Biographical Note on Friedrich Eisenlohr

[quadibloc]

I have added a small note to my page on the August Conformal Projection of the World on a Two-Cusped Epicycloid.
It turns out that in addition to the mathematician Friedrich Eisenlohr (1831-1904) who devised the Eisenlohr projection, there was also another famous individual of the same name, Friedrich Eisenlohr (1805-1854)... who perfected the modern form of the cuckoo clock!

[daan]

I almost think I would rather be known for perfecting the cuckoo clock…!

Nice write-up, John; here is the link for other readers.

Do you want the formula for the projection as a complex function? I derived that. It’s easier to work with for analysis.

— daan

[quadibloc]

Oh, certainly. For August's conformal, the complex formula is very simple: z^3 - z. (Actually, that formula only maps from the Lagrange conformal, not the sphere, though.) It would be interesting to know what it is for the Eisenlohr.

[daan]

Given:
𝜓 = tan(π/4 + 𝜑/2) * (cos 𝜆 + 𝑖 sin 𝜆), (= stereographic projection centered on south pole, central scale = ½)
𝜑 is latitude
𝜆 is longitude

Then:
E(isenlohr) = 2∙(3+2√2) [(𝜓–1) / (1+𝜓+√[2𝜓]) – arctan([𝜓–1] / [1+𝜓+2√(2𝜓)])]
dE/d𝜓 = 2∙(3+2√2) / (1+𝜓+√(2𝜓))²

So, it’s fairly simple. Central scale by this formulation is 1. The difference in the denominators in the two terms of E is correct.

— daan

[quadibloc]

E(isenlohr) = 2∙(3+2√2) [(𝜓–1) / (1+𝜓+√(2𝜓) – arctan([𝜓–1] / [1+𝜓+2√(2𝜓)])]

The difference in the denominators in the two terms of E is correct.

I'm confused. E seems to have only one term, but its denominator has two terms. The second term in the denominator somewhat resembles E itself. Is there a misplaced parenthesis somewhere?

[daan]

Ugh. I’ve updated my posting to insert the missing bracket.

I’m merely calling attention to the fact that the two denominators (1+𝜓+√[2𝜓]) and [1+𝜓+2√(2𝜓)] are not supposed to be the same, in case the difference seems suspicious. The missing parenthesis turned out to be much more suspicious!

— daan