Experimental projections

[quadibloc]

Talking about experiments... what do you think about this one?

I like it a great deal, but of course I could be biased.

[Piotr]

tristandard projection — a pseudocylindrical projection that uses a fitted quartic polynomial to achieve three standard parallels (-60, 0, 60) that have no shearing distortion whatsoever.

[quadibloc]

Of course it looks funny, but as it's just presented as an experimental projection, that's not really a valid criticism.

However, it doesn't seem to be equal-area. With that constraint removed, it's trivially easy to do so much better:



No shearing distortion anywhere!

[daan]

Miljenko Lapaine has a presentation about an azimuthal projection with three standard parallels. It illustrates the fallacy in using projective geometry as a way to think about map projections. Of course, he does not suggest that the projection is otherwise useful.

— daan

[daan]

The ever-flexible Hufnagel gives an equal-area solution for the pseudocylindric case.

A = -0.664
B = 0.376
Aspect ratio = 0.363636
φ₀ = 52°

I didn’t spend the effort to converge perfectly on to three standard parallels, but it’s very close, and it’s obvious that it’s possible.

— daan

Hufnagel3.jpg (51.57 KiB) Viewed 5710 times

Hufnagel3dist.jpg (50.98 KiB) Viewed 5710 times

[Piotr]

Of course it looks funny, but as it's just presented as an experimental projection, that's not really a valid criticism.

However, it doesn't seem to be equal-area. With that constraint removed, it's trivially easy to do so much better:



No shearing distortion anywhere!

I mean the tristandard projection has no scale distortion either on the standard parallels. A standard parallel is one that has no distortion, relative to the map scale. It's possible to generalize this to any amount of standard parallels, with extreme angular distortion between the standard parallels, which makes it experimental.

[quadibloc]

I mean the tristandard projection has no scale distortion either on the standard parallels.

I'll have to admit that I missed that important point.
I could say that Daniel Strebe's example, using the Hufnagel projection, does have yours beat. But I think that it's funny-looking too.
At least it's obvious what is happening: the projection starts out like the Sinusoidal around the Equator, then changes so as to become like the Peters projection at the other standard parallels and above without a kink.
To my mind, it seems that all that projection demonstrates is that there are worse things than the Peters projection. But it still serves a purpose. By asking ourselves why we find it strange, and are inclined to dislike it, we learn something about what makes a projection attractive.
Basically, if you compare this unusual case of the Hufnagel to the Sinusoidal, the Sinusoidal also has no error in two places, the Equator, its standard parallel, and its central standard meridian. They have an organic relationship, the projection follows a simple rule, and is consistent everywhere, so one has a basis for expecting what the error will be anywhere on the map. Interrupting the projection can produce more error-free lines and reduce maximum error elsewhere.
The Hufnagel, on the other hand, is contrived. It's good in three places, but there's no particular reason one would want a map good in those three places and not elsewhere.
If, for example, I wanted a map that showed the area from 35 degrees N to 50 degrees N to the best advantage, then I would want to make the distortion low within that whole band, and have it vary continuously. So instead of something like this, I'd be inclined to consider a Bonne's projection with a standard parallel in the middle of that region.
Not that I would necessarily be satisfied with that either, since it would make it awkward to interrupt the map, and interruptions might bring more benefits than optimizing the standard parallel.

[Piotr]

And the tristandard experimental projection makes this false:

In a pseudocylindric projection, the only parallel that could be conformal is the equator. The only other points that could be conformal would be along the central meridian. This series is pseudocylindric. Therefore none of them are conformal along the ±45°parallel or any other parallel that is not the equator.

— daan

And the projection at https://mapthematics.com/forums/viewtop ... t=50#p1043 has an entire range of conformal parallels (but only one or two or zero standard parallels depending on the standard parallels of underlying Mercator)

[daan]

And the tristandard experimental projection makes this false:

I don’t object to a definition of pseudocylindric that permits higher latitudes not to be strictly shorter than lower latitudes, but that’s not the definition in play in the context.

The more common notion of pseudocylindric is described here:

The pseudocylindrical group is universally accepted, being those projections with meridians curving toward the poles and straight, scale-invariant parallels (the scale of any given parallel is constant along its entire length).

And here.

Hufnagel and HEALPix are modern projections that relax the requirement of monotonically decreasing parallel length. Denoyer relaxes the constraint of constant scale along parallels. Relaxing that latter constraint is not a definition I would use, ever.

— daan

[quadibloc]

Hufnagel and HEALPix are modern projections that relax the requirement of monotonically decreasing parallel length. Denoyer relaxes the constraint of constant scale along parallels. Relaxing that latter constraint is not a definition I would use, ever.

I agree about the Denoyer projection not being pseudocylindrical. While all the common well-known pseudocylincrical projections have well-behaved shapes, though, I hadn't thought that had anything to do with the definition of the class of projections.