Eisenlohr’s optimal conformal map of the world

[daan]

I know that I saw a description of the Eisenlohr projection somewhere that actually showed the "beta" projection.

Naturally I would be very interested if you could find that!

Cheers,
— daan

[quadibloc]

I thought I had seen it in something by Snyder, but it was not in Flattening the Earth. There is at least a mention of the beta projection in An Album of Map Projections.

[daan]

There is at least a mention of the beta projection in An Album of Map Projections.

I don’t see anything in the Eisenlohr entry in Album. Where should I look?

Cheers,
— daan

[quadibloc]

Yes, you're right. It's later on in the book; not on page 235, where the formula is given, and, as you say, not on page 184, where the projection is described.
No, I must be mistaken, as a search does not turn it up. And Map Projections: A Working Manual does not even mention the projection at all. Could it have been in Flattening the Earth?
In any case, I have now acted to go whatever book it was I thought I once saw one better... on my web site, not only do I picture the beta component of the Eisenlohr:

but also the alpha component, from which it is subtracted:

[daan]

Nice.

To be clear, did you have the β projection up on your site already before I posted about it?

— daan

[Milo]

I can't find it on the website even now. It's not on this page, where both August and Eisenlohr are discussed.

[quadibloc]

Yes it is; scroll about halfway down. I double-checked by going to the site myself with your link, to be sure it was uploaded.

To be clear, did you have the β projection up on your site already before I posted about it?

No. It was in response to this discussion that I decided there was a need to make these images available.

[Milo]

Yes it is; scroll about halfway down. I double-checked by going to the site myself with your link, to be sure it was uploaded.

Ah, apparently I still had it cached and had to hit reload.

[daan]

Looking into the area of E…

(Assuming I haven’t messed something up,) It comes down to 2π – 8∫₀^π/2 φ sin(φ)/√(1 + sec(φ)) dφ. The integral stumps Maxima, Mathematica, and Maple, so I presume it’s not integrable in closed form. There are ways of evaluating definite integrals that I don’t know much about but that those programs sometimes know things about. They are still stumped. I can’t say for sure that the case is closed, but it’s not looking good. I have a suspicion that, if there is a closed form, it includes a √2 factor, and once factoring that out, what’s left is not a rational multiple of π or π². I get that from the Taylor series expansion.

— daan

Shockingly, this does have a closed form, but what I have is gigantic, incredibly messy, and is subject to branch-cut difficulties when evaluating. Since the imaginary portions evaporate in the end, there must be some simpler way to express it than what I have. I’ll continue working on it, but… wow. What a mess.

To a bunch of digits:
A_E = 2.42715905403482204509

With strong confidence in all digits.

— daan

[daan]

A_E = π [ln(16) - 2]

The end.

— daan