat 60 as the pole, and puts higher-latitude places above or below the map, and how it has Greenland, evidently inverted, over Eurasia, and Svalbard & Franz-Josef Land over North America. I could really have fun with automated map-making if I ever found out how to do it.

These power-function maps present a different problem, compared to most other maps. Most all map proposals are already available as images, but this one is a formula that hasn't been imaged yet. So, finding out what it looks like presents a different kind of problem.

Of course if you can make an image of PF8.32-1, from its formulas, then you can make an image of any map from its formulas.

I wouldn't even know how to start trying to make an a computer image of a map from its formulas.

...which is why I'm about to draw it on paper and digitally photograph it, and try to post that.

But, again, I appreciate your work on this. You've got the image, except for that one latitude-bug. You'll probably have a good computer image of PF8.32-1 before I get it drawn, photographed & posted.

Thanks for your help.

Michael

[RogerOwens]

Tobias--

The Baker Dinomic projection sounds (& looks) like a great idea! Mercator up to lat 45, and then something with a point-pole. Do you know what projection Baker used above lat 45?

Liking & valuing Mercator as I do, and preferring a point-pole if possible, then I want a map like the Baker Dinomic on the wall.

Mine will use Mercator up to the Arctic Circle, and then use Sinusoidal. ...because Sinusoidal, having the right parallel-lengths, and equidistant parallels, has a kind of accuracy. Mercator & Sinusoidal can be regarded as the two natural ways to fix the Cylindrical-Equidistant's east-west magnification in high latitudes. ...make the parallels the right length, or expand the vertical scale to match the horizontal scale.

I'm going to draw, photograph, & post a PF8.32-1 map.

But I can say something about how PF8.32-1 will look:

Other than in the arctic, it will look a lot like Gall Isographic, because its meridians wait till a high latitude before they begin to curve inward, and because, like Gall isographic, PF8.32 is conformal at (lat 45, lon 0) (Of course Gall Isographic is conformal at lat 45 at all longitudes.)

In the Arctic, it might look a bit like Sinusoidal, but with meridians more nearly horizontal. Maybe more like Quartic? As I remember, the meridians will reach the pole with a slope-rise of only about 1/16. So, with the map centered on the Greenwich Meridian, I'd expect Alaska to look very shear-distorted.

But, when I did the calculations some time ago, it seems to me that the meridians are fairly vertical below the arctic, even near the outer meridians.

When I draw the map, I'll only draw the regions that often have a shape-problem when the map is centered on the Greenwich Meridian: Africa (it will be east-west shrunk by a factor of .709), Australia, New Zealand, Japan, U.S., & Canada (With their non-arctic latitude, they should look pretty good), Alaska (which, as I said, might be strongly sheared), and Greenland, Svalbard & Franz-Josef Land (which might be better-shown, because they aren't as close to the outer meridian).

The reason why I say that Alaska might be strongly sheared is that it was some time ago that I did the calculations for such things.

Michael Ossipoff

[RogerOwens]

When I said that my version of Baker Dinomic would use Sinusoidal for the Arctic & Antarctic, I meant, of course, that the Sinusoidal would be horizontally-sized to fit the Mercator that it's being grafted to.

The only remaining unspecified variable is the vertical scale of that Sinusoidal. My first inclination would be to make it conformal at the central meridian (as ordinary Sinusoidal is). ...unless that makes the map inconveniently tall.

Michael Ossipoff

[RogerOwens]

Some slants, scales & areas with PF8.32-1

Before an image of a paper-drawn PF8.32-1 map can be posted, I'd like to give some numerical information about the appearance, in PF8.32-1, of some regions whose shape &/or area can be difficult because of being near the outer meridian, or at high latitude, or both. Additionally, I'm including the two countries where reside the 3 people who visit this forum. (Germany & the U.S.)

The best way to present this information would be in the form of a table, and so I'll try that, even though tables don't always post well (the columns sometimes aren't straight).

The 1st column is meridian dx/dy, the slant of the meridian expressed as X distance divided by Y distance.

The 2nd column is that same slant, expressed in degrees from the vertical.

The 3rd column is meridian scale divided by east-west scale.

The 4th column is area magnification in comparison to regions at the equator.

As for the places listed, sometimes a particular city or other point is specified. Otherwise, the reference refers to the center of the region named.

Here is the table:

------------ dx/dy-------------Slant Degrees---------Meridian Scale/E-W Scale-------Area mag

E. Austr.-----.00195-----------------.11------------------------1.25--------------------------------------1.1325
N.Z.------------.0385-------------------2.2-----------------------1.061-------------------------------------1.33
Tokyo----------.0112-------------------.64-----------------------1.1412-----------------------------------1.235
Africa-----------0------------------------0--------------------------1.41445-----------------------------1.0038
Bering Strt-----1.445-------------48.855------------------------.943----------------------------------2.272
Svalbard--------.391---------------21.359----------------------.45222--------------------------------3.347
Franz-Josef L.--1.633------------58.525-----------------------.72967--------------------------------3.7
Greenland------.56657------------29.5346---------------------.574375------------------------------2.821
Germany--------.01077------------.617135---------------------.8952---------------------------------1.575
U.S.---------------.008---------------.4605------------------------1.139------------------------------1.2376
W. Hudson Bay---.316859----------17.58128-------------------.7657-------------------------------1.93

Yes, the precision of the numbers is more than is justified by the precision of the specification of the places.

PF8.32-1 doesn't have any problem with high-latitude meridian-cale exaggeration.

The magnification at high latitudes is moderate for a linear high space-efficiency map.

Franz-Josef L. stands for Franz-Josef Land
W. Hudson bay is the westmost end of that bay in Canada
E. Austr. refers to the eastmost point in mainland Australia
Bering Strt. is Bering Strait, halfway between Alaska & the Chukchi Peninsula.
N.Z. is New Zealand.

The map will be posted soon.

Michael Ossipoff

[RogerOwens]

Some of my results for Hudson Bay don't look right. I might have made a calcululator-error. For me, it usually seems more work to have a computer do a problem like that, than to do it by calculator. But those table-entries have lots of opportunities for calculator-error, and so I'd better have the more error-free computer do it.

Michael Ossipoff

[RogerOwens]

I did the calculations for West Hudson Bay again, and, this time, got reasonable answers.

Area magnification: 1.93
Meridian-scale/E-W scale: .7657

I've edited the table with that correction.

By the way, as I said, I like a version of Baker Dinomic with Mercator up to the Arctic Circle, and with the Sinusoidal at higher latitudes. Of course the Sinusoidal is east-west sized to match the width of the Mercator, and I'd leave the Sinusoidal's dimensions as-is, so that it remains conformal on the central meridian.

What I'd like to add is that the area-magnification of that Sinusoidal part of the map (in comparison to regions at the equator) would be about 6.3

There's no question about what it would look like, since it's just Mercator & Sinusoidal.

Michael Ossipoff

[Atarimaster]

By the way, as I said, I like a version of Baker Dinomic with Mercator up to the Arctic Circle, and with the Sinusoidal at higher latitudes.

You mean, something like this?
(That image is not a real projection, I’ve just copied & pasted a Mercator image and two parts of a Sinusoidal image together.)

[RogerOwens]

By the way, as I said, I like a version of Baker Dinomic with Mercator up to the Arctic Circle, and with the Sinusoidal at higher latitudes.

You mean, something like this?

Yes--Thanks for posting it. That's what I like for a combination of Mercator & Sinusoidal.

(That image is not a real projection, I’ve just copied & pasted a Mercator image and two parts of a Sinusoidal image together.)

It's a genuine projection. It's the projection that I spoke of, a graft of Sinusoidal on Mercator.

...Sinusoidal from the Arctic Circle, grafted to Mercator up to the Arctic Circle (and sized to fit its width).

That's what you posted, and that's exactly how I defined my suggested & preferred version of Baker Dinomic.

Of course it's clear why Baker chose to do the graft at lat 45 (instead of at a higher latitude)--in order to avoid an abrupt change in the meridians' direction.

This version that I prefer, and that you posted an image of, has a pronounced abrupt change in the meridians' direction, but, to me, that's justified by the desirability of showing nearly all of the inhabited Earth in Mercator.

Thanks for posting an image of my preferred version of Baker Dinomic !

Michael Ossipoff

[Atarimaster]

(That image is not a real projection, I’ve just copied & pasted a Mercator image and two parts of a Sinusoidal image together.)

It's a genuine projection. It's the projection that I spoke of, a graft of Sinusoidal on Mercator.

What I was trying to say, was:
The image wasn’t generated by a formula, or by an application like Geocart. I just copied, rescaled and pasted parts of two projection images.
So this image is not a projection, although someone could of course come up with a formula that creates the same result – only cleaner at the lines where I pasted the parts together.

[RogerOwens]

(That image is not a real projection, I’ve just copied & pasted a Mercator image and two parts of a Sinusoidal image together.)

It's a genuine projection. It's the projection that I spoke of, a graft of Sinusoidal on Mercator.

What I was trying to say, was:
The image wasn’t generated by a formula, or by an application like Geocart. I just copied, rescaled and pasted parts of two projection images.
So this image is not a projection, although someone could of course come up with a formula that creates the same result – only cleaner at the lines where I pasted the parts together.

Sure, but that formula would just be a combination of the formulas of Mercator & Sinusoidal--Mercator's formula in the domain from lat 0 to the arctic circle's latitude; and Sinusoidal's formula in the domain from the arctic circle's latitude to the lat 90.

If there's any discontinuity at the arctic &/or antarctic circle, I didn't notice it. It looks fine.

Regarding the choice of graft-latitude:

Of course Baker's motivation was understandable, to want a smooth transition between the two projections, explaining his choice of lat 45 for the graft-latitude. I'm more interested in worldwide usefulness than smooth appearance, which is why I prefer to make the graft at the arctic circle.

Besides, though Greenland's abrupt bend at the arctic circle, in the version that I like, emphasizes a cylinder-&-cones shape for the Earth, of course there are lots of projections, including Baker's, and both projections that it uses, that don't show the Earth as round. And, in fact, Greenland's abrupt bend at the arctic circle, in the version that I like, gives the map a solid 3D impression, which, arguably, enhances realism, in a way.

Michael Ossipoff