Chalker Continental Composites

[dchalker]

Thank you, Pete. Yes, that clicked much better. Still trying to go from 0 to 100 on a subject I knew nothing about a week ago! My math background topped out at graduate biostatistics, but very little geometry, so I'm having to revisit some retired neurons I haven't talked to in a decade or two. On first pass, that looks like everything I need, so I'll work on writing in your specs. Since you are declining authorship I'll include "Special thanks to Peter Denner for his invaluable assistance drafting technical portions of this paper."

Potential meridian selection, the Diomedian Antimeridian:
"The preferred antimeridian for Continental World Maps is the Diomedian Antimeridian, the International Date Line at 168°58′37″W, between the Diomede Islands also known as Yesterday and Tomorrow Island. The antipodal central meridian is the Nuremberg Meridian, 11°1′23″ East of Greenwich."

I find intentionally splitting these islands aesthetically and functionally preferable for defining the "ideal" form of the class (and nicely compatible with that monument!). On any map large enough to show them, they can also provide the eye an approximate measuring stick for comparing the two sides of the map. If you glance back and forth from Alaska to Chukchi on a large enough projection, this configuration naturally aids in perceiving how the sides would stitch together, visually connecting latitudes from the other side of the map (even on complex geometries). Gotta be pretty big to see though; St. Lawrence does this as well, helping visualize the wrap. Works well on conic, enhancing one of the better members of the class a bit further. Name sorta fits function... I doubt I'm going to find another Bering option combining such interesting characteristics as this. I welcome feedback on this selection. Maybe it's just another "obvious" one but I've searched around online and found no other proposals to use this, no "Diomedian Antimeridian" or "Antidiomedian Meridian".

Got to add Bartholomew. Apparently he is credited with the first azimuthal Antarctic maps, and thus the inset itself.

[PeteD]

By the way, the rolling thing works for the Collignon projection if you stretch it horizontally and compress it vertically, and not just at 60° S but at any latitude you choose for the interruption:

Collignon for rolling.png (125.72 KiB) Viewed 13999 times

You can also make a version with true scale along the central meridian that works with the azimuthal equidistant, again at any latitude you choose for the interruption:

Collignon for rolling ESP.png (124.97 KiB) Viewed 13999 times

Not that either of them look any good ...

[PeteD]

I've had a quick look to see whether there are any more projections for which the rolling thing works because so far you have a choice between vertically stretched cylindricals and a horizontally stretched Collignon. I couldn't find any that fit exactly for your preferred interruption latitude of 60° S. The Craster parabolic comes closest, and it turns out you need to shift the interruption to 59.00° S, which I presume would be OK:

Craster.png (193.35 KiB) Viewed 13996 times

The equivalent of the Craster parabolic with constant scale along the central meridian is Putnins P3, but then it turns out you need to shift the interruption all the way to 51.37° S, separating Tierra del Fuego from the rest of South America. However, a modified Putnins P3 can be slightly horizontally compressed so that the rolling thing works with your preferred interruption latitude of 60° S:

modified Putnins P3.png (198.6 KiB) Viewed 13996 times

This modified Putnins P3 is given by:

x = 0.3 π λ {1 − [(2 ϕ)/π]²}
y = ϕ

[PeteD]

Of course, Putnins P3 and the Craster parabolic do a terrible job of the high latitudes in the northern hemisphere. One option that's already been discussed would be to introduce a second azimuthal inset for the northern hemisphere. Another option would be to replace the top part of the northern hemisphere of the Craster parabolic with the Mollweide projection, as in the Goode homolosine, but I don't like the resulting kink, so I prefer to replace the whole northern hemisphere with a Mollweide projection modified to match the width of the Craster parabolic at the equator (which actually makes it closer to the Bromley projection than to the original Mollweide):

Craster and modified Mollweide.png (184.83 KiB) Viewed 13992 times

This modified Mollweide is given by:

x = sqrt(3/π) λ cos θ
y = [4/sqrt(3 π)] sin θ ,

where 2 θ + sin 2 θ = π sin ϕ as usual.

The equivalent of the Mollweide with true scale along the central meridian is Apian II, so we can match the modified Putnins P3 from the previous post with an Apian II projection modified to match its width at the equator:

modified Putnins P3 and modified Apian II.png (202.82 KiB) Viewed 13992 times

This modified Apian II is given by:

x = 0.3 π λ sqrt{1 − [(2 ϕ)/π]²}
y = ϕ ,

i.e. the only difference between the formulae for the top and bottom halves of the map is the square root.

[PeteD]

With a little hint from Tobias, I've worked out how to implement the interruption in d3 without it getting too confused about where to draw the boundary:

Collignon stretched for rolling.png (119.37 KiB) Viewed 13953 times

Collignon ESP stretched for rolling.png (117 KiB) Viewed 13953 times

Craster.png (128.31 KiB) Viewed 13953 times

[PeteD]

As you can see, it still draws a line to the South Pole, which isn't ideal.

[PeteD]

Putnins P3 stretched for rolling.png (126.91 KiB) Viewed 13953 times

Craster and stretched Mollweide.png (122.48 KiB) Viewed 13953 times

stretched Putnins and stretched Apian II.png (129.72 KiB) Viewed 13953 times

[PeteD]

I can't believe I didn't think of the loximuthal projection for this sooner. For any interruption latitude, you can choose the central latitude such that the rolling thing works. For an interruption latitude of 60° S, it turns out you need a central latitude of 56.79° S.

[PeteD]

loximuthal.png (124.44 KiB) Viewed 13942 times

[dchalker]

That's fantastic, man. I'm excited to feature some truly optimized representatives of the class in the upcoming work. What software were you using for the previous set, showing distortion? (I'll need that functionality to prove any benefit of north-biasing the parameters of the bases...) I'll probably need a mix of illustrations for different purposes so looks like I need to license more than one tool before it's all said and done.

I am especially fond of the "stretched Putnins and stretched Apian II". Looks great. Some of these have the best representation of Greenland I've seen, highly compatible with the design goals.

I've got a giant series of maps here that do not all need my name on them individually. Which one should I feature as the Denner Continental World Map?

I've refocused the paper just a bit now, if anyone is interested in that sort of thing:

chalkerabstract.jpg (77.45 KiB) Viewed 13928 times