Which projections you favor?

[mapnerd2022]

Yes, I saw them. Thanks for sharing your work!

It's no problem, really. Thank you for looking at them!

[PeteD]

If my calculations are correct, the Kavrayskiy-type (i.e. log squared) angular distortion is minimized for phi0 = arccos(2/e) = 42.63° for both the cylindrical equal-area and equirectangular projections.

And for the cylindrical stereographic projection*, angular distortion is minimized for phi0 = 2*arctan(sqrt(1 - 4*e^(-pi/2))) = 44.63°. It's interesting that although the value is different, the first two decimal places are the same.

*I mean the projection where phi0 = 0 gives the Braun stereographic, phi0 = 30° gives the BSAM and phi0 = 45° gives the Gall stereographic.

[PeteD]

Of course, 44.63° is so close to Gall's 45° that these two versions of the projection are probably indistinguishable.

[mapnerd2022]

Also, the Fahey Projection is close to a modification of the BSAM(Cylindrical Stereographic Projection with standard parallels at 30° N and S) since it's spacing of the parallels
is like the one in a Cylindrical Stereographic secant at the 35° parallels N and S, since obviously 35° is close to 30°.

[PeteD]

That's interesting. Are the meridians semiellipses?

The Gall–Bomford and Times projections are also modifications of the Gall stereographic

[mapnerd2022]

Yes, they are. I do also know about the two modified Galls( The Times and Gall-Bomford/Bomford Modified Gall).

[PeteD]

And there's even an equal-area projection with the same spacing of the parallels – the Foucaut projection – though I can't work out why Monsieur Foucaut ever thought it would be a good idea. I can't decide which looks worse between this and the Lambert cylindrical equal-area!

[Milo]

And there's even an equal-area projection with the same spacing of the parallels – the Foucaut projection – though I can't work out why Monsieur Foucaut ever thought it would be a good idea. I can't decide which looks worse between this and the Lambert cylindrical equal-area!

I'd have to say Foucaut. The Lambert cylindrical has a less-than-ideal choice of standard parallel, but at least it looks pretty good in the tropics, and even the temperate zones are only moderately squashed. It's only the polar regions where few people live that are blatantly terrible. Meanwhile, the Foucaut projection severely mangles even tropical regions away from the central meridian, such as South America. Even Africa, which is close to dead center (when centered on the Greenwich meridian, as most easy-to-come-by example maps are), is more distorted than in the Lambert cylindrical. Cylindrical projections have the advantage of working equally well regardless of longitude, so that, for example, Australia is distorted no worse than South Africa, something that pseudocylindrical or lenticular projections, even otherwise-good ones, can't claim.

The images I used to arrive to this conclusion:
https://map-projections.net/single-view/foucault
https://map-projections.net/single-view/lambert

That said, the Foucaut has something going for it... a little. A while back, I was interested in trying to figure out which pseudocylindrical equal-area projection would have the "least" angle distortion, by a minimax criterion. I never managed to produce any complete results, but I did figure that it would need to be an angled-pole projection, like the sinusoidal or Foucaut projections (as opposed to a flat-pole projection, like Equal Earth, or a rounded-pole projection, like Mollweide, as both of those would reach infinite angle distortion at the poles and thus clearly fail a minimax criterion). Furthermore, using some mathematical mumbo-jumbo that I could probably reproduce if I took some time but that I don't remember accurately enough to post without thouroughly recalculating it, I concluded that the angle distortion at the poles (and therefore over the whole projection, if you assume that distortion elsewhere can be kept lower than at the poles) is minimized when the pole angle is about 87°, significantly less than the sinusoidal projection, but much closer to the Foucaut projection. Unfortunately, the Foucaut's weird zig-zag countour means it probably fails the "distortion elsewhere is kept lower than at the poles" criterion, so you're probably better off with just a rescaled sinusoidal projection or something. Also, either way, we saw from the circular Hammer projection that minimaxing angle distortion is not a particulaly good way to produce good-looking equal-area maps.

[mapnerd2022]

To me, the Foucault Stereographic Equivalent looks like an even worse Sinusoidal. And the Tobler-Mercator much much worse.

[PeteD]

A while back, I was interested in trying to figure out which pseudocylindrical equal-area projection would have the "least" angle distortion, by a minimax criterion.

How does this minimax criterion differ from the one that gives the circular Hammer?