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A loxodrome on the surface of the earth is a curve of constant bearing: it meets every parallel of latitude at the same angle. Suppose its bearing is $\theta$ north of east, i.e. due east is $\theta=0$; due north is $\theta=\text{a right angle}$; due west is $\theta=\text{a half circle}$. Its whole length as it goes from the south pole to the north pole is fairly routinely seen to be $\pi R\csc\theta$ where $R$ is the radius of the earth (in particular if it goes straight east, it circles the earth infinitely many times without getting closer to either pole, so its length is $\infty$). Let this loxodrome pass through the point whose longitude and latitude are both $0$. That is a point in the Atlantic Ocean off the coast of Africa. I had a ninth-grade teacher who told the class that that is "nowhere". So suppose one starts at "nowhere" and travels a certain distance in a certain direction along this loxodrome and arrives at geographic location $p$. Let $f(p)$ be the point in the $(x,y)$-plane reached by going that same distance in that same direction from $(0,0)$. So $f(p)\in\mathbb{R}\times[-\pi R/2,\pi R/2]$. More than one loxodrome can take you from "nowhere" to $p$, but there is a unique shortest one: the one that doesn't cross the $180^\circ$ meridian on its way from "nowhere" to $p$. If we keep all of them, making $f$ "multiple-valued", then infinitely many copies of the earth get mapped into that strip; if we keep only the shortest one, we map the earth into a sort of oval centered at $(0,0)$ and mapping the poles to $(0,\pm \pi R/2)$.

This map projection is readily seen to be far from conformal. It preserves distances along the shortest loxodrome from "nowhere" to $p$, but I doubt it preserves lengths of other loxodromes or great-circle distances.

What is this map projection called?

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As far as I have been able to determine, this projection is known as a loximuthal projection with central latitude 0. It is hard to find good references for this. The best mathematical description I have been able to locate is in this PDF.

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Thank you. I find that Wikipedia has no article titled "loximuthal projection", but it has more than 50 articles that link to that title. –  Michael Hardy Nov 1 '12 at 3:07
    
I've now created a rather stubby new article with that title. –  Michael Hardy Nov 1 '12 at 3:29
    
Maybe you can add the description you gave in the question to the Wikipedia article? It's the clearest one I have seen so far. –  Niels Diepeveen Nov 1 '12 at 23:40
    
There seems to be an image with the desired centre –  Henry Nov 2 '12 at 1:36
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