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Mercator projection

Actually the distortion was made with paper, pen, a ruler and compasses. Then I simply reported the coordinates from my paper to Publisher.

Actually, the distortion isn't entirely geometric. When doing it geometric, the hexagons spread between two triangles appear like bean-shaped concave octagones; so I twisted them a bit (basicaly replacing the two sides of the concave angle by a single side, along with doing so with the two sides opposite to the concave angle) to get convex hexagons.

Along the same lines, the pentagons forming the angles of the isocaedron were distorted by my projection and became heptagons. Again, I twisted the thing a bit to get pentagons back.
 
As for the construction:

We'll be working on one of the upper triangle of the isocaedric map: abc (with a the upper angle and b the lower left).

Let's suppose we project a point x (anyone of the points were three hexagons join).

1) Draw a line passing trough a and x. That ax line intersects bc in a point we call y.

2) From y draw a line perpendicular to bc.

3) From x draw a line parrallel to bc.

4) The two lines from steps 2 and 3 intersect in a point we call z. That point z is the projection of point x.

5) Repeat the steps 1 to 4 for every point of the triangle and you'll get a matrix of point, spread in a square.

6) Joint those point to redraw the hexagons, now distorded.

7) Repeat the operation for all 10 'outer' triangles of the map (drag and drop is fine too at this stage).

Appart from the minor twist I explain above, that's it.
 
Are you sure that's Mercator? I thought Mercator wasn't supposed to work for the polar regions (your map does however look like what you'd get if you'd made a Simple Cylindrical projection - let me check...).

Yep.In fact, it IS Simple Cylindrical. I just rendered it in Povray and it came out fine. But that's a good thing - it means that maps can now be rendered onto spheres without distortion, which is kinda useful on the visualisation front
. Here's what it looks like:

icosahedral.jpg


The only problem is that there's some wackiness going on at the corners of the icosahedral triangles (you'll note the hexes there aren't hexagonal). I dunno if that's fixable though.

Apart from that minor thing, it looks great. Nice work!


Is there any chance you could make a larger version of that available somewhere (maybe even at the T20 file library?) - that could be very useful!

Now, for your next trick, try a Mollweide projection
(I'm kidding ;)
file_23.gif
).
 
Mercator is cylindrical.

As for the distorted hexes (note that they still are hexagons, just not regular ones) at the edges of the triangles it as two reasons.

1) The minor twist I had to make to avoid the bean-shaped-octagons, as I explainded earlier.

2) And that is the main reason for the distortion. The map I drew is a projection not of a sphere, but of an icosaedron.

So when starting with the actual planet that's about spherical, you distord it once by making it an icoasedron, the a second time by projecting the isocaedron on a rectangle. If you then paste the rectangle back on a sphere, you end with distortions caused by the twists made at both stages. Note that the worst distortions occures at the edges of the triangle, just were the icosaedron is farer from the sphere.

My source document being vectorial, it shouldn't be a problem to make it as large as you want. As for posting it on the T20 file library, someone will have to explain it to me.

And Malenfant, I NEED to know how you did to paste my map on a sphere!
 
They're both cylindrical projections, but Mercator and Simple Cylindrical are not the same (I know this. I studied map projections for a while when reprojecting planetary images).

A Mercator projection stretches out to infinity at the poles. The grid looks square at the equator and gets more and more elongated in a vertical direction near the poles (it gets too stretched out to be useful around 70 degrees latitude). This is why you need separate polar-centred projections to view the high latitudes if you're given a Mercator map.

A Simple Cylindrical (AKA Plate Caree) projection can show the poles, but instead of being elongated vertically towards the poles the map squares are compressed.

Take a look at this link:
http://www.3dsoftware.com/Cartography/USGS/MapProjections/

The images there should illustrate the difference between the two projections. (this is actually the book I was using at work).

I think I'm right in saying that to make a Mercator projection, you project each point on the sphere as seen from the centre of the planet onto a flat sheet that is tangential to the equator (which is why you can't see the poles in it - they wouldn't intersect the sheet). Whereas to make a Simple Cylindrical projection, you look at the sphere from outside and project each point back onto a sheet along the line of sight. (I dunno if you can visualise that easily...)


That's fair enough re: the distortions. You're never going to get it 100% right when converting between projections anyway, it's inevitable that such things will creep in. As it is your work is still very good and useful
.


As for how I projected your map onto a sphere, simple - I used POVray. It's a free raytracer that uses a kind of programming language to design scenes. I simply made a sphere in it, used your map as a texture for it, and took a picture
. You can take a look at some of my relatively amateurish space art that I've done for it on my Art Hub, but there are plenty of other far superior images around made using POVray. It's a very versatile program and can do pretty much everything that the expensive renderers can.
 
Originally posted by Malenfant:
Are you sure that's Mercator? I thought Mercator wasn't supposed to work for the polar regions (your map does however look like what you'd get if you'd made a Simple Cylindrical projection - let me check...).

Yep.In fact, it IS Simple Cylindrical. I just rendered it in Povray and it came out fine. But that's a good thing - it means that maps can now be rendered onto spheres without distortion, which is kinda useful on the visualisation front
. Here's what it looks like:

Apart from that minor thing, it looks great. Nice work!


Is there any chance you could make a larger version of that available somewhere (maybe even at the T20 file library?) - that could be very useful!

Now, for your next trick, try a Mollweide projection
(I'm kidding ;)
file_23.gif
).
Ah that's really interesting. I tried the same thing in Bryce and got much the same results. The hex's near the poles seem to be a bit distorted, but overall it looks pretty good.

Then I tried replacing one of the default textures in Celestia but could not seem to get it to work. The .jpg files in Celestia look like they should map the same but all I get is a featureless grey sphere. :( I will have to research the documentation and see what the format should be. Putting traveller worldmaps into Celestia would be a nice visual aid.
 
Then I tried replacing one of the default textures in Celestia but could not seem to get it to work. The .jpg files in Celestia look like they should map the same but all I get is a featureless grey sphere. I will have to research the documentation and see what the format should be. Putting traveller worldmaps into Celestia would be a nice visual aid.
Another Celestia user, eh? ;)

PM me with the STC file you're using if you're stuck, I may be able to to figure out what the problem is (it might also be that the celestia.cfg file isn't pointing to your extras directory where the texture is stored?)
 
Originally posted by Malenfant:

Another Celestia user, eh? ;)

PM me with the STC file you're using if you're stuck, I may be able to to figure out what the problem is (it might also be that the celestia.cfg file isn't pointing to your extras directory where the texture is stored?)
Thanks for the offer and I might take you up on it. I havn't had any time to look at it again.

Actually I was not creating anything new. I just renamed the Mercator map to jupiter.jpg and put it into the default data directories after renaming the real jupiter.jpg. I thought that would work, but somehow it didn't. :( Creating new things for Celestia is something I have been wanting to look into, but just havn't got around to yet. Damn, I wish there was 36 hours in a day!
 
Mercator is a little more complex than a radial projection to the equatorial tangent. For that the 60° lattitude would be 1.732 times the distance from the equator as the 45° lattitude, but on a Mercator it is only about 1½.
 
Originally posted by Straybow:
Mercator is a little more complex than a radial projection to the equatorial tangent. For that the 60° lattitude would be 1.732 times the distance from the equator as the 45° lattitude, but on a Mercator it is only about 1½.
That's explained here, if anyone's interested.
 
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