Wagner’s Eckert IV: An obsolete projection

[Atarimaster]

Hello,

again, it was Dr. Böhm’s website that drew my attention to a certain map projection…

In Kartografische Netzentwürfe (p. 222, 1949) Karlheinz Wagner presents an alternative to Eckert IV.
Why did he do that? – Eckert IV is determined by a transcendental equation that is solvable by iteration only. And I guess that was an uncomfortable thing to do back in those days. Wagner’s alternative however is solvable algebraically. Moreover, Wagner allowed to set standard parallels.

This is Wagner’s alternative that closely approximates Eckert IV:

wagnereckert-n.png (64.47 KiB) Viewed 1840 times

This is the result with standard parallels at 50° N/S:

wagnereckert-50deg.png (65.41 KiB) Viewed 1957 times

And here’s the original Eckert IV (red lines), layered over Wagner’s close approximation (grey):

wagnereckert-compare.png (83.69 KiB) Viewed 1840 times

In this illustration, the differences are quite obvious. Nonetheless, I’m fairly certain that if you’d stumble across Wagner’s alternative in an atlas or on a wall map, most people would think that it’s the genuine Eckert IV – of course, counting only those people who actually know what Eckert IV is after all.

The formula is straightforward enough that even I had no problem at all to transcribe it to JavaScript syntax for d3-geo-projection.
For the variant that only approximates Eckert IV without the ability to set standard parallels, you need the constant n:

Code: Select all

function wagnerEckert4StaticRaw(lambda, phi) {
    var n = 0.851;

    var x = n * lambda * pow(cos(phi/2), 2),
    y = (2/n) * (phi - tan(phi/2));

    return [ x, y ];
}

In the other case, the desired standard parallel is passed to the function, so the source code’s a bit different. I’ve used a default value that results in n ≈ 0.851, like in the function above:

Code: Select all

function wagnerEckert4Raw(phi0) {
    var n = ( cos(phi0) / pow(cos(phi0/2),2) );
    
    function forward(lambda, phi) {
        var x = n * lambda * pow(cos(phi/2), 2),
        y = (2/n) * (phi - tan(phi/2));
        return [ x, y ];
    }
    
    return forward;
}
var wagnerEckert4 = function() {
  return parallel1(wagnerEckert4Raw)
      .parallel(42.21369) // standard parallels
      .scale(180.739);
};

(Notes: I’m not certain how the d3 guys come up with the default scale factor. So in this case, I simply copied & pasted the value they use for the genuine Eckert IV.
And usually, the d3 functions also include the inverse formula, but that’s beyond my mathematic capabilities.)

I guess Wagner’s alternative is obsolete nowadays, because having computers to render map projections, an iteration is the least of all problems. And setting an arbitrary standard parallel can be done by other means, as daan pointed out in a different thread.
But I think it’s interesting from a historical point of view. And that’s why I’m posting this (I’m not asking to add it to Geocart).
Hope you find it a bit interesting, too.

Kind regards,
Tobias

[daan]

Hello Tobias.

I didn't know about that projection. Thanks for calling my attention to it. You don't mention it specifically, but the Wagner alternative is, in fact, equal-area.

—daan

[Atarimaster]

Thanks for calling my attention to it.

You’re welcome!

You don't mention it specifically, but the Wagner alternative is, in fact, equal-area.

I suspected as much (since it is very similar to Eckert IV), but Wagner himself didn’t mention it specifically so I couldn’t be sure if it’s really equal-area or just close to. I guess Wagner just thought he doesn’t have to mention it…

Regards,
Tobias

[daan]

I added this to Geocart, but in the course of the implementation, I recognized Wagner’s formulation to be identical to the Nell-Hammer, which Hammer formulated in 1900 as a modification to Nell’s pseudocylindric. Hence this is not original to Wagner. Hammer only discussed the form having the equator as the standard parallel, but of course any parallel can be made distortionless at the central meridian by the techniques already discussed.

I abandoned the addition.


Maybe you should inform Dr. Böhm, Tobias?

Best,
— daan

[Atarimaster]

Thank you, that’s very interesting!
Just for the record, Wagner didn’t claim that this is a new projection or his own development, but neither he mentioned that it actually is the Nell-Hammer projection with different standard parallels. He introduced it by saying that you can avoid the difficulties of Eckert IV »with a different projection that resembles Eckert’s projection on the surface« (loosely translated).

I wonder why he neglected to credit this projection to Nell & Hammer, but, well, I guess we’ll never know…

Regards,
Tobias

[Atarimaster]

any parallel can be made distortionless at the central meridian by the techniques already discussed.

… and oh, I just realized something – I’m probably stating the obvious here, but in this case, I don’t even have to do that »k scale factor in the info palette« bit, because the width multiplier I have to enter in Geocart’s »Stretch and Rotation« dialog is Wagner’s n, namely 0.851 in the first example above (and of course using the Equal Area option).