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A backwards map projection question

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#1
ScottM

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Be gentle, I'm new at this and may get terms wrong. :rolleyes:


(Attempting to attach the map in question seems to have failed, even though as a .jpg it's small enough. So look at
http://picasaweb.goo...577941279990930
and use Picasa's download option, because the usual right-click/download just gives you a useless thumbnail.)

I have a map of a (mythical) world, represented as a bitmap. The world is a ball and the map is rectangular. The bitmap may not cover every point on the sphere, but anything not represented is ocean. So, you ask, what projection was used on this map? Well, despite having drawn it, I don't know.

I know that, like any other flat map of a curved place, the map has distortions. What I'd really like to be able to do is draw this map on a ball, so I could know what the continents "really" look like. Lacking a printer that writes on spheres, I'd at least like to be able to transform it to other, more standard projections (ideally one that I can cut up and glue to a ball).

I know the following about the map: the grid marks off squares, 300 miles on a side. I'd like that measurement to remain as accurate as possible, at least over a small number of squares. The sides of the squares are not N-S, E-W - obvious from the point that both the north and south pole are marked on the map and are not even at the edges. The circumference of the sphere is 24000 mi, but that's adjustable a bit if it makes things easier. If possible, I'd like to preserve "local shape", in that I'd like any small section of this map to represent the right approximate shape of things, even if overall it's entirely wrong. My life won't end if that's not possible.

I'll be writing my own code to read this existing bitmap and do whatever transformation makes sense. I'm not afraid to code arcane formulas, or fazed by reading and writing bitmap pixels from software. I have at least a high school grasp of trig and can dust off higher math skills if absolutely necessary.

I realize the problem may be ambiguous or underspecified, and it's ok if there are several solutions that work - I'll pick whatever I think looks best. No scientific accuracy will be harmed in the conversion of this map, because there was none involved to begin with.

Formulas and insight appreciated. If the problem is completely meaningless because there's just too many variables to work with, knowing that is fine, too. Thanks in advance.

#2
mdsumner

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This is a total stab in the dark, but try loading this kml (with tif) into Google Earth, if you have it. I can't test as I don't have GE.

You can just wrap an image around a globe like GE with a little bit of coercion.

I tried with WorldWind but I've not followed that for a while and couldn't figure it out. I created this KML/TIF using Manifold - I just assume your image covers LonLat [-180, 180, -90, 90] and exported to KML.

If you can find a globe that overlays your image the way you want, you can use tools like Manifold or GDAL to georeference the image, or create world files etc. Whether that is the right interpretation of the image is another question, but you can probably find a projection that suits, and warp it if need be to get it onto a globe viewer.

If that seems promising and no one else has a better idea, I can do this with the real image, or guide you through using the tools. (You only really need the free GDAL as KML can be hacked up by hand).

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#3
woneil

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I can tell you from looking at it that there is no formula for transforming your map onto a spehere such that the positions of the poles are preserved and the grid points all remain 300 mi apart. Something is going to have to give, and there is no getting around it. You have what a mathematician calls an overdetermined system.

One thing you might want to do is to get a copy of the late John P. Snyder's Map Projections: A Working Manual. (It may be downloaded from http://pubs.er.usgs....spubs/pp/pp1395 .) Looking through it should give you an idea of what your options are.

Another thing that might help would be to experiment with Bernhard Jenny's wonderful FlexProjector < http://www.flexprojector.com/ > which will allow you to try various projections very easily.

It's not really necessary to contruct a physical sphere to see the shapes of the landmasses. There are shape-preserving projections you can use, once you get your map in a standard form. You can also project it onto a speherical model and then examine the spehere from various angles on your monitor.

Good luck.
Will O'Neil
Author and amateur cartographer

http://analysis.williamdoneil.com/w.d.oneil@pobox.com

#4
Hans van der Maarel

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I agree with woneil, it doesn't seem to be possible to construct a sphere out of this without either compromising the position of the poles or the grid.

The only projection that comes close to that map, as far as I know, is Two-point Equidistant (and that covers a circle, not a square)
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#5
ScottM

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I agree with woneil, it doesn't seem to be possible to construct a sphere out of this without either compromising the position of the poles or the grid.

The only projection that comes close to that map, as far as I know, is Two-point Equidistant (and that covers a circle, not a square)


Alright... I can see that. The one immutable point then, is the location of the poles. I'd also like to preserve the sphere's circumference, at 24000mi. The grid can go, as long as it is pretty accurate in the "straight line" between the N and S poles, which I'll call longitude zero. Other than that, I'd like a projection which is as close to conformant as possible, in that as far as coastlines are concerned, I'd like the shapes in at least small areas to roughly match. Some distortion in that regard is ok. So I gather what's going to change most radically is the apparent area of the continent shapes. That's fine.

The other critical point is that the map, as given, is only a rectangle because bitmaps are. If some of the water on the map's edges doesn't actually correspond to real places, or if there's areas of ocean on the sphere is off the edges of the map, that's entirely acceptable. Of course, it would be simple enough if I could state that the sphere was a million miles in diameter, this map represented a tiny (and therefore essentially flat) portion of the sphere, and distortion is negligible. But I want to stick to that 24000mi circumference, or at least be close.

So with these looser needs, what projection makes the most sense? I was on wikipedia and looking at Polyconic projection. It doesn't look very conformant, though. Stereographic looks interesting but I gather that it *really* distorts area near the projection point. But maybe if the projection point is in the area of ocean I don't care about...




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