What's a pseudoazimuthal projection?

[daan]

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.

I might be guilty of inventing a few purely decorative projections.

Curiously, a variant of Wiechel is useful. Invented by Leszek Pernarowski in 1970 and rediscovered by Yours Truly in the early 2000s, the isoperimetric pseudoazimuthal has true scale along all parallels and meridians! I was boggled to discover that I could realize this property when I (also belatedly) first came up with the isoperic cordiform. The term isoperimetric is due to Pernarowski. His paper is obscure; I had never seen it referenced, nor ever seen any implementations of his projections until I rediscovered them.

— dean

[Milo]

Curiously, a variant of Wiechel is useful. Invented by Leszek Pernarowski in 1970 and rediscovered by Yours Truly in the early 2000s, the isoperimetric pseudoazimuthal has true scale along all parallels and meridians!

Ah, so it's relative to the orthographic projection what Wiechel is relative to the Lambert azimuthal.

[PeteD]

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.

According to daan's quotation of Bugayevskiy and Snyder, the requirement is "parallels are ... circular arcs", not "parallels are circles", though I suppose it's possible that they actually meant to write "circles".

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.

Yes, this is the part that I was less sure about. In my opinion, if this is what they meant to say, then "intersecting" would have been a more appropriate word than "converging" since lines can converge without ever actually meeting.

[Milo]

Yes, this is the part that I was less sure about. In my opinion, if this is what they meant to say, then "intersecting" would have been a more appropriate word than "converging" since lines can converge without ever actually meeting.

But for meridians in the shape of arbitrary curves rather than straight lines, their extension beyond the actual projection isn't defined in any meaningful sense. You can just scribble in whatever shape you want that's continuous with the existing projection and eventually meets at the center, and who's to say you're wrong?

[PeteD]

Absolutely. For pseudoconic projections, where the meridians are curves, one standard parallel must be at the pole. I said as much in a previous post:

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?

[Milo]

Yeah, but even if the meridians are straight lines for the full length of the projection, what's to say that they might not suddenly do an about-turn and go on some other curve once the projection is over?

[PeteD]

Well, at least in optics, if a light beam is focused at a point and then you put an optical element in front of that point, e.g. a collimating lens and/or a bent optical fibre etc., such that the light rays no longer actually meet at the focal point, then we'd still say that in the region in front of that optical element, the light rays are converging at the focal point. Not being a cartographer, I can't say whether or not cartographers would use the word "converging" in exactly the same way, but it would seem to me that if it can be used even when we know for certain that the light rays do an about-turn (e.g. in the case of a bent optical fibre) after leaving the region of interest, then we should at least be able to say "converging" when there's merely a hypothetical possibility of straight lines bending after leaving the region of interest.

Of course, I don't know whether or not Bugayevskiy and Snyder intended to include conic projections with neither standard parallel at the pole in their definition of pseudoazimuthal projections, but the way they worded their definition at least leaves it open to that interpretation.

[mapnerd2022]

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.

I might be guilty of inventing a few purely decorative projections.

Curiously, a variant of Wiechel is useful. Invented by Leszek Pernarowski in 1970 and rediscovered by Yours Truly in the early 2000s, the isoperimetric pseudoazimuthal has true scale along all parallels and meridians! I was boggled to discover that I could realize this property when I (also belatedly) first came up with the isoperic cordiform. The term isoperimetric is due to Pernarowski. His paper is obscure; I had never seen it referenced, nor ever seen any implementations of his projections until I rediscovered them.

— dean

So it's the orthographic counterpart of the Wiechel?

[daan]

That’s right, as Milo noted earlier.

Cheers,
— daan

[mapnerd2022]

Ok, thank you for the clarification, Mr Strebe and Milo!