What's a pseudoazimuthal projection?

[PeteD]

Hello and Happy New Year to everyone!

I recently came across the term "pseudoazimuthal" in this paper of daan's.

According to Wikipedia, "in standard presentation, pseudoazimuthal projections map the equator and central meridian to perpendicular, intersecting straight lines. They map parallels to complex curves bowing away from the equator and meridians to complex curves bowing in towards the central meridian."

This sounds like lenticular projections. Is "pseudoazimuthal" just a synonym for "lenticular"? Or is there more to it than that?

[daan]

I’m guilty of propagating the confusion, I think. Sorry about that.

I don’t think there’s a universally recognized definition for “pseudoazimuthal”. Snyder dodges the problem in An album of map projections by not using the term “pseudoazimuthal” but instead using “modified azimuthal”. However, that sets up a different problem: a lot of the projections he puts into the category of “modified azimuthal” are not modifications of azimuthal projections in some meaningful sense; nor does the designation say much about the projection’s appearance.

In my paper and in the Wikipedia article, the definition used is synonymous with my “lenticular” category. The reason this happens is that “lenticular” is not widely recognized. Wikipedia can’t really use “lenticular” because of that, but it also shouldn’t give an arbitrary definition the way that it does. The real problem is that these projections are not identified in the literature in any consistent way.

The closest thing to a rigorous definition that I can find appears in Bugayevskiy and Snyder 1995 (Map Projections — A Reference Manual). They define “pseudoazimuthal” to mean “On polar aspects of pseudoazimuthal projections, parallels are represented by concentric circular arcs, and meridians are shown as curves or straight lines, converging in the center of the parallels…”. By that definition, Aitoff is not lenticular.

Happy muddle!
— daan

[Milo]

The closest thing to a rigorous definition that I can find appears in Bugayevskiy and Snyder 1995 (Map Projections — A Reference Manual). They define “pseudoazimuthal” to mean “On polar aspects of pseudoazimuthal projections, parallels are represented by concentric circular arcs, and meridians are shown as curves or straight lines, converging in the center of the parallels…”.

That at least makes sense in terms of being consistent with the definition of pseudocylindrical projections.

A pseudocylindrical projection is one in which parallels are still projected the same way as in a cylindrical projection, but meridians are more complicated. So defining a pseudoazimuthal projection as one in which parallels are still projected the same way as in an azimuthal projection, but meridians are more complicated. means a pseudoazimuthal projection has the same type of relationship to an azimuthal projection as a pseudocylindrical projection has to a cylindrical projection, giving the impression that the "pseudo-" prefix has a consistent meaning in cartography. (To completely perfect the analogy, you would also have to specify that parallels in a pseudoazimuthal projection aren't just concentric circles, but distance-preserving circles.)

However, I'm not sure it's actually an all that useful definition. I mean, what would even count as a pseudoazimuthal projection under that definition? The Wiechel projection? As you note, a lot of the common inspired-by-azimuthal-but-not-actually-azimuthal projections that people actually use, such as Aitoff, wouldn't count under that definition.

What all of this really adds up to, in my opinion, is "pseudoazimuthal" being a rather poor term that's better off being avoided.

Wikipedia's use of "pseudoazimuthal" to mean what's also called "lenticular" is rather dubious. As Wikipedia's own article admits ("Listed here after pseudocylindrical as generally similar to them in shape and purpose."), lenticular projections tend to actually have more in common with cylindrical and pseudocylindrical projections than with azimuthal ones in spirit, even if some (such as Aitoff and Hammer), but by no means all, are inspired by azimuthal projections in construction. Of course, the definition of "lenticular" is also somewhat vague and variable, compared to the well-accepted definitions of "cylindrical" and "pseudocylindrical", but I at least tend to think of lenticular projections as being somehow "pseudopseudocylindrical": the next step of abstraction up, relaxing the requirements while still including pseudocylindrical projections as a special case.

The real reason that pseudocylindrical projections work as a category, though, I think, is that the pseudocylindricality property (straight-line parallels) is a useful property in itself, similar to conformality or area-preservation, even aside from the mathematical convenience it offers. None of the definitions I've seen for pseudoazimuthal projections really trigger the same "yeah, that's nice to have" thought, so ultimately it's just an arbitrary way to group together projections that don't really have that much in common.

[mapnerd2022]

Hello and Happy New Year to everyone!

Happy new year!

[mapnerd2022]

[ The Wiechel projection?
[/quote]
I really like the Wiechel projection, sure, everything looks twisted but at least it still has a purpose: to be used as a decorative map... «that in it's polar aspect, has semicircular meridians arranged in a pinwheel.» I don't just like useful projections, I also like «fun to look at» projections. So even a novelty projection can have value, contrary to what someone who has only seen the most useful projections might think. Or even what a layman might have seen.

[PeteD]

They define “pseudoazimuthal” to mean “On polar aspects of pseudoazimuthal projections, parallels are represented by concentric circular arcs, and meridians are shown as curves or straight lines, converging in the center of the parallels…”.

Thanks for the explanation!

I mean, what would even count as a pseudoazimuthal projection under that definition? The Wiechel projection?

Am I right in thinking that in addition to the Wiechel projection, all pseudoconic projections (in the case of curved meridians) with one standard parallel at the pole (so that the meridians converge at the common centre of the concentric parallels) and all conic projections (in the case of straight meridians) would also fall under this definition?

More obviously pseudoazimuthal projections according to this definition would include the Spilhaus projection, the Hellerick boreal triaxial projection, many of the projections presented on this page and all of our flat-earth projections.

[daan]

Bugayevskiy and Snyer continue their explanation of “pseudoazimuthal” by noting Ginzburg projections in that camp, and end the short section with the Wiechel, which they describe as the first pseudoazimuthal.

— daan

[daan]

I note a term that I coined (reported in A bevy of area-preserving transforms for map projection designers) for a related but distinct category: “quasiazimuthal”: Projections with straight meridians in polar aspect whose meridians are not necessarily spaced equiangularly, and not generally having circular parallels. The apple quasiazimuthal equal-area projection is one example; equal-area projections in polygons are more examples. As I state there:

I use “quasiazimuthal” to mean a projection which, in polar aspect, has straight meridians without constant angular separation between them. Recog- nizing that a region boundary on an equal-area projection says nothing about the region’s interior, it follows that an equal-area projection in some particular shape, such as an ellipse or a square or an apple, is not unique. Qualifying the description with “quasiazimuthal” specifies which among an unlimited number of projections it is.

Cheers,
— daan

[mapnerd2022]

Really great explanations Mr Strebe! Very clear, interesting and enlightening!

[Milo]

Am I right in thinking that in addition to the Wiechel projection, all pseudoconic projections (in the case of curved meridians) with one standard parallel at the pole (so that the meridians converge at the common centre of the concentric parallels) and all conic projections (in the case of straight meridians) would also fall under this definition?

The thought had occured to me as well. It depends on if you count the "parallels are circles" requirement to be satisfied when the parallels are circular arcs that don't reach the full way around.

Probably you would also need to specify, for conic projections, that either one standard parallel is at the pole, or it's the Lambert conic, which converges to one point regardless. Otherwise you have meridians which, due to being straight lines, can be interpolated as being aimed at a single point, but never actually reach that point.

I note a term that I coined (reported in A bevy of area-preserving transforms for map projection designers) for a related but distinct category: “quasiazimuthal”: Projections with straight meridians in polar aspect whose meridians are not necessarily spaced equiangularly, and not generally having circular parallels.

So, basically the opposite of the "pseudo-" modifier: meridians are still projected the same way, but parallels are more complicated.

This is an easy method of making simple polygonal projections, although doing that will result in obvious kinks at the diagonals, so it doesn't suffice to come up with really good ones.