Tobler’s hyperelliptical projection

[daan]

In the "real" scenario I'm modelling, you wouldn't actually know the behavior of 𝜃. You're trying to calculate ∫ 𝑓(𝑦) d𝑦, but because of inaccuracies in your numerical integration algorithm, you're instead calculating ∫ 𝑓(𝜃(𝑦)) d𝑦. Because these inaccuracies are inherent to your integration algorithm, they cannot be separated from each other. You can calculate 𝑓(𝑦), or you can calculate ∫ 𝑓(𝜃(𝑦)) d𝑦 (and presumably you can also calculate the inverses of those expressions, through numerical solution if nothing else, if you're trying to figure out the value of 𝑦), but you cannot calculate 𝑓(𝜃(𝑦)) or ∫ 𝑓(𝑦) d𝑦 (or their respective inverses).

The scenario you're describing would only make sense if the numerical inaccuracies take place inside the calculation of 𝑓, rather than in the integration algorithm.

The equation describing 𝑥 in terms of 𝑦 is extremely simple and does not involve integration at all, and therefore you're unlikely to miscalculate that equation itself to any significant degree, even if you're plugging in the wrong value for 𝑦. Therefore, I don't see how you're getting your "two wrongs make a right" result.

I agree with that. However, that is how I arrived at my result, so the two wrongs would have been in my thinking rather than in the maths.

— daan

[justlikeoldtimes]

I've been experimenting with hyperellipticals in GeoCart, and I've skimmed this thread and Tobler's paper, and I'm wondering what exact parameters need to be entered for the "World within a hypocycloid". More specially, an astroid with a 2:1 aspect ratio.

I tried the parameters in the paper itself, but it the results were not I was looking for. I'd like to create one that could essentially "fit" inside four 2:1 Mollweides, if that makes sense. (Does it?)

Also, is there a reason why hyperellipticals like Tobler's preferred parameters can't be interrupted in Geocart? I like interrupting projections into hemispheres just so I can better visually understand what's happening with the underlying geometry. My novice mind believes there shouldn't be a reason why the Tobler's hyperelliptical can't have gores when the Sinusoidal and Mollweide projections can. But I'd understand there's already a lot of complicated calculations happening, and interrupting just adds one more layer of unneeded complexity.

[daan]

Also, is there a reason why hyperellipticals like Tobler's preferred parameters can't be interrupted in Geocart? I like interrupting projections into hemispheres just so I can better visually understand what's happening with the underlying geometry. My novice mind believes there shouldn't be a reason why the Tobler's hyperelliptical can't have gores when the Sinusoidal and Mollweide projections can. But I'd understand there's already a lot of complicated calculations happening, and interrupting just adds one more layer of unneeded complexity.

Hello. I don’t have any difficulty interrupting the hyperelliptic; it’s just like any other pseudocylindric that way:

hyperelliptic.jpg (90.99 KiB) Viewed 8085 times

What version of Geocart are you on?

Cheers,
— daan

[daan]

I've been experimenting with hyperellipticals in GeoCart, and I've skimmed this thread and Tobler's paper, and I'm wondering what exact parameters need to be entered for the "World within a hypocycloid". More specially, an astroid with a 2:1 aspect ratio.

I tried the parameters in the paper itself, but it the results were not I was looking for.

Tobler’s paper has some typos and his implementation suffers from poor numerical accuracy (it was 1973, after all), which might account for some problems. This is what I get using his parameters, and it’s very close to a 2:1 astroid:

astroid.jpg (122.69 KiB) Viewed 8083 times

Is that not what you want?

I'd like to create one that could essentially "fit" inside four 2:1 Mollweides, if that makes sense. (Does it?)

Not yet, to me.

[justlikeoldtimes]

3.3.6 win.64 on my Windows 10 laptop. i guess it was more of an issue than projection discussion.

how it normally looks (but downscaled in an another app)

a.gif (349.33 KiB) Viewed 8079 times

how it looks in "hemispheres" (it's on the top left) it weirdly shrank.

b.gif (2.69 KiB) Viewed 8079 times

actually it looks like gores sometimes works? but it's a minimum of 3, not 2.

c.gif (350 KiB) Viewed 8079 times

[justlikeoldtimes]

This is how it looks when I enter the parameters (0, 6, 1/2), compared to the original:

d.gif (266.83 KiB) Viewed 8079 times

It falls slightly short of 2:1.

But it's not what I was looking for anyway? I guess I have to find the correct parameters.

e.gif (427.29 KiB) Viewed 8079 times

[Milo]

I don't know about Geocart, but I have no trouble producing astroid maps, with the caveat that my program doesn't do graticules:

astroid1.png (186.03 KiB) Viewed 8077 times

astroid2.png (185.78 KiB) Viewed 8077 times

(I used a central meridian of 11°E for the uninterrupted one and 20°W for the two-hemisphere one, rather than just going with the prime meridian like you did, because I personally think those work best for those aspects.)

I tried the parameters in the paper itself, but it the results were not I was looking for. I'd like to create one that could essentially "fit" inside four 2:1 Mollweides, if that makes sense. (Does it?)

I don't get what you mean by this either. Wait, I think I figured it out. Unfortunately, that's not how astroids actually work.

Although the edges of an astroid look very similar to circles/ellipses "in reverse", they're actually not.

You can get even closer if you use a hyperelliptic exponent of log(½)/log(1-√½) ≈ 0.5645, but it still isn't a perfect match, as you can see if you zoom in.

plot.png (36.61 KiB) Viewed 8077 times

That said, it's easy enough to make a projection that's actually a rectangle with four quarter-ellipses cut out of it. It just won't be a hyperellipse. I'll show it in my next post, since we're limited to a maximum of three attachments per post...

[Milo]

And here they are!

inverse_mollweide.png (141.71 KiB) Viewed 8077 times

inverse_mollweide_packed.png (490.74 KiB) Viewed 8077 times

The antialiasing at the edges is a little sloppy since I glued them together in a general-purpose image-editing program, but I'm sure you can't tell by looking at it.

[justlikeoldtimes]

Thanks. I wasn't sure about that precisely because of the imprecise 1970s illustrations. Like, if something was (or this case, was not) what I thought it was. I guess I'm showing the limits of my understanding of geometry lol.

This is obviously an impractical projection, but still a thing I wanted to see for aesthetic and conceptual reasons.

[daan]

how it looks in "hemispheres" (it's on the top left) it weirdly shrank.

Puzzling. I don’t have any problem on Windows 10. Does that happen at any scale and resolution?

Thanks.
— daan