Tuesday, September 6, 2011

Projection

I've had a long Labor Day weekend, invested mostly in furniture building - which I am bad at - and mapping India.  I've made it to the southern tip and I've just finished mapping Lanka, which is Sri Lanka in modern parlance and Ceylon on English colonial maps.  As it happens, in 1650 none of the island was controlled by the English, and was in fact in dispute between the Dutch and the Portuguese on the coast, and the Kingdom of Kandy in the interior.  The Portuguese would hang on to portions of the island until 1650.

Lanka is officially the most southern part of the world that I have mapped.  I am beginning to realize that there are certain flaws in the projection of using a flat hex map to create the half dome of the northern hemisphere, causing my world to be somewhat fatter around the equator ... or rather, that the hexes themselves at the equator are in fact only 13 and a third miles in diameter, while still technically 20 miles high.  This was unavoidable, and in fact doesn't bother me in the least.

Its the same distortion that occurs on the map below, which makes Africa and the small part of South America much larger in comparison with Northern Asia than they have a right to be.  Personally I can live with the distortion, since this is D&D and the principle aim is to create a game world, not to devise flight plans for real world aircraft.

Distorted.
It does come back to the argument I made before and before that my world is anything but 'perfect' or 'realistic.'  What it remains is extremely workable and gratifying, and thus excellent for my purposes in D&D.  If the party wants to go to Lanka, it is there ... with all the internet entries necessary to make it come alive.

The bigger problem is the Southern Hemisphere, in that it too is a flat disk like the one above, the two flat disks coming together at the equator.  The players aren't supposed to be able to tell, naturally (this involves hand-waving), but on the map itself the reverse becomes difficult to show.  Remembering that the map 'turns' every 60 degrees, the flip side on the northern view then becomes six 'leaves' that extend out from the center map towards the South Pole.  A simple version shows below:


Only much, much bigger.
If you imagine the lightest ring is the equator, and the north pole is at the centre, then everything that is south of the equator diverges into six different points.  With my actual map, the equator ring is 1,866 hexes in circumference (the actual Earth should be about 1,233), but the principle is the same.

If you're looking at a map somewhere in the middle of where one of the points meets the equator (on the larger version), it is easy to forget that the equator in the middle distorts the map.  But where the map is at 30, 90, 150 degrees and so on, it is very evident.

However, the Southern Hemisphere can also be the center of the map, with the Northern broken into six points.  This, obviously, is how I intend to represent it, since it is much easier for travel distances.

I wasn't going to go all into this, I was originally intending just to write one front paragraph and move onto other things, but what the heck.  I ran into this on the weekend when I was preparing the map for the Maldives Islands, which extends south of India ... and to get the staggering right on the next ring of large hex maps (Lanka is split between H 14 and H 15), which would be 'I', I was sketching out the shape of the map for the Great Rift valley in Africa, which is on the 30th Meridian.

I get the sneaking suspicion, however, that there's something wrong with all this, and its been bugging me since Saturday.  If someone wants to give me a poke and let me know what it is, don't hesitate.  I think I've calculated something wrong in the projection split between the points ... but I honestly don't know what it is, IF it is anything at all.  I'd like some reassurance that I've got it right, or some correction if its out there.

We all make mistakes.

UPDATE:

Kees de Kunder sent me the necessary correction to the map above:


I'll leave mine up so the difference can be noted ... and as proof positive of my lack of perfection. Thank you Kees.

2 comments:

gilgamec said...

If I remember your projection correctly, you have a twenty-mile hex centered at the north pole, then work your way south in rings of increasing number of hexes to the equator. When I tried to reproduce it, the problem I found was that (of course) it's not an even multiple of 10 miles from the pole to the equator, so you don't get a ring centered on the equator, but a partial ring. If you try to attach the maps centered on the north and south poles, then, you get a ring of something that aren't hexes at the equator. I don't know if you have a solution to this; in my case, I just slightly increased the size of the Earth to make it an even number of hexes from the pole to the equator.

Besides that, you seem to be describing the projection accurately: the biggest distortion is indeed the east-west stretching (which reaches something like 50% at the equator), with all of the angular distortion stuck along the 30-90-etc. degree lines of longitude. (Fortunately, the southern hemisphere has almost no land along these lines, so it'll be easier to navigate down there.)

Is your sneaking suspicion assuaged?

Alexis said...

Thank you Gilgamec,

Kees fixed it yesterday and I was in the middle of updating the post today when I saw your comment.

You're right about the equator. When you count down from the poles you get a fraction of a hex around the equator, which north and south makes that ring of hexes a little larger than reality. But the problem is solved by shrinking the earth a tiny bit.

Being a DM I can do that.

That angular distortion you talk about it why a lot of the maps look a bit odd ... India, I am finding, drifts towards the right the further south you go. But the solution is to remember that north is not at the "top" of the map you're looking at, but always towards the center ring.