Help with Eisenlohr's German paper

[daan]

Friends,

I’m trying to figure out the relationship between Snyder’s formulas for the Eisenlohr as reported in his An Album of Map Projections, p. 235, and Eisenlohr's original paper. I don’t know where Snyder’s formulation comes from; I can’t find anything like it in the literature. Actually, the literature on Eisenlohr seems to be extremely sparse.

I’m having trouble with Eisenlohr’s notation, and not reading German doesn’t help, although Google Translate does reasonably well. On p. 146, he states that t = 𝜑 is the longitude and a is the logarithm of half the tangent of the distance of the point from the north pole. On p. 148, he states that 𝜗 is the latitude. I believe he gives the x and y formulation for his projection on p. 149. We see that it is a function of a and 𝜗 only. Nowhere is the longitude evident. In the formulation for y, we see a mysterious l. after 2/√(27) whose meaning I can’t find. Next, in the description of w, he seems to use 𝜗 as a function, rather than a variable, which has me completely confused. Meanwhile, this formulation isn’t sufficiently similar to Snyder’s to tell me that Snyder’s comes from it, even factoring in trig identities. Obviously, the lack of longitude isn’t helping.

Help?

Cheers,
— daan

[PeteD]

a is the logarithm of half the tangent of the distance of the point from the north pole.

α is the logarithm of half the tangent of half the distance of the point from the north pole.

It doesn't specify which pole. If it's the south pole, then α corresponds to the y values of the Mercator projection.

I believe he gives the x and y formulation for his projection on p. 149.

Equations (5a) and (5b) on p. 148 are for the whole globe ("Setzt man b = 2 π, so erhält man eine Darstellung der ganzen Kugel"). Equation (6) and equation (7), which starts on p. 148 and continues to include the x and y formulation on p. 149, seem to be for a spherical lune of width π / 3 ("Die Annahme b = π / 3 giebt eine sehr gute Darstellung des sechsten Theils der Erdoberfläche"), which I think is where all the √27's come from.

In equation (5b), tan u is identical to Snyder's T, and if we assume that the l is a natural logarithm (which elsewhere is confusingly abbreviated lg), then v is identical to the logarithm of Snyder's V.

Solving the right-hand part of the first line of equation (5b) gives me something very similar to, but not quite the same as, Snyder's x and y formulation. There's a good chance the differences are due to me making a mistake or two with the algebra. I don't have time to recheck my working now.

Next, in the description of w, he seems to use 𝜗 as a function, rather than a variable, which has me completely confused.

I presume that refers to θ functions.

[PeteD]

I've found the time to take a second look at this and have derived the x and y formulation from equation (5b) again. I get the same result as I did the first time, so at least there are probably no random errors in my algebra. I get:

x = 2 arctan T + 2 C (V - 1/V)
y = -2 ln V - 2 C T (V + 1/V)

I don't have time to plot this right now, though.

[daan]

Many thanks, Pete. This is very helpful. I missed the similarities on Page 148, and it’s encouraging to see the longitude involved there.

The l. notation is very, very strange. Notice the l.2 in Formula 6.…

— daan

[PeteD]

I get:

x = 2 arctan T + 2 C (V - 1/V)
y = -2 ln V - 2 C T (V + 1/V)

This is what I get when I plot this:

Eisenlohr.png (331.35 KiB) Viewed 1686 times

so that's obviously wrong.

As far as I can tell, for Eisenlohr's x and y formulation to agree with Snyder's, equation (5b) would have to be changed from

x + y i = 2/i (v + u i) + 2 √2 [exp(v - u i) - exp(-(v - u i))]

to

x + y i = -2 (v + u i) + √2 [exp(v - u i) - exp(-(v - u i))] .

Either I've made a mistake somewhere or Eisenlohr has.

[daan]

As far as I can tell, for Eisenlohr's x and y formulation to agree with Snyder's, equation (5b) would have to be changed from

x + y i = 2/i (v + u i) + 2 √2 [exp(v - u i) - exp(-(v - u i))]

to

x + y i = -2 (v + u i) + √2 [exp(v - u i) - exp(-(v - u i))] .

Either I've made a mistake somewhere or Eisenlohr has.

Okay. I had a few minutes to look at this last night. Eisenlohr’s paper has a lot of typos. What I get is:
x + y i = –2 (v + u i) + √2 [exp(v + u i) – exp(-(v + u i))]. Snyder must have corrected these errors.

Thanks, Pete.

— daan

[PeteD]

What I get is:
x + y i = –2 (v + u i) + √2 [exp(v + u i) – exp(-(v + u i))].

Quite right. The fact that Snyder has + 2 C T (V + 1/V) rather than - 2 C T (V + 1/V) in his y formulation escaped my attention.

Thanks, Pete.

No problem. Glad I could help!