Projection-Proposal: Behrmann-Eckert IV graft.

[daan]

Hyperelliptic with similar characteristic. Outer edges are shaped a little differently, but not to any particular consequence.

Hyperelliptic.jpg (82.92 KiB) Viewed 1924 times

α = -7
γ = 1.3
k = 7

— daan

[Atarimaster]

It always amazes me to see what the Hyperelliptic projection can do when you use the right parameters.
Geocart’s manual says Try and err… or succeed so I tried a lot but mostly, I’ve been on the err side.

[daan]

It always amazes me to see what the Hyperelliptic projection can do when you use the right parameters.
Geocart’s manual says Try and err… or succeed so I tried a lot but mostly, I’ve been on the err side.

We’re all in that camp. I need a parameter explorer in Geocart.

By the way, I did scale that hyperelliptic horizontally and vertically to give the same distortion as Behrmann at the equator. Thanks to Atarimaster for pointing that out privately.

— daan

[RogerOwens]

daan--

Thanks for the image. It looks a lot like I expect BME (Behrmann & Modified-Eckert IV) to look.

BME's pole-line is about 85% as long as its equator.

That long pole-line suggests that BME's amount of meridian-convergence won't bring much improvement. Maybe it will add more distortion than it will avoid. So I don't know if BME will bring any improvement.

Michael Ossipoff

[RogerOwens]

The Scandinavian Peninsula looks more NS/EW shape-proprtionate, than in Behrmann, but Alaska looks thinner in its portrayal's shortest dimension.

Michael Ossipoff

[RogerOwens]

...but Alaska looks more realistic in the BME approximating-image than in Behrmann.

Michael Ossipoff

[RogerOwens]

Evidently, at the central meridian, Rounded Behrmann only gains a 3% improvement over Behrmann's min scale, at lat 60, and a 7% improvement over Behrmann's min scale at the northernmost Arctic Ocean coast of Europe.

The similar map that daan posted, above on this page of this thread, does considerably better, because of its much shorter pole-line.

But that increases distortion, and lowers min scale, at the more outer longitudes, and there's question about whether one would want to favor middle longitudes at the expense of outer longitudes, on a world map. After all, a world map is intended to try to show all the world well, and, if one wants to show one particular region well, then one can best achieve that with a larger-scale map of that region only.

Michael Ossipoff

[quadibloc]

(Admittedly Eckert IV’s construction would require that the hearer be willing to listen to a bit more talk. But though not as brief, the construction-explanation is still completely explainable to someone who is curious enough to be willing to listen to all of it.)

Oh, yes. As it happens, I explained how to construct the Eckert IV projection on my web page about map projections. Basically, one has to use an iterative method to find where the parallels should go - it's basically the same method as used for the Mollweide, just with a different formula for the area.

[RogerOwens]

Yes, Mollweide and Eckert IV require iterative equation-solving to place their parallels, but the formulas, those equations, can be derived geometrically.

...using Cylindrical-Equal-Area's geometric-construction, and a purely geometric derivation of the area of a half-circle minus a segment at the top (a region cut-off by a horizontal line across the circle).

Michael Ossipoff
August 18th
34 Su
1831 UTC