Tobler’s hyperelliptical projection

[Atarimaster]

Can someone who has access to Tobler's original work check if Wikipedia's formulae are correct?

Well – you, too, can have access.

[Milo]

(Incidentally, may I note that the symbols for γ and y look annoyingly similar to each other?)

To address this, I decided to use ɣ instead of γ in this post, even though it isn't technically a Greek letter as far as Unicode is concerned.

Well – you, too, can have access.

Thanks!

The link on Wikipedia didn't work, but that one does.

Whoever scanned the presumably-originally-in-print article and converted it to PDF appears to have gotten a little sloppy (for example, whether variables are italicized or not is inconsistent, and the formula at the bottom of the second page (not counting the ResearchGate splash page) failed to superscript its exponents), but the intent is still clear.

Unfortunately, the formulae are equivalent (up to rearranging of terms) with Wikipedia's ones. (EDIT: Actually, I think Tobler's version also has a mixed-up sign that's fixed in Wikipedia?)

I am still fairly certain that the formula for y is, in both cases, wrong. The listed indefinite integral (from 0 to y) should be normalized by dividing it by the corresponding definite integral (from 0 to ɣ), instead of dividing it by ɣ itself. This definite integral would have a value which is proportional to ɣ² (not ɣ itself), but which also depends on k. I can't explain how Tobler managed to draw functional projections if his formulae were wrong, however.

Tobler does describe his hyperelliptic meridians as (x/λ)^k + (y/ɣ)^k = 1, which confirms my interpretation that an aspect ratio of π/ɣ is intended (keeping in mind that λ = π at the edges of the projection).

On the first page:

In the usual notation the requirement that the parallels be horizontal lines requires that ∂y/∂λ = 0; the equal area condition is then, for a unit sphere, ∂y/∂φ ∂x/∂λ = cos φ. Two procedures are available to solve this equation, with the usual requirement for symmetry about the equator and about the mid-meridian. The simpler approach is to arbitrarily choose the spacing of the parallels on the central meridian and then to solve for the shape of the remaining meridians. Conversely, by choosing the shape of the meridians, one can solve for the spacing of the parallels.

My pixel-counting algorithm works great for any projection formulated using the second method (the one that Tobler calls more complicated). Not so much for projections formulated using the first method, but as Tobler notes, there are generally easier ways.

[daan]

Plugging in y = 0, this cancels out to:
d sin φ / dy = 1

Likewise, plugging in y = 0 to the formula for x gives x = λ.

Taken at face value, this means that the scale at the equator in both the x and the y directions is independent of the value of γ, which is clearly nonsensical if γ is supposed to control the aspect ratio. And just looking at the formula for x on its own makes it clear that it does.

Before I respond in depth, I need to understand what you’re getting at here. What you’ve shown is that the Tissot indicatrix at the equator and central meridian is undistorted. But, doesn’t say anything about what happens in the y direction away from the equator, and so that doesn’t say anything about the projection’s aspect ratio. What am I missing?

I go by the formulas in Snyder’s Flattening the Earth, in which the only substantive difference from Tobler’s paper is the sign correction that you note. The Wikipedia article matches Snyder. Of course, I don’t really know what Tobler’s intended formulas are, given that the paper could have typos, but I get results that appear to match Tobler’s… as long as we also allow for x/y rescalings of the sort that Tobler himself notes for his preferred parameterization.

— daan

[Milo]

Before I respond in depth, I need to understand what you’re getting at here. What you’ve shown is that the Tissot indicatrix at the equator and central meridian is undistorted. But, doesn’t say anything about what happens in the y direction away from the equator, and so that doesn’t say anything about the projection’s aspect ratio. What am I missing?

If you perform an affine scaling on a map projection (i.e., multiplying the x and y coordinates by different values), then both the aspect ratio of the projection as a whole, and the aspect ratio of a Tissot ellipse on the equator, will change proportionally to each other. The constant of proportionality differs depending on the underlying projection, but it remains the case that changing one changes the other, and therefore neither can be constant if the other isn't.

All pseudocylindrical equal-area projections that have the shape of a hyperellipse with the same exponent(s) must, necessarily, be affine scalings of each other. Since they can have varying aspect ratios, they must therefore also have varying distortion at the equator.

(Also, I was talking only about the equator and didn't say anything about the central meridian. The projection can't be undistorted at both the equator and central meridian, because that would be the sinusoidal projection.)

The only other thing I can see that might explain this is if Tobler did not, in fact, intend for his projections to have the shape of a hyperellipse. The formulae as written might work if the intent is to have a projection in the shape of a cropped hyperellipse:

cropped_ellipse.png (224.54 KiB) Viewed 9860 times

(Or cropped normal ellipse, in the case of this example. I made this by generating an 800:500 ellipse, then cropping it to 800:400, and then using that as the shape for a pseudocylindrical projection.)

This means that the projection would almost always have a pole line, unless ɣ has a very specific value, depending on k, whose value isn't obvious.

as long as we also allow for x/y rescalings of the sort that Tobler himself notes for his preferred parameterization.

I must have skimmed over the part that mentions that.

If postprocessing rescaling is allowed, then I must conclude that despite initial appearances, ɣ is not intended to set the projection's aspect ratio (which is done by this postprocessing instead), but rather to set the location of the pole line.

And, on closer reading, this does in fact turn out to be the case. As written on the sentence crossing the second and third pages:

The value of ɣ controls the length of the pole line,

Despite the fact that α also controls the length of the pole line in a different way.

The problem here is that the examples Tobler shows in his paper don't seem particularly inclined to having pole lines, unless they also have a nonzero α. Presumably, he just preferred cases without a pole line and carefully chose his ɣ values to avoid it, but then why have such a convoluted parameter!?

In this case, your observation that Tobler's ɣ is near the lower limit isn't an accident. To avoid having a pole line, you have to set ɣ to exactly its minimum permissible value for a given k, or as close as your floating point accuracy will permit.

[daan]

The only other thing I can see that might explain this is if Tobler did not, in fact, intend for his projections to have the shape of a hyperellipse.

That’s the answer to your riddle. He intended to use Lamé curves as a basis, but he did not intend for the results to be hyperellipses. I find Tobler’s papers to be a little breezy sometimes — such as omitting 𝛾 in mentioning the Mollweide.

As far as aspect ratio goes, I infer that your statement “proving” that the formulas are wrong presupposes that the projection is intended to be a hyperellipse, in which case, I now understand why you were confused, why I was confused about your confusion, and why your statement about aspect ratio made sense after all.

— daan

[PeteD]

That said, it's interesting to note that there is at least one important projection whose usual definition is technically a Strebe homotopy: the Lagrange projection.

I don't know what projections you consider to be important, but if you allow different values of k for each of the x and y coordinates, and if you allow postprocessing rescaling, to borrow your term, then there are lots of projections that are technically a Strebe homotopy, many of which are well-known.

In particular, the Umbeziffern transformation that preserves parallel spacing is a Strebe homotopy where the initial projection A is the plate carrée and the final projection B is the parent projection, while the Umbeziffern transformation that preserves area is a Strebe homotopy where the initial projection A is the Lambert cylindrical equal-area and the final projection B is again the parent projection. In both cases, the values of k used for the x and y coordinates of the Strebe homotopy are the n and m of the Umbeziffern transformation, respectively.

This gives a total of over 30 named projections that are technically a Strebe homotopy (again, if you allow different values of k for each of the x and y coordinates, and if you allow postprocessing rescaling).

[daan]

That said, it's interesting to note that there is at least one important projection whose usual definition is technically a Strebe homotopy: the Lagrange projection.

In particular, the Umbeziffern transformation that preserves parallel spacing is a Strebe homotopy where the initial projection A is the plate carrée and the final projection B is the parent projection, while the Umbeziffern transformation that preserves area is a Strebe homotopy where the initial projection A is the Lambert cylindrical equal-area and the final projection B is again the parent projection. In both cases, the values of k used for the x and y coordinates of the Strebe homotopy are the n and m of the Umbeziffern transformation, respectively.

Sometimes I am dense. I did not recognize this until you pointed it out. It now surprises me that Siemon and Wagner did not make it all the way to the general homotopy method that I described.

— daan

[PeteD]

Sometimes I am dense. I did not recognize this until you pointed it out. It now surprises me that Siemon and Wagner did not make it all the way to the general homotopy method that I described.

Lots of things seem obvious in hindsight. We landed on the Moon before we put wheels on suitcases.

[daan]

Lots of things seem obvious in hindsight. We landed on the Moon before we put wheels on suitcases.

[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 projections from actually feeling like a proper "midpoint".

That’s been a source of dissatisfaction to me, but I haven’t found a method that yields symmetry.

— daan