Small World (J. Richard Gott III)

[quadibloc]

I decided that since the Gott projection isn't itself pseudocylindrical, I would follow the precedent set by where I placed the discussion of certain other projections on my web site, and so I moved the discussion of it from the page on the Mollweide projection to that discussing the Dietrich-Kitada projection (since it was in the same section as also included the discussion of the Hammer projection, of a similar general type).

I did this after encountering the name of J. Richard Gott III in another context entirely.

It turns out he was responsible for popularizing, after independently re-inventing, the so-called "Doomsday Argument"; I added a discussion of that, along with Bayesian statistics, to my web site, and then checked the name.

[Atarimaster]

Nice work again, thanks a lot!
And I never heard about the Doomsday Argument, so thanks again for pointing me to an interesting theory.

By the way – I hope you don’t mind if I stray a bit off-topic here –, since you mention the Atlantis projection in your discussion, there’s one thing that I’ve always been curious about:
Did Bartholomew ever explain why he chose 45°N/30°W as the projection center?
The thing is, if you slightly shift the center to 50°N (and 30°W again), New Zealand doesn’t get interrupted anymore, and I can see no striking disadvantage. Of course, some areas (e.g. the Antilles) slide into an area having a greater amount of angular distortion… but not much.

Did he simply not care about New Zealand’s interruption – or is there some advantage of his configuration that I’m missing?

[quadibloc]

If Bartholemew did explain, I didn't see it, but then I haven't read much of what he wrote.

I think you raise a very good question. However, remember that the Mollweide is a pseudocylindrical projection, so nothing is stopping it from being continued beyond its normal edges to draw the rest of New Zealand, albeit with a little more distortion, in the existing aspect.

And, in fact, I purchased a copy of the relevant edition of the Edinburgh Atlas of Advanced Geography from AbeBooks because I wanted to be sure of the original design of the Regional Projection, and I see Bartholemew did exactly that. As well, the map for which he devised the Atlantis projection... was not a particularly detailed map. It basically just showed the outlines of the continents... and world air routes.

Personally, from your diagrams, I'd be inclined to say that 55° N and 30° W would probably be even better (but 60° N would be going too far).

[daan]

Did Bartholomew ever explain why he chose 45°N/30°W as the projection center?

I am going to guess it’s because calculations are easier at 45°.

Your caption reads 20°, but that must be a typo. The map is at ~50°.

— daan

[daan]

I did this after encountering the name of J. Richard Gott III in another context entirely.

It turns out he was responsible for popularizing, after independently re-inventing, the so-called "Doomsday Argument"…

I hadn’t connected the two. Thanks!

— daan

[Atarimaster]

However, remember that the Mollweide is a pseudocylindrical projection, so nothing is stopping it from being continued beyond its normal edges to draw the rest of New Zealand, albeit with a little more distortion, in the existing aspect.
And, in fact, I purchased a copy of the relevant edition of the Edinburgh Atlas of Advanced Geography from AbeBooks because I wanted to be sure of the original design of the Regional Projection, and I see Bartholemew did exactly that.

Meanwhile I’ve googled a bit and found this map, which indeed has a little “extension” for New Zealand. At first, I wondered why he did that instead of chossing a slightly different map center, but…

I am going to guess it’s because calculations are easier at 45°.

Sounds reasonable. After all, they didn’t have computers which will do the dirty work for you.

Your caption reads 20°, but that must be a typo. The map is at ~50°.

Ooops, you’re right. I just corrected that, thank you!

Kind regards,
Tobias

[quadibloc]

I am going to guess it’s because calculations are easier at 45°.

I'm inclined to agree with you for the following reason: The oblique Mollweide devised by Fairgreve in 1928, and the oblique Mollweide in the orientation later used for Bartholomew's Nordic projection by O. M. Miller in 1944, also both used an inclination of 45 degrees.

But I had to also ask myself: in what way is an inclination of 45 degrees easier to calculate?

When drawing the graticule, for any inclination, symmetries would allow calculating only 1/4 of the points, so it doesn't seem as if a new symmetry halves the number of points to be calculated.

But the sine and cosine of 45 degrees are both equal to half the square root of 2. So if one calculates the right two points at once, the multipications can be shared between them.

[daan]

But the sine and cosine of 45 degrees are both equal to half the square root of 2. So if one calculates the right two points at once, the multipications can be shared between them.

And more. In order to calculate the transformed longitude of the oblique projection, the tangent form (See Snyder, Map projections—a working manual, p. 31) gets to use the computed sine of the transformed latitude as its divisor for the 45° case but not in the general case, due to the equality of sin and cos.

— daan