Improving the Goode Homolosine

[quadibloc]

The Goode Homolosine improves on the Mollweide projection by replacing the part where the land is stretched vertically by the Sinusoidal projection. But that creates a kink where the two different projections are joined; at the kink, the angular distortion is infinite.
Is there a way to retain its advantages and get rid of the kink?
One way would be to mix the Sinusoidal and Mollweide projections gradually, using some formula to make the region of the join still also be equal-area.
Another way that causes a more extreme alteration of the appearance of the projection, but which is mathematically simpler, has occurred to me.
South of 40 degrees, 44 minutes, and 12 seconds North latitude... use Bonne's projection, with 40 degrees, 44 minutes, and 12 seconds North latitude as its standard parallel. Use a bent-to-fit Mollweide north of that, and drop the use of the Mollweide on the southern side.

[PeteD]

Interesting idea. Any chance you could upload an image?

A couple of years ago, I came up with a more conservative solution for getting rid of the kink and posted it here. I had to upload the images again because they've all disappeared.

[quadibloc]

Interesting idea. Any chance you could upload an image?

I'm going to try; hopefully, it won't be too complicated for me.

[Milo]

The main thing that interests me about Goode's projection is his choice of interruptions, which do a good job of showing the continents (minus Greenland and Antarctica), with only Eurasia being wide enough to suffer a particularly great amount of distortion. The actual "homolosine" idea is rather stupid and I never saw any point to it over just using either the Mollweide or the sinusoidal projection all the way through. With sinusoidal probably being better, as Mollweide's main advantage (looking better in the far northwest/northeast/southeast/southwest) is less useful in an interrupted aspect, so maintaining correct shapes near the equator is more valuable. You could also try Bromley, which looks slightly better at first glance, although that's partly just because it has the same sort of distortion that we're used to seeing on many maps (such as a squashed Scandinavia).

Ultimately, pseudocylindrical equal-area projections are pretty simple. If the outer contour is smooth (except at the poles), then the projection as a whole will be too. As such, there are only so many shapes you can experiment with. What you're essentially asking is for a geometric shape that forms a compromise between an ellipse and a paired sine wave. I have to wonder what you're trying to do. If you start with an unstretched equator, then your options for extending beyond that are limited: either your meridians curve in too quickly and you get severe shear of shapes that should be vertical (like the sinusoidal projection), or your meridians curve in too slowly and your far north gets squashed (like Lambert cylindrical). Anything between those two extremes is a compromise, and I don't think the existing state-of-the-art can be greatly improved on.

[PeteD]

I have to wonder what you're trying to do.

I'm not sure whether or not this was rhetorical and whether it was only directed to quadibloc or also to me. Obviously I can't speak for anyone else, but all I was trying to do was get rid of an annoyingly unnecessary kink.

I'm not a big fan of projections interrupted along more than one meridian (except this one and variants thereof), and if I needed a pseudocylindrical equal-area projection with point poles, I'd use Canters W34, Tobler hyperelliptical or Hufnagel 2, 3 or 4 ahead of any variant of Goode homolosine or Érdi-Krausz, with or without a kink. So I didn't have any use for the resulting projection, but the whole pointlessness of the unnecessary kink still annoyed me, and I had fun doing the maths to get rid of it.

[Milo]

I'm not a big fan of projections interrupted along more than one meridian

Fair enough. Everything's a tradeoff. In this case, it improves the appearance of each individual continent in exchange for obscuring the relative positioning of the continents. You could argue that beyond a certain point, there's no meaningful advantage over just showing the eastern and western hemispheres in completely separate maps.

if I needed a pseudocylindrical equal-area projection with point poles, I'd use Canters W34, Tobler hyperelliptical, or Hufnagel 2, 3, or 4 ahead of any variant of Goode homolosine or Érdi-Krausz, with or without a kink

Which is kind of my point.

You can't "get rid of" the kink because it is inherent to the definition of the Goode homolosine projection. If you don't want a kink, you're better off just not using the Goode homolosine projection, rather than trying to "fix" it, for some arbitrary definition of what counts as removing the kink while still staying close enough to the spirit of the original homolosine projection that you can say your version is "based" on it.

But the Goode homolosine projection specifically makes the tradeoffs of improving the internal appearance of both the equatorial lands (by keeping the equator distortion-free) and the polar ones (by avoiding excessive shear or squashing), in exchange for having severe distortion where the two meet. The actual Goode homolosine projection in fact has infinite distortion along the seam, but even you created a variant that "spreads it out" by having high-but-finite distortion over a wider area near that region, the basic principle remains: distortion is greatest near the seam. If you don't have that, then you no longer have a homolosine projection in any meaningful sense.

[quadibloc]

But the Goode homolosine projection specifically makes the tradeoffs of improving the internal appearance of both the equatorial lands (by keeping the equator distortion-free) and the polar ones (by avoiding excessive shear or squashing), in exchange for having severe distortion where the two meet.

And so it does. That is not an argument against trying to see if one can get something for nothing.
Of course, you can't usually get something for nothing. Tradeoffs have to be made, since the surface of the sphere is not reducible.
But it's possible that one could modify the Sinusoidal in a different way than abruptly changing to the Mollweide to avoid the problems at the poles; that the severe distortion where the two projections meet was a mistaken unnecessary sacrifice, in addition to the tradeoffs that are genuinely necessary to achieve the goal.
The Wagner IV projection is based on the Mollweide, and the Wagner IV projection is based on the Hammer. A similar projection, based on the Sinusoidal, and pinned to 40 degrees North and South instead of the Equator... if that were used for the caps, that would be mathematically simpler, and might do the trick.
I see the projection I'm thinking of is the Eckert VI, not something by Wagner. Except the Eckert IV, of course, uses an iterated procedure for the latitude, unlike the Wagner projections which just use the mathematics of the cylindrical equal-area, so the Eckert VI may not be suitable as a basis, since my intent is to use a Wagner-style transform.

[Milo]

Eckert VI looks superficially inspired by the sinusoidal projection, in that its boundaries are sine waves, but the nature of pseudocylindrical equal-area projections is that any change to the map's overall area propagates to the behavior over the entire projection, so Eckert VI looks different from sinusoidal even at the equator. This makes it not much of an analogue to Goode homolosine, where the entire point is that near the equator, it is exactly identical to the sinusoidal projection, and thus has benefits such as not excessively stretching Africa, which many other pseudocylindrical projections (including the fairly good ones that Pete listed) fail at.

[PeteD]

the basic principle remains: distortion is greatest near the seam. If you don't have that, then you no longer have a homolosine projection in any meaningful sense.

the entire point is that near the equator, it is exactly identical to the sinusoidal projection, and thus has benefits such as not excessively stretching Africa ...

Fair point. To be honest, it was mostly the kink in the Érdi-Krausz projection that was bothering me. Replacing the sinusoidal projection with a Wagner I variant brought the disadvantage of no longer having the sinusoidal projection near the equator. The only advantage of doing this that I could see was to get rid of the kink. However, Érdi-Krausz set the values of the free parameters of the Wagner I not only such that the kink is still there but also such that the scale factor of the two parts of the map no longer matches at the join, so rescaling is required to get the meridians to line up and the map is no longer equal-area. I found this frustrating.

[PeteD]

Eckert VI looks superficially inspired by the sinusoidal projection...

I'd say more than just superficially. Its meridians are the mean of the sinusoidal projection and the plate carrée.