Györffy's minimum-distortion pseudocylindrical projection

[daan]

PeteD, I very much enjoyed your comparisons showing the effects of optimizing according to the two different criteria. I’d like to see a lot more of that, particularly for projections that aren’t pseudocylindrical. Airy-Kavrayskiy is gelling toward my criterion of choice, based on many papers over recent years.

— daan

[PeteD]

PeteD, I very much enjoyed your comparisons showing the effects of optimizing according to the two different criteria.

Thanks very much!

I’d like to see a lot more of that, particularly for projections that aren’t pseudocylindrical.

Wagner VII and IX variants are low-hanging fruit since, like the Snyder minimum-error projections, they only have three free parameters, meaning the optimization isn't too difficult. (Wagner VIII variants have four free parameters, which is already significantly more time-consuming than three, while several other lenticular projections such as Györffy D–F, the Laskowski tri-optimal and Canters W10–W14 have many more free parameters.)

In fact, Francula has already done the work for us. Compare:

Francula III vs. Francula XII (similar to Wagner IX variants but derived from the American polyconic rather than the azimuthal equidistant)
Francula IV vs. Francula XIII (Wagner IX variants)
Francula V vs. Francula XIV (Wagner VII variants)

In each case, the projection with the lower number is optimized according to the Airy criterion, while the projection with the higher number is optimized according to the Airy–Kavrayskiy criterion. You can also compare Francula IV and XIII with Canters W09, which is the Wagner IX variant optimized according to Canters's metric.

Since the Airy criterion gives more weight to regions of higher distortion than the Airy–Kavrayskiy criterion does, the projections optimized according to the Airy criterion are taller and have shorter pole lines in order to depict higher latitudes more accurately at the expense of the rest of the globe.

Out of the equal-area projections, I prefer Francula XIV, which is also my favourite of all equal-area projections, but Francula V isn't far behind.

For the compromise projections, on the other hand, it isn't so clear-cut. The reason for the shorter pole lines of the projections optimized according to the Airy criterion may be to depict higher latitudes more accurately, but these shorter pole lines also have the side effect of tipping the balance between areal and angular distortion slightly towards low areal distortion, which I prefer.

This is why I prefer a weighted Airy–Kavrayskiy criterion. My current preferred weighting is 60.5 : 39.5 in favour of areal distortion. Optimizing Wagner IX according to this weighting results in a projection very similar to Tobias's Wagner IX autobiographical, combining a pole line length similar to that of Francula III with an axial ratio similar to that of Francula XIV, thereby improving (in my opinion) on both of these projections.

Airy-Kavrayskiy is gelling toward my criterion of choice, based on many papers over recent years.

This paper by Kerkovits makes a compelling case for the (weighted) Airy–Kavrayskiy criterion. Is this one of the papers that you're referring to? I'd be interested to know what the others are.

[Atarimaster]

In fact, Francula has already done the work for us. Compare:

Francula III vs. Francula XII (similar to Wagner IX variants but derived from the American polyconic rather than the azimuthal equidistant)
Francula IV vs. Francula XIII (Wagner IX variants)
Francula V vs. Francula XIV (Wagner VII variants)

Aaah, you’ve beaten me to it. I was gong to post that, too, but didn’t have the time earlier.

I just like to add that Francula was going for “advantageous projections for atlas cartography”, so it’s reasonable to assume that he kept the parameters within certain limits in order not to end up with a projection like Canters’ optimised version of Wagner VII, which has low distortions according to a certain metric, but does not seem very appropriate for general reference world maps.

So it’s possible that optimized Wagner variants without that restriction may look different. I doubt that but I’ve got no way of knowing.

Kind regards,
Tobias

[PeteD]

I just like to add that Francula was going for “advantageous projections for atlas cartography”, so it’s reasonable to assume that he kept the parameters within certain limits in order not to end up with a projection like Canters’ optimised version of Wagner VII, which has low distortions according to a certain metric, but does not seem very appropriate for general reference world maps.

No, Canters W07 and W08 ended up like that because the calculations were restricted to the continents excluding Antarctica, meaning the projections weren't penalized for having wacky boundaries. If you use a decent metric and carry out the calculations over the whole globe, then stuff like that doesn't happen and there's no need to artificially constrain parameter ranges.

This is one reason why I don't like restricting calculations to the continents. The other reason is that your result becomes dependent on the particular spherical body that you're projecting, and even if you only consider the Earth, it's still dependent on the particular aspect or central meridian that you choose.

For example, Canters carried out his calculations over the continents in equatorial aspect using the prime meridian as the central meridian. While the location of the geographical poles is determined by the Earth's rotation and it makes sense to put them at the projection poles, the location of the prime meridian ultimately goes back to where someone decided to build a certain observatory in the 17th century (although this convention wasn't definitively settled on until 1884), which is much more arbitrary. If Canters had instead set the central meridian to your 11°33' E (which I'm still calling the Oktoberfest meridian), this would have not only decreased the severing of landmasses but also reduced the distortion according to his metric by shifting Afro-Eurasia a little way towards the centre of the map.

[Atarimaster]

If you use a decent metric and carry out the calculations over the whole globe, then stuff like that doesn't happen

I assumed so (that’s why I said “I doubt that”) but it’s good to be confirmed, thank you!

and there's no need to artificially constrain parameter ranges

No need, but considering that he carried out the calculations on a 1971 computer (and possibly having only temporary access) it makes sense to, for example, dismiss values above roundabout 100 for λ₁ when you’re aiming at atlas cartography… and especially when you know that they won’t result in a good score.

This is one reason why I don't like restricting calculations to the continents. The other reason is that your result becomes dependent on the particular spherical body that you're projecting, and even if you only consider the Earth, it's still dependent on the particular aspect or central meridian that you choose.

If Canters had instead set the central meridian to your 11°33' E (which I'm still calling the Oktoberfest meridian), this would have not only decreased the severing of landmasses but also reduced the distortion according to his metric by shifting Afro-Eurasia a little way towards the centre of the map.

Aah, another good reason to use it!
And well, I can live with the name “Oktoberfest meridian” although I’m not a Oktoberfest friend at all.

Kind regards,
Tobias

[PeteD]

No need, but considering that he carried out the calculations on a 1971 computer (and possibly having only temporary access) it makes sense to, for example, dismiss values above roundabout 100 for λ₁ when you’re aiming at atlas cartography… and especially when you know that they won’t result in a good score.

Yes, even on a 2023 computer, you wouldn't try values that you know won't result in a good score.

In the case of Francula I and II, it seems that the results did conflict with Francula's idea of what constitutes a good projection for atlas cartography, so he repeated the calculations with the standard parallel fixed at 40° to obtain Francula VI and VII. However, he still published Francula I and II, so I think it very unlikely that he found the need to constrain parameter ranges for his lenticular projections just to make the results suitable for atlas cartography.

And well, I can live with the name “Oktoberfest meridian” although I’m not a Oktoberfest friend at all.

While I do enjoy beer festivals, I'm also not a particular fan of the Oktoberfest, our favourite beer festival being the Erlanger Bergkirchweih at 11°00'19" E. Unfortunately, it's not located at an integer number of arcminutes, and using this as your central meridian will cause the eastern tip of St Lawrence Island to be severed (although the cut will go very nicely between the two Diomede Islands). In any case, it's probably not a good idea to base your central meridian on things like your favourite beer festival or the largest city along the river that shares your first name etc.

[Milo]

I just like to add that Francula was going for “advantageous projections for atlas cartography”, so it’s reasonable to assume that he kept the parameters within certain limits in order not to end up with a projection like Canters’ optimised version of Wagner VII, which has low distortions according to a certain metric, but does not seem very appropriate for general reference world maps.

Hey, those big curly gaps would be a good place to put a legend

No, Canters W07 and W08 ended up like that because the calculations were restricted to the continents excluding Antarctica, meaning the projections weren't penalized for having wacky boundaries. If you use a decent metric and carry out the calculations over the whole globe, then stuff like that doesn't happen and there's no need to artificially constrain parameter ranges.

Well, I can see the argument both ways. Antarctica is always going to look bad in any (normal-aspect) lenticular projection, so it makes sense to sacrifice it for better results elsewhere. But like you suggest, a well-designed metric should be able to just recognize that Antarctica counts for a relatively small portion of the map without resorting to special pleading.

This is one reason why I don't like restricting calculations to the continents. The other reason is that your result becomes dependent on the particular spherical body that you're projecting, and even if you only consider the Earth, it's still dependent on the particular aspect or central meridian that you choose.

A worthwhile point to consider! Even on this forum, we usually demonstrate our projections using Earth, because the math is the same no matter what world you're drawing, but it's interesting to think about which kinds of bodies would warrant using different projections. Even just Earth can have a lot of variety to it. For example, if I had to make a map of Cambrian-era Earth, when most of the land was in a big supercontinent approximately centered on the south pole, I wouldn't use a lenticular projection! (...Or would I? After all, all macroscopic life was in the sea at the time, so maybe focusing on the continent is missing the point? But then, even sea life likes the shallows near the coastline.)

Other solar system bodies do have the complication that it's hard to decide what to even care about. On Earth, we usually design our maps to look best over the major continents where people and animals live, while sacrificing barren places like Antarctica and Greenland where nothing interesting lives anyway. But on Mars, there isn't any life anywhere, so what do you prioritize? If you like science fiction, you could try to imagine what a terraformed Mars might look like, filling in the lowlands with restored oceans and imagining where the most vibrant habitats would be, but such flights of fancy have little bearing on modern Mars rovers, which can probably do approximately equally-useful science all over the planet (and indeed, we've dropped them in all sorts of places). On the moon, who's to say the South Pole-Aitken Basin isn't the patch of geography that interests you most?

It's impossible to make a map projection that doesn't make sacrifices somewhere.

For example, Canters carried out his calculations over the continents in equatorial aspect using the prime meridian as the central meridian.

It doesn't even do that great of a job of maximizing the continents, since it's still symmetric. A true continent-biased map wouldn't be symmetric because Earth (in the present day) has more land in the northern hemisphere than the southern hemisphere, with important coastlines significantly closer to the north pole than the south one (except for Antarctica).

[daan]

Other solar system bodies do have the complication that it's hard to decide what to even care about. On Earth, we usually design our maps to look best over the major continents where people and animals live, while sacrificing barren places like Antarctica and Greenland where nothing interesting lives anyway. But on Mars, there isn't any life anywhere, so what do you prioritize? If you like science fiction, you could try to imagine what a terraformed Mars might look like, filling in the lowlands with restored oceans and imagining where the most vibrant habitats would be, but such flights of fancy have little bearing on modern Mars rovers, which can probably do approximately equally-useful science all over the planet (and indeed, we've dropped them in all sorts of places). On the moon, who's to say the South Pole-Aitken Basin isn't the patch of geography that interests you most?

I overwhelmingly prefer stereographic (double hemispheric) for the moon and Mars. Because, circular craters dominating the landscape. And for the moon it makes double sense; its tidal locking presents us with a nearly fixed hemisphere always.

— daan

[mapnerd2022]

Yes, it is the only logical choice, since it preserves circles. A circle on the globe is a circle on the map. And the circles passing through the center are a special case, for since they are straight lines, they're circles of infinite diameter.

[mapnerd2022]

Of course, unless you want a map of the Moon as seen from space, for which you would use the Azimutal Ortographic, or the Vertical Perspective.