The Orthographic-Aitoff Projection

[RogerOwens]

Orthographic-Aitoff

Maybe this has been proposed, but, for globe-appearance, I suggest a world map consisting of the Orthographic projection, in equatorial aspect, Aitoff-expanded horizontally.

It would be the Orthographic counterpart to Aitoff and Hammer.

It would be the Orthographic picture of the Earth, with just one modification, to allow it to show the entire Earth—Aitoff’s horizontal expansion.

Motivation and justification for that:

When it’s desired to have an un-interrupted equal-area or linear world map, in equatorial aspect, and if globe-realism is desired too, then the usual choice is Apianus II, Mollweide, or Hammer.

But, with equal-area global-realism, there’s a problem with that: It doesn’t really make sense for a map to be a picture of the Earth, and also be equal-area. Obviously, on a globe’s curved surface, the land areas will be foreshortened, and so they won’t really show apparent-sizes in proportion to their actual area. So the global realism of an equal-area elliptical map isn’t very convincing or correct-appearing.

So, if equal-area is desired, wouldn’t it be better to give up the goal of looking like a picture of the Earth?

What’s the simple and obvious way to transfer a globe’s surface to a flat paper? Remove a thin latitude-strip from the globe, along and all the way around the equator, and paste it to a flat piece of paper. Then remove a similar thin latitude-strip adjacently north of the first strip, and paste it on the paper, adjacent to the first strip.

…and so one, until the entire surface of the globe has been transferred to the paper.

Of course that gives you the Sinusoidal Projection. (if the strips are infinitesimally thin).

Since we’re giving up making an equal-area map look like a picture of the Earth, and since, in fact, it’s better that it not look like a picture of the Earth (since there’d then be something incorrect about the equal-area appearance of the continents), then why not use the completely obvious and natural Sinusoidal Projection—a projection whose equal-area is implied and expected (rather than being contrived by manipulation of an elliptical map) from its natural and obvious construction.

Yes, some people don’t like Sinusoidal, because of the great “shear-distortion”. But what you call “shear-distortion”, I call “pole-swept-ness”. It gives the map dynamism. I’ve found that I’m not the only person who likes the look of the Sinusoidal.

But I don’t deny that the elliptical compromise might often be desirable, because many don’t like the Sinusoidal. Compromises like that can be necessary, though I, myself, don’t care for compromised maps.

Though I'm not objecting to linear elliptical projections (Apianus II), it can be said for Sinusoidal that it is linear without differing scales on the various parallels. Just as Sinusoidal is the natural equal-area pseudocylindrical, it’s also the natural linear pseudocylindrical.

Of course Sinusoidal is the only linear equal-area map.
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But, if it’s global-realism that’s desired, then Orthographic-Aitoff would achieve that.

Sinusoidal and Orthographic-Aitoff are for when one wants one of those two extremes, instead of a compromise between them.
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Another possibility for globe-realism would be to map the Earth’s surface on one side of a 2:1 oblate spheroid. That’s probably been suggested and done.
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Speaking of compromises, the Orthographic-Aitoff could be partly compromised by making a pseudocylindrical in a 2:1 ellipse as the boundary, and spacing the parallels as they’d be in an equatorial-aspect Orthographic.

Michael Ossipoff

[daan]

I tried that back in the 90s when I was playing around with what i called “meridian duplication”. It turns out meridian duplication has a name as given by Wagner: Umbeziffern, which means “renumbering”. The projection itself I called pseudorthographic.

pseudorthographic

pseudorthographic.jpg (107.77 KiB) Viewed 1579 times

pseudorthographic distortion

pseudorthographic distortion.jpg (78.9 KiB) Viewed 1579 times

Best,
— daan

[RogerOwens]

Thanks for showing what PseudoOrthographic/Orthographic-Aitoff looks like.

The idea sounds good, and it does have a globe-realistic look.

But, because it mashes the U.S. and Australia significantly more than Sinusoidal would, I guess it would amount to completely trading usefulness for globe-looks.

...maybe as a logo, or for purely decorative use.

But another good purpose that it serves is that it makes Sinusoidal look not so bad after-all, by comparison, in regards to that mashing.

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About that area-and-foreshortening problem with Mollweide and Hammer--That had occurred to me a long time ago, and I'm sure that it's occurred to you as well.

The problem is that their illusory globe-realism could give someone a subconscious impression that the peripheral regions must be foreshortened on the globular surface. ...giving a subconscious impression that they're actually larger than they appear. ...defeating the purpose of an equal-area projection.

...or, or in addition to, spoiling their globe-realism.

Since Sinusoidal doesn't have the appearance of trying for globe-realism, neither of those concerns is a concern with Sinusoidal.

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Sinusoidal's natural idealness, alone, is enough to make it worthwhile.

...though I realize that, for practical use, the shapes-problem could sometimes be a bit inconvenient. (but not as bad as it looked without Orthographic-Aitoff for comparison).

A Scientific-American article, some time ago, used (un-interrupted) Sinusoidal to map Mars, showing that someone considers it practically useful. And it's often been used for star-maps.

Michael Ossipoff

[Atarimaster]

I tried that back in the 90s when I was playing around with what i called “meridian duplication”. It turns out meridian duplication has a name as given by Wagner: Umbeziffern, which means “renumbering”. The projection itself I called pseudorthographic.

Ummm, isn’t that really close to the Raiz Half Ellipsoidal in the equatorial aspect (see below)?
Although I might miss a few details here…

Here’s a larger image of the Raisz Half Ellipsoidal (about 2000px in width, 1 MB file size).

Regards,
Tobias

Edit: Instead of in the equatorial aspect I should have said: when the »tilt angle« is set to 0°.

[daan]

Huh. I have never seen an image of this particular Raisz development. Snyder seems to describe it, though he notes that the ellipsoid is tilted before projecting. It does seem to be the same projection. Thanks for pointing that out!

— daan

[daan]

Well. This image suggests not orthographic, but a closer perspective, so it is not clear to me that the image you posted correctly represents the Raisz development. (Or possibly vice versa.) I’ll look more into it.

Best,
— daan

[Atarimaster]

Snyder seems to describe it, though he notes that the ellipsoid is tilted before projecting.

It usually is, and then it looks different of course (see below or here for the large version), but G.Projector allows to set the tilt angle arbitrarily. As far as I know, Raisz prefered 20°.

It does seem to be the same projection.

Okay, I wasn’t sure because on low-res images, it’s easy to miss some details.

Thanks for pointing that out!

You’re welcome!
Tobias

[daan]

No, it turns out to be a different projection. Now that I know what to look for, I can see the differences even on maps of this resolution.

The Raisz form is a closer perspective than the orthographic is. It doesn’t show the entire earth. That is consistent with the projection geometry from the armadillo as well. I don’t have Raisz’s paper on “Orthoapsidal Maps”, but I’ll try to acquire it. Also, Raisz’s usual technique is geometrically projective, which is nothing like Umbeziffern even though the equatorial aspect looks so similar.

Best,
— daan

[Atarimaster]

No, it turns out to be a different projection.

Aaah, okay – thanks for pointing that out!

Kind regards,
Tobias

[daan]

The large image you posted, Atarimaster, underlays precisely beneath the graticule I generated for the “pseudorthographic”. It really is the same projection.

However, the smaller image is wrong. The outer perimeter is not an ellipse. It has been trimmed in some odd way. Beyond that, it’s also a very different projection than the equatorial image that you provided. It is not just a coordinate transformation. There is something more going on here. I do not think G.Projector is behaving well. Does it have a perspective distance parameter, and does that change when you recenter the projection?

— daan