Tobler’s hyperelliptical projection

[justlikeoldtimes]

It looks like it can do hemispheres on smaller resolutions, but not larger ones?

Tried existing saved files and changing it to Hyper, then Hemispheres, and the problem happens.

Started a new file (with the default settings), then changing it to Hyper, then Hemispheres, increased the resolution, and the same problem happens still.

The issue seems to be unique for hyperelliptical. Hemispheres work fine for Sinusoidal, Eckerts, Mollweide, Equal Earth, etc.

[quadibloc]

There is a reason that the exponent of 2.5 was "Tobler's favorite"; that's because it's the exponent used by Piet Hein when he famously proposed the hyperellipse as a shape intermediate between the true ellipse and the rectangle.

[justlikeoldtimes]

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
Is that not what you want?

It's been some time, I've accepted that the astroid is the more "organic" shape than the space between four touching ellipses (which Google does not provide a name for, if any shape name exists).

The parameters in the paper only approximate a 2:1 astroid; it does not give me the exact 2:1 shape. I'm a pixel counter. Is there an easy way to figure out what those exact parameters would be? Mathematical development hasn't been my strong suit.

I've been fascinated by the entire family of projections that could be lumped together into the hyperelliptical category. I'm trying to think what other shapes besides the rectangle, ellipse, diamond, sinusoid, and astroid are "natural" choices for similar equal areas projections. (And also their equidistant counterparts). But I think that's about it?

[Milo]

It's been some time, I've accepted that the astroid is the more "organic" shape than the space between four touching ellipses (which Google does not provide a name for, if any shape name exists).

Eh, I don't there there's anything particularly "natural" about an astroid, or hyperellipses in general. It's taking the formula for a circle/ellipse and going "hmm, what if we plug a different number instead of a 2 there?", but that's really an abuse of the formula that doesn't have any theoretical justification beyond "I can do it, so why not?". You don't encounter hyperellipses much in actual nature, unlike plain ellipses.

The parameters in the paper only approximate a 2:1 astroid; it does not give me the exact 2:1 shape. I'm a pixel counter. Is there an easy way to figure out what those exact parameters would be? Mathematical development hasn't been my strong suit.

I don't remember exactly what parameters Tobler's definition of his projection uses, and I don't feel like looking it up. As I've discussed before, I think his parametrization is somewhat awkward and more complicated than it needs to be. It's not that hard to make a "Tobler-in-spirit" projection that meets your requirements without using his exact formulae, though it probably requires numerical integration.

I've been fascinated by the entire family of projections that could be lumped together into the hyperelliptical category. I'm trying to think what other shapes besides the rectangle, ellipse, diamond, sinusoid, and astroid are "natural" choices for similar equal areas projections. (And also their equidistant counterparts). But I think that's about it?

Are you just talking about pseudocylindrical projections? Because a sinusoid isn't a hyperellipse.

Pseudocylindrical equal-area projections can be made in the shape of anything. Take a look at this one. (Arguably, a pseudocylindrical projection isn't the best choice for that shape, since it lacks sixfold symmetry... but it's what I have!)

[justlikeoldtimes]

I wasn't sure about Sinusoidal. I didn't mean to imply that it definitely was a hyperelliptical. But I think, like Mollweide, Lambert EA (and Smyth ES), Collignon, there's an appealing simplicity to it even if it's not immediately aesthetically pleasing.

I'm all for alternative ways generating hyperelliptical projections (as well as other equal area maps); but I'm sticking with the tools I have for now, mainly GeoCart.

[daan]

The parameters in the paper only approximate a 2:1 astroid; it does not give me the exact 2:1 shape. I'm a pixel counter. Is there an easy way to figure out what those exact parameters would be? Mathematical development hasn't been my strong suit.

Being equal-area, you can stretch it in one direction and compress it reciprocally in the other to retain overall area and differential area, as well as get your exact 2:1 ratio. Maybe that doesn’t fit your idea of an astroid…?

— daan

[justlikeoldtimes]

Yeah it does. Sorry.

[Milo]

That looks like a hyperellipse with exponent ½, which would make sense to call an "astroid" in as far as that it looks starlike, but Wikipedia claims that the name "astroid" is officially reserved specifically for the hyperellipse with exponent ⅔, and not similar-looking shapes. The exponent ⅔ at least has the unique property that in addition to being a hyperellipse, it is also a hypocycloid, which hyperellipses in general are not.

(Neither exponent will produce an "inverted ellipse" shape, so don't worry about that!)

[daan]

I’d call it obsolete after my homotopy, letting you blend any two projections.

I've read about that one. It's pretty clever, but I do find it a little inelegant due to its lack of symmetry, which prevents a 50% blend of two

You know what? I used to agree, but I have wasted far too much time on the symmetry “problem” and after all that, I think that the quest for symmetry is poorly motivated: There are good topological reasons to want asymmetry. Consider a cylindric projection (and therefore each pole is a line the width of the map) and a pseudocylindric, with each pole being a point, as the two terminal projections in the homotopy that I created. When A is the cylindric and B is the pseudocylindric, the pole-line on (k – 1) × A + k × B merely shrinks as k grows, but there is always a pole-line until k = 1. If you swap the two projections for A and B, then you have never have a pole line as k grows until k = 1. Let’s say, now, that we had a symmetrical homotopy. In that case, what is the topology? What happens to that pole-line? If it’s always a pole-line, then it’s not different from the first case in any interesting way. If it’s never a pole-line, then it’s not different from the second case in any interesting way. Whereas, if it’s a pole-line partway through the progression but then begins deviating (where, at k = ½?), that seems very strange indeed, and even then, that’s not different than just swapping the two halfway along the way. Presumably you could do that smoothly in a hypothetical symmetrical method, but still, it smells arbitrary.

More generally, dealing with singularities doesn’t have any symmetrically coherent solution. I think my method more honestly acknowledges the topology and lets you choose your biases for resolving such problems.

Cheers,
— daan

[Milo]

Well yeah, there are multiple ways to blend maps together. But then, I could also ignore both directions of your homotopy and use some other method entirely.

So long as it unavoidably comes in multiple variations, you can't really claim it's the One True Way of blending maps and that all other ways are obsolete.