Some compromises between cylindrical equal-area maps:
First, a few compromises that I’ve mentioned before:
A merit of Behrmann is that half of the Earth’s surface is closer to the equator than lat 30, and half of the Earth’s surface is farther from the equator than lat 30. In that way, a standard-parallel of lat 30 could be called a compromise. …even if that isn’t really what could be called an optimization.
Behrmann’s standard parallel of lat 30 is a shape-compromise between the equator and lat 41.4 By that I mean that the Shape-Accuracy (defined immediately below) at those two latitudes is the same.
I measure Shape-Accuracy at a point by min scale divided by max scale, at that point.
I abbreviate ”Shape Accuracy” as “ShAcc”.
It varies from 0 to 1.
A shape-compromise between the northernmost arctic coast of Europe, and the equator, would be achieved by a CEA that’s almost identical to Tobler CEA.
Gall Orthographic, with standard parallel at lat 45, is a shape-compromise between the equator and lat 60.
Compromise based on a global average:
The latitude at which cos^2(lat) equals the average cos^2(lat), over all the Earth’s surface is about 35.264
That qualifies as a compromise-choice for a standard parallel. An actual optimization, finding the standard-parallel that would maximize the average ShAcc over the Earth’s surface, would require solution of an equation whose solution would probably require a numerical solution-method.
…but that optimal standard-parallel is likely not much different from 35.264
So: For one thing, as mentioned, that 35.264 standard parallel, though it’s a good compromise, isn’t an optimization.
Additionally, it’s determination didn’t use a subjective rating-function applied to ShAcc.
The same compromise, estimated over land-area only:
I wanted to also calculate a compromise standard-parallel, based on an average of cos^(lat), averaged only over land.
To do that accurately would require a numerical-integration, with measurements, along many parallels, of the amount of land on each of those various parallels.
I wanted a quicker, more immediately-available estimate.
I made a very rough estimate of that, by only looking at the biggest collection of land-masses, and making a very rough assumption:
For greatest simplicity, this very rough approximating, simplifying, assumption looks only at the largest collection of the largest land-masses—the Northern Hemisphere.
At the equator, a relatively small percentage of the equator is on land. As we go north from there, Asia and Africa rapidly add to the percentage of a parallel that’s on land. Soon North America, too, becomes part of that percentage increase, as Asia continues to rapidly increase its contribution to the land-percentage of the parallels.
At the latitude of the Chukchi peninsula, that percentage is at its highest.
In other words, the percentage of a parallel that’s over land starts out small at the equator, and then steadily increases as we go north, reaching a maximum at the latitude of the Chukchi peninsula.
In other words, as the parallels become shorter, due to higher latitude, the percentage of land on those parallels is steadily and greatly increasing.
Those two effects, at least roughly, tend toward cancelling eachother out.
So, for this rough estimate, I disregarded the differences in the lengths of the parallels (...only for the purpose of calculating areas. For the calculation of area, I assumed the Earth to be a cylinder with a height of pi Earth-radii.)
With that very rough (not claimed to be accurate) assumption, the latitude at which cos^2(lat) equals the average cos^2(lat), averaged over land-area, is lat 45. So, by that questionable assumption, a standard parallel of lat 45, the standard-parallel of Gall-Orthographic, would be a good compromise (but not an optimization) for ShAcc, over the Earth’s land-area.
I’m certainly not suggesting that a map should be drawn based on that questionable assumption about land area at various latitudes. Of course the numerical integration based on actual measurement of the length of land along each of various parallels would be needed, to provide the necessary information for a good compromise standard-parallel based on land-area.
Also, of course an actual optimization, a maximization of the average ShAcc over all of the Earth (or over all of the Earth’s land-surface, as determined by measurement), as opposed to just a compromise with a standard-parallel at the latitude with the average cos^2(lat), would be much better, for making a map.
And it would be interesting, and arguably better, to apply a subjective rating-function to ShAcc at each point.
For this post, I'd have done the optimization that maximizes the average ShAcc over the entire Earth, instead of just the compromise consisting of a standard parallel for which cos^2(lat) equals the Earth's average cos^2(lat). But that requires the solution of a large equation that can only be solved iteratively.
Eventually I'll have time to do that. ...a project for later.
Likewise, I'd have done the measurements of land along various parallels, for a numerical integration to accurately maximize average ShAcc on land. But that, too, can only be a project for later, when more time is available.
Michael Ossipoff