Talk:Littrow projection

File:Littrow projection SW.JPG to appear as POTD

Hello! This is a note to let the editors of this article know that File:Littrow projection SW.JPG will be appearing as picture of the day on March 11, 2015. You can view and edit the POTD blurb at Template:POTD/2015-03-11. If this article needs any attention or maintenance, it would be preferable if that could be done before its appearance on the Main Page. Thanks! — Crisco 1492 (talk) 00:59, 21 February 2015 (UTC)

Picture of the day

The Littrow projection, created by Joseph Johann Littrow in 1838, is a conformal retroazimuthal map projection. It allows direct measurement of the azimuth from any point on the map to the center.Map: Strebe, using Geocart

Done, and fixed up the article a bit. Thanks, Crisco 1492.

Where did India go ?

This seems to be a useless map. Where did India go ? — Preceding unsigned comment added by Lathamibird (talk • contribs) 14:47, 11 March 2015 (UTC)

Does that mean a map of Antarctica is useless if it does not show India? Strebe (talk) 15:30, 11 March 2015 (UTC)

Uselessness under suspicion? Well... Denmark seems larger than Madagascar, so it is just another example of Eurocentric mindset. — Preceding unsigned comment added by 12.157.119.82 (talk) 16:13, 11 March 2015 (UTC)

This map is pretty blatantly Afrocentric. Ntsimp (talk) 16:22, 11 March 2015 (UTC)

"Useless" may be a little WP:POV, but it probably *is* relevant to the article to discuss why a good chunk of east and central Asia, as well as North and South America are missing from the displayed map. It also might be instructive as to the properties of the projection to show what happens when one recenters the map over different areas of the globe. From the text it looks like that is easily possible and commonly done, although given only the one figure one does not necessarily get that point. (e.g. the standard Mercator projection pretty much forces you to center on the equator - the "recenter-able-ness" of the Littrow projection is a feature which should be highlighted.). -- 160.129.138.186 (talk) 17:47, 11 March 2015 (UTC)

It might help to not look at it as a map, but as a projection. It is not a world map; it's not even a hemispheric map. In any useful application, it would only cover a few hundred square km or less. The reason the maximum extent possible for the projection is given here is so that you can see how the entire projection behaves. Strebe (talk) 01:43, 12 March 2015 (UTC)

The answer is with complex analysis. Any such conformal projection can be interpreted as holomorphic function between two Riemann surfaces. In this case, the Littrow projection has the form

$L:{\hat {\mathbb {C} }}\rightarrow \mathbb {C} ,\ L(z)=i{\frac {z^{2}-1}{2z}}$

(derivation is complicated) from the Riemann sphere ${\hat {\mathbb {C} }}$ (here taken as representing the surface of the Earth, with the South Pole at $z=0$ and the North Pole at $z=\infty$, the point $z=1$ is geographic coordinates 0N/S 0W/E, the unit circle $|z|=1$ is the Equator, and $\mathrm {arg} (z)$ increases as you go East. With that coordinate setup, the given function should produce the map shown unless we made a mistake somewhere.) to the plane, $\mathbb {C}$ (here taken as the sheet on which the map is to be projected). This function covers the plane twice, that is, the map actually is "doubled up" (so should really look like two maps superimposed with a transparency filter, if you will), thus its pullback, used to generate a legible world map, is given by

$L^{-1}(w)=-iw\pm {\sqrt {1-w^{2}}}$

taking in a point on the drawn map, $w$, and spitting out the geographical location of that point on the Earth, namely $L^{-1}(w)$. And you can see that is double-valued by virtue of the $\pm$. Depending on which sign you choose, you will get two partial maps or "leaves", with "seams" (which are branch cuts) as in the picture, depending on the ones chosen for the square root itself (usually, taken as cut along $(-\infty ,0]$). One of these maps is the one you see in the picture. The other one contains everything else it "misses", including India, Central Asia, and most of the Americas. mike4ty4 (talk) 00:06, 14 May 2017 (UTC)

EDIT: Darnit! India is on there, for the most part -- just very distorted and scrunched up near one of the two branch singularities. Look very closely at the right singularity -- where that arc of the Himalayas is, and to the left of that, you can see it. Bangladesh and the easternmost part of India (Arunachal Pradesh, etc.) are not, though, they slipped across the right branch cut onto the other leaf. mike4ty4 (talk) 23:59, 13 May 2017 (UTC)