Eisenlohr’s optimal conformal map of the world

[daan]

I’ve written a paper on this, not yet submitted for publication.

The gist of it is, I noticed that Eisenlohr’s projection (E) can be expressed in complex-valued functions as:
E = 2i (√2 sin β – β) <<< UPDATED with 2i multiplier

which yields a simple, fast Newton’s iteration for the inverse:
δ[n] = (√2 sin β[n] – β[n] – z/[2i]) / (1 – √2 cos β[n]) <<< UPDATED with i divisor
β[n+1] = β[n] + δ[n].

There, z is the x + i y coordinate in E to find the latitude φ and longitude λ for, and once you have β:
ψ = (1 ± 2 sin β cos β) / (1 + √2 sin β)²
φ = π/2 – 2 arctan |ψ|
λ = arg(ψ)

Letting r = 2 sin β cos β, the sign for ± in computing ψ is determined by noting these relationships:
{sgn λ = sgn x = sgn ℝ(z)} ≠ sgn 𝕀(r)
{sgn φ = sgn y = sgn 𝕀(z)} = sgn ℝ(r)

If sgn equalities don’t match in the computed r using +, then negate the offending ones.

A seed value of 0 works for all cases and converges to 13 digits in 7 iterations almost 97% of the time, and in 12 iterations maximum. If ever |δ[n+1]| ≥ |δ[n]|, you can discard the point as being outside of E’s range. However, not every point that converges is within E’s range. My paper describes a reasonably simple test for determining whether z is outside the range, which I omit here.

To calculate the forward case, first project the point to the simple polar stereographic:
ψ = tan(π/4 – φ/2) ∙ (cos λ + i sin λ)

Then,
β = arctan([1 – ψ] / [1 + ψ + 2√(2ψ)])

and, with enough tedious derivation, you find that you can compute √2 sin β with very little extra calculation:
√2 sin β = [1 – ψ] / [1 + ψ + √(2ψ)].

The principal square root is used in √(2ψ). Finally, as noted first:
E = 2i (√2 sin β – β) <<< UPDATED with 2i multiplier

Eisenlohr’s scale factor s at a point is given by:
s = |2 [cos λ + i sin λ] / [(sin φ + 1) ∙ (√(2ψ) + ψ + 1)²]| <<< UPDATED with 2 multiplier

There is probably no great value in the forward case over what Snyder gives in An album of map projections, where he eliminates the complex-valued calculations. The inverse calculation, however, is usefully novel, and I also have not seen any other method for computing scale value.

Final note: This expression of E gives a scale factor of 1 on the outer meridians, as Eisenlohr presented it. Usually you’d want the scale factor to be 1 at the center. To get that, you would scale the result by 3 + √8. Don’t forget to unscale when computing the inverse!

Thanks to PeteD in this thread for helping decipher Eisenlohr’s German and helping to confirm the typos in the formulas.

Cheers,
— daan

[Milo]

Might be easier to read if you explained what the different variable names you're using stand for...

[daan]

Ouch. I think it’s all in there, though? Just not all up front, necessarily. Let me know if you have any questions.

— daan

[Milo]

Ah, I think I get it. Assuming that φ and λ are latitude and longitude on the original sphere, as they usually are, the following two lines explain the meaning of ψ, and I can then work backwards from there.

φ = π/2 – 2 arctan |ψ|
λ = arg(ψ)

Presumably E is the Euclidean coordinates of the projection, i.e. x + i y. Confusingly you refer to this same value as z elsewhere (when discussing signs).

β is then an arbitrary intermediate value whose relationship is described with either E or ψ.

I'm going to try to see if I can implement this later. I'm especially interested in the scale factor bit, as I've been wanting to compare the scale factors of Eisenlohr and Lagrange.

As I side note, I found the formula for φ suspicious, so I determined that it can also be described as:
φ = -gd(ℝ(ln(ψ)))
λ = 𝕀(ln(ψ))
in reference to the Gudermannian function, which, unsurprisingly, turns up a lot in conformal projections. So essentially ln(ψ) is the Mercator projection (although with east being up and north being left, if charted with the usual orientation of the complex axes).

EDIT: And come to think of it, while -i ln(ψ) is the Mercator projection, ψ itself is the stereographic projection (centered on the north pole).

[daan]

Presumably E is the Euclidean coordinates of the projection, i.e. x + i y. Confusingly you refer to this same value as z elsewhere (when discussing signs).

I’d like folks’s opinion on that. I refer to E as the range of the projection, or as the projection definition for all values of 𝜑 and λ, whereas I use z to mean a specific point being calculated against on the x,y plane — which point, by the way, may not even be in E.

I don’t really know how to talk about this properly, and I should probably get it sorted out before I tender the paper.

Thanks.
— daan

[Milo]

Well, you seem to be using β as a single point rather than a range, so...

[daan]

Although, in my paper, I talk about β as a projection in its own right, so…
‾\_(ツ)_/‾

Maybe I’m just being incoherent. Maybe I should use E throughout and let it mean the projection or any point in it.

— daan

[Milo]

Although, in my paper, I talk about β as a projection in its own right, so…

Huh. Got a picture of that projection?

[daan]

Although, in my paper, I talk about β as a projection in its own right, so…

Huh. Got a picture of that projection?

Funny you should ask.

Cheers,
— daan

β map

beta_map_lo_res.png (167.57 KiB) Viewed 63778 times

β distortion

beta_distortion_lo_res.png (101.27 KiB) Viewed 63778 times

β map, rotated to convention

beta_map_rotate_lo_res.png (163.55 KiB) Viewed 63778 times

β distortion, rotated to convention

beta_distortion_rotate_lo_res.png (93.84 KiB) Viewed 63778 times

[Milo]

Hmm. It looks similar to Lagrange, but elliptic rather than circular? Presumably the exact aspect ratio of the ellipse is important for some deep reason.

Incidentally, you know you're on a cartography forum when the forum search page tells you "The following words in your search query were ignored because they are too common words: lagrange.".