Talk:Hammer retroazimuthal projection

File:Hammer retroazimuthal projection combined2.jpg

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Picture of the day

The Hammer retroazimuthal projection is a modified azimuthal proposed by Ernst Hermann Heinrich Hammer in 1910. As a retroazimuthal projection, azimuths (directions) are correct from any point to the designated center point. In whole-world presentation, the back and front hemispheres overlap, making the projection a surjective function. Here, the frontside and backside hemispheres, both with a 15° graticule and center point of 45°N, 90°W, are presented side-by-side.Map: Strebe, using Geocart

Purpose?

Perhaps some mention could be made of why one would use such a projection? Because the provided example image seems like a rather silly way of plotting a world map. (Although I fully admit that I may be missing some useful property of the projection.) -- 162.238.240.55 (talk) 02:00, 5 May 2014 (UTC)

As noted in the article text, As a retroazimuthal projection, azimuths (directions) are correct from any point to the designated center point. Strebe (talk) 07:28, 5 May 2014 (UTC)

The same question had occurred to me, so perhaps I might rephrase it in a slightly more direct way than 162, on the offchance that we might elicit a somewhat more comprehensive and illuminatory answer than a repetition of the text that most people would infer we have already read.

What the fuck does that mean? --77.102.114.99 (talk) 23:12, 5 May 2014 (UTC)

Despite the verbiage of that colorful interrogatory: A flat map doesn't make it obvious what the shortest path from here to there is. For instance, the direction of the shortest path from New York to China is north, not east. This map shows the direction you start out with, although it could change (the New Yorker is going south by the time he gets near China). But if you know the starting direction, you could correct that direction using the map as you progress. Why would you need the direction in the first place? 1. If Mecca is the center point, Muslims could use such a map to know which way to face during prayers. 2. If a traveler's home port is the center point, he could use such a map to find the fastest way home, although GPS software presumably handles that better these days. 3. If a transmitting antenna is the center point, others could use such a map to determine which way to aim their receiving antennas. Mentioned here. But I didn't find evidence that such maps are actually used for any of those reasons. Art LaPella (talk) 23:45, 5 May 2014 (UTC)

I applaud Art LaPella for his considerable effort on behalf of those who fantasize that strangers are obliged to and capable of exercising telepathy in divining just what about an explanation does not suffice for their understanding. Perhaps “he” succeeded. Perhaps not. For my own part, those who are too lazy and abusive to bother articulating their needs while expecting others to provide for them can simply live with their ignorance. Strebe (talk) 03:24, 6 May 2014 (UTC)

OK. I'd rather be like Bill Nye myself. Art LaPella (talk) 04:13, 6 May 2014 (UTC)

Surjectivity

"the back and front hemispheres overlap, making the projection a surjective function"

Can anyone justify calling this projection "surjective", especially after the context that part of the two hemispheres overlap? In my opinion, the first part of the sentence does not implies surjectivity. Rather, I would say the projection (a function mapping sphere surface to a plane) is only "non-injective"; that is, two points on the sphere surface can map to a same point on the plane — Peterwhy 10:24, 22 July 2014 (UTC)

If the hemispheres didn't overlap, as in a normal Mercator projection, then the function would still be technically surjective but it would also be injective. Is that what you mean? If so, then in that sense you are right; it isn't the overlap that makes it technically surjective. I say "technically" because if the projection were injective, one would be unlikely to go out of one's way to say it's surjective, because anything injective is automatically surjective. But to be hyper-correct, it wouldn't be bad to edit it to say "making the projection a surjective function but not an injective function". Art LaPella (talk) 14:32, 22 July 2014 (UTC)

Yes that is what I mean, and I assume every map projection is surjective by common understanding.* Then, is it really necessary to say this projection is surjective? Isn't it misleading to say that "the back and front hemisphere overlap" makes the projection surjective? So I suggested above, "non-injective" might be the word the original author looked for.

* Technical note: surjective function requires a definition of the co-domain, that is, a subset area on a paper. If the paper is an infinite strip of rectangle, then Mercator projection might be surjective, but Hammer retroazimuthal projection is not. Even if the co-domain paper is a circular disk, still there is a "rainbow"-shaped area that no location on sphere maps to there, hence I wouldn't even say definitely the projection is surjective. This is another minor technical reason I think surjectivity is misleading. — Peterwhy 15:37, 22 July 2014 (UTC)

• The Hammer retroazimuthal projection is surjective if the co-domain is defined to be the function's image.
• I don't think the existing text is misleading, because Strebe wrote it and I understood it. The point is that different points can be mapped to the same place. By analogy, our family has four people and two animals, even though all six would be considered Animalia if we were comparing our cells to plant cells.
• That said, I don't really object to your text either, so I'm editing it to say "non-injective". Art LaPella (talk) 18:15, 22 July 2014 (UTC)

I don't understand how your analogy relates, but anyway, thanks! — Peterwhy 15:02, 23 July 2014 (UTC)