Mapping Issues: Curvature, Latitude and Global Projection

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Havard » Sun Mar 29, 2009 9:10 pm

Pretty interesting work Thorf. It will take some time for me to get used to the continents appearing so different from what I am used to though...

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Mon Mar 30, 2009 1:44 am

Cthulhudrew wrote:I'm not sure I'm understanding what you're doing here exactly, at least in regards to the Hollow World. Are you warping the maps to match the way they'd fit if they were on the inside of a sphere, or as they'd appear due to curvature on the outside of a sphere (as the Outer World would)?
That's a very fair question. Does the curvature change if we're talking about the inside of a sphere rather than the outside? The official cartographers seem to have taken the position that it doesn't. I can't think how it would be any different, other than that the surface is interior rather than exterior - but it's still the same surface. (Well, for Mystara it's not, but that's only a matter of distance, not proportion.) The polar regions are another matter entirely, but for now I haven't been taking them into account.

Since quite a few people seem to be having a little trouble working out what exactly I'm trying to do, let me summarise the issues we're dealing with here.

One World, Three World Maps, Two Projections and a d20
As we all know, there are three different world maps detailing Mystara: the Master Set Outer World map, the Hollow World Set Outer World map, and the Hollow World Set Hollow World map.

The Master Set map was apparently presented in a latitude/longitude projection (aka equirectangular projection), which is to say that it just maps coordinates onto a grid of squares. It's sometimes called "unprojected", and (for us at least) it's the easiest form of map to convert into others. This is the form of surface map that we are probably most familiar with from working with computers, and with a ratio of 2:1 it is perfect for applying as a texture to a sphere in 3D programs. It has the advantage that direction is always true, but area is not - while the area near the equator is okay, the map gets more and more distorted as you move away from it. The polar regions of the map are especially warped. The only caveat with the Master Set map is that it is not in a ratio of 2:1. But since it was based on maps of the world millions of years ago, we have strong evidence that it should be. For this reason, we seem to have come to a consensus that the Far End Ocean is much larger than we have previously thought.

With the release of the Hollow World Set, the Outer World was redrawn, and we had our first - and only - glimpse of the Hollow World. Both maps are drawn in the same style, in what looks to be a Robinson projection (see also the Wikipedia article). This projection has the advantage that it presents a pleasing view of the world, but the view is pretty warped. Direction is only true along the parallels (latitude lines) and meridians (longitude lines), which means that north only points up in the dead centre of the map; at the outer edges, north points in the direction of the curved of the meridian, so that for example in the extreme west of the map it points up and right. Area is also warped, and again the polar regions suffer the most. The most important thing about this map is the terrain detail it provides, and its topographical labels.

The Hollow World map is the only view we have of the inner world, so we are stuck with it, whether we like it or not. What I have done in my recent posts is change the projection from Robinson to unprojected - the same style as the Master Set map. This allows us to see the map with true directions (north is up, south is down, east and west are left and right - inverted for the Hollow World as usual) for the first time. It doesn't solve the problems with area or scale, but we are now free to change the projection and generate new maps as we need them.

The Outer World map is a different story. It bears a strong resemblance to the Master Set version of the same map, but it introduces numerous problems too. To start with, this map is a proper projection, which means it covers the whole surface of the globe. The Master Set map did not, which means that the continents were stretched to cover this map. The original cartographers may not have realised this. The reason we are going with the Master Set map rather than this stretched map is that it seems the hex maps were all based on the "unprojected" Master Set map.

With our cartography becoming more and more accurate, I have a strong desire to fix these problems. With the Outer World, this means expanding the Far End Ocean on the Master Set map and then projecting it in whatever way we choose. For hex maps, we have chosen to go with an icosahedral projection, which is to say we're treating the world as if it were a d20. It is possible for us to generate other projections too, of course. With the Hollow World map, the first step was unprojecting the map; next we will need to reproject it, presumably using an icosahedral projection to match the Outer World. We may however need to think about how the inverted nature of the Hollow World - being the inside rather than the outside of the sphere - affects projections.

I hope this summary helps everyone understand the somewhat complex topics involved. :) Please don't hesitate to ask further questions if there's something you don't understand - or to point out mistakes, for that matter! :D

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Mon Mar 30, 2009 4:43 am

The next thing I want to talk about is the Master Set map. I'd like to move forward with it, so that we can work towards finalising our view of the world and adapting our maps to take curvature into account.

I can see a number of problems that we're going to have to solve before we can do this.

Polar Openings
The state of the polar openings can in some respects be left until later, and adapted to fit the map as necessary. But the 60-90 degree problem and the position of the start of the lip both impact the world maps quite a lot, because they determine the amount of stretching and squashing that is going on in the upper parts of the map. This is relevant to both the Hollow World and the Outer World. I'll try to boil this down into some questions we need to answer.
  1. The Lip - At what latitude does the lip begin? Is it the same latitude in both the Outer World and the Hollow World?
  2. The 60-90 Degree Problem - How are we going to handle the degrees for the polar regions? There is less land between 60 and 90 on Mystara than on Earth. Where does 90 fall on Mystara: at the lip of the polar openings; at the central point of the curve between the worlds; or at the top and bottom of the globe where this is no land?
  3. Mapping the Polar Openings - How are we going to map the polar openings? There is no standard defined way of doing so, since most cartographers don't have to deal with such strangeness.
  4. Polar Opening Size - How big are the polar openings? I believe that this can be left until later and fit to the size of the world as necessary.
Latitude
LoZompatore did a great map showing the Latitude Problem with official maps. The official info conflicts in all sorts of ways. Add to this the fact that the Hollow World Set maps are drawings, not accurately-plotted maps. At some point we will have to choose where the latitude lines actually fall - and warp the map accordingly to reflect our decisions. For the Hollow World map, I have gone with the latitudes marked on the map, and extrapolated more lines that fit roughly in between the ones provided on the map.

Outer World Maps
We have concluded that the Master Set map is an unprojected map; that the Gazetteer maps and other hex maps are based on it rather than the Hollow World Set map; that the Master Set map does not cover the whole world, and the Far End Ocean needs to be expanded to compensate; and that the world was stretched for the Hollow World Set map to fit the pseudo-Robinson projection.

Master Set Map Revisions
We need to work out how we want to fix the Master Set map; specifically, we need to create a new view of the world, and place the latitude and longitude lines appropriately. This is not hard, but I think creating a view centred on 0 longitude would be nice. This of course means centring the map on the meridian of Sundsvall.
  1. Dimensions - Do the borders of the map define the extent of the world? The map border was identical for both appearances of the map, in the Master Set and Dragon 153. We will be expanding it east-west, but are north and south acceptable as they are? Additionally, any land shown in the extreme north/south will need to be folded into the polar opening.
  2. Scale - What is the scale of the map, and what is the exact size of the world? In answering this question, we need to consider the 60-90 degree problem, and measure the area covered by the official hex maps. We can then extrapolate the size of the whole world using this information.
  3. Corrections/Updating - The map badly needs to be updated with more detailed coastlines derived from the hex maps. Some parts are way out of date, or just plain wrong - look at Alphatia, for example, the northern part of which is simply not there! This is related to working out the scale, and the precise position of the official hex maps needs to be decided and locked in before any corrections are made.
Hollow World Set Map Revisions
Despite the unhelpful stretching present on this map, we still need to use it. Aside from anything else, it gives development details and place names for many areas of the world, and it is also the key to relating the Outer World to the Hollow World. At the same time, this map suffers from some of the same problems as the Master Set map.
  1. Projection and Stretching - The map needs to be unprojected to match the Master Set map, and squashed to restore it to the proper aspect ratio to fit with the Master Set map. It may well not match up with it properly, but even a reasonably loose fit will still allow us to make use of the map's details.
  2. Corrections/Updating - This map may be some help in updating the Master Set map, especially in areas not covered by hex maps.
Preapocalyptic Map
This map in the Hollow World Set depicting the Outer World before Blackmoor's demise is the only such map published in official sources. It would be nice to keep it in line with the revised official maps, but in order to do this it will need to be processed in the same way as the regular Hollow World Set Outer World map as described above (i.e. unprojected and unstretched).

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Cthulhudrew » Mon Mar 30, 2009 9:10 am

Thorf wrote:That's a very fair question. Does the curvature change if we're talking about the inside of a sphere rather than the outside? The official cartographers seem to have taken the position that it doesn't. I can't think how it would be any different, other than that the surface is interior rather than exterior - but it's still the same surface.
I guess I was thinking that on the Outer World, the landmasses will be stretched in a different way (around) than they would in the Hollow World (where they'd be more scrunched), and that would affect the actual flattening. Ultimately, I suppose, there would be the same landmass involved, but it just seems that you'd want to use your projections for flattening in opposite ways to achieve the same ends. But maybe I'm just not seeing it correctly.
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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Mon Mar 30, 2009 10:57 am

Cthulhudrew wrote:
Thorf wrote:That's a very fair question. Does the curvature change if we're talking about the inside of a sphere rather than the outside? The official cartographers seem to have taken the position that it doesn't. I can't think how it would be any different, other than that the surface is interior rather than exterior - but it's still the same surface.
I guess I was thinking that on the Outer World, the landmasses will be stretched in a different way (around) than they would in the Hollow World (where they'd be more scrunched), and that would affect the actual flattening. Ultimately, I suppose, there would be the same landmass involved, but it just seems that you'd want to use your projections for flattening in opposite ways to achieve the same ends. But maybe I'm just not seeing it correctly.
I agree it's hard to get your head round - to the point that I'm not entirely sure I'm right. :?

But take a look at this orthographic ("view from space") projection:

Image

At a glance it looks like a normal globe, right? But that perception is all in our heads. Look again, and this time try to imagine it sinking away from you rather than coming up towards you. Isn't it the same thing? Wouldn't we normally distinguish something like this by things like lighting?

(I tried to work out a better way to show this, for example by putting lens blur in the middle part, but I couldn't get it looking like I wanted it to.)

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Cthulhudrew » Mon Mar 30, 2009 11:01 am

Thorf wrote:I agree it's hard to get your head round - to the point that I'm not entirely sure I'm right. :?
I think I see what you're saying better with that illustration; it's sort of like cutting the globe along one longitude and unrolling it to make it flat. So you'd unroll it the same way to get the inside or the outside. It's just weird to visualize, and seems counterintuitive. But I think you are right.

(Of course, now that I'm thinking about it again, I keep seeing it differently- the squishing and the squooshing to flatten things- and I'm beginning to doubt myself again. :lol: )
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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Mon Mar 30, 2009 11:19 am

Okay, I found a projection that makes it much easier to visualise.

Image

This projection really brings the Hollow World concept to life, don't you think? :D It shows more than it would ever be possible to see, of course, but it's still really cool.

Edit: I made the image bigger so you can see it a bit better. (I'm a bit leery about posting all these manipulated images... It would be good to have a replica to replace them with, so I'll have to get working on it as soon as I can.)

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Zendrolion » Mon Mar 30, 2009 1:10 pm

Astonishing results! :o

And a great job with projections, Thorf. 8-)

I long to see what the Outer World will be like in this fashion... :twisted:

About latitude issue:
Thorf wrote:The 60-90 Degree Problem - How are we going to handle the degrees for the polar regions? There is less land between 60 and 90 on Mystara than on Earth. Where does 90 fall on Mystara: at the lip of the polar openings; at the central point of the curve between the worlds; or at the top and bottom of the globe where this is no land?
I'd prefer to go with option (3): put 90-degrees latitude on the top/bottom of the globe where there's not land and adopt a special latitude marking for areas between 90-OW and 90-HW. This way latitudes on the outer world would keep some resemblance to those of our Earth (BTW, it was a solution also LoZompatore adopted time ago on the Italian MMB ;) ).
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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Gawain_VIII » Mon Mar 30, 2009 2:29 pm

Zendrolion wrote:
Thorf wrote:The 60-90 Degree Problem - How are we going to handle the degrees for the polar regions? There is less land between 60 and 90 on Mystara than on Earth. Where does 90 fall on Mystara: at the lip of the polar openings; at the central point of the curve between the worlds; or at the top and bottom of the globe where this is no land?
I'd prefer to go with option (3): put 90-degrees latitude on the top/bottom of the globe where there's not land and adopt a special latitude marking for areas between 90-OW and 90-HW. This way latitudes on the outer world would keep some resemblance to those of our Earth (BTW, it was a solution also LoZompatore adopted time ago on the Italian MMB ;) ).
I agree. What you're measureing is the degree of slope of the spherical globe--not the surface the globe sits on. So 90 degrees should be located at a point in the center of the polar opening, where the north pole would normally be if it weren't for the openings. The spot where 90 degrees is located on the surface area" of the poles would then be folded into the lip... causing (I think) for that spot to be located at the center of the lip between OW and HW. That point, however would end up being stretched in area to cover the entire line of the lip.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Mon Mar 30, 2009 2:42 pm

Gawain_VIII wrote:
Zendrolion wrote:I'd prefer to go with option (3): put 90-degrees latitude on the top/bottom of the globe where there's not land and adopt a special latitude marking for areas between 90-OW and 90-HW. This way latitudes on the outer world would keep some resemblance to those of our Earth (BTW, it was a solution also LoZompatore adopted time ago on the Italian MMB ;) ).
I agree. What you're measureing is the degree of slope of the spherical globe--not the surface the globe sits on. So 90 degrees should be located at a point in the center of the polar opening, where the north pole would normally be if it weren't for the openings. The spot where 90 degrees is located on the surface area" of the poles would then be folded into the lip... causing (I think) for that spot to be located at the center of the lip between OW and HW. That point, however would end up being stretched in area to cover the entire line of the lip.
You know, perhaps we can use the fuzziness of the (official) polar openings to our advantage here; the official maps never defined the starting latitude for the lip, which leaves us free to calculate the exact point as necessary. Moreover, we can choose roughly how much land of each world is folded into the opening (hopefully going by the official maps where possible) to fit the world size that we end up with. This may mean that we don't have to do any rescaling for the 60-90 degree areas, which I was worried we would end up having to do.

Incidentally, I am also strongly leaning towards Mystara being exactly the same size as Earth, because it really opens up so many possibilities with using real-world cartography tools (which are calibrated for the real world, obviously ;) ). I'm still not sure if it will be feasible, but we'll have to wait and see.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Mon Mar 30, 2009 2:53 pm

Zendrolion wrote:Astonishing results! :o

And a great job with projections, Thorf. 8-)
Thanks! I'm really impressed with what this program lets me do - it was definitely worth buying. :oops: :geek:

I try not to spend money on mapping too much, but in this case it was just too much to say no. Now I have to make sure I get my money's worth and make good use of it, though.
I long to see what the Outer World will be like in this fashion... :twisted:
I actually did a mock-up earlier today at work (it's spring holidays here but I still have to go to work... doesn't mean I actually do any though - nor do I have any to do! :roll: ), but it was just before I left and I didn't have time to save it and upload it. I'll do that tomorrow, hopefully.

Funnily enough it worked out looking very much like the ancient earth map Roger linked to: all ocean with all the land bunched up in a small area that seems like less than half the map. I think we will need to develop something to make the enormous ocean more interesting - even if it's just some islands.

The other thing I did today was research our icosahedral projection - also known as Fuller's Dymaxion Projection. I can't decide if it's going to be correct to put the world map onto the icosahedral net in the way I proposed before or not; the pages I read (including Wikipedia) all said that Fuller used a Gnomonic Projection to project the globe onto the icosahedron. The other problem is that for some reason I can't work out the icosahedral net is marginally smaller than the unprojected (2:1 flattened) map. So we have a little bit too much ocean. If we cut this ocean, the proportions will change. If we squash the whole map horizontally (so it's 1.9:1 or something like that) we will also change the map. And we can't scale it vertically, because then it will be too small. It's a bit of a conundrum. I will post images of this also tomorrow to make the problem clearer.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Tue Mar 31, 2009 3:40 am

Here are the images I promised last night.

First, the Master Set map. I centred the map on the meridian of Sundsvall, which means that the land is off-centre. It doesn't really matter where we put the land, of course - multiple views are equally valid, depending on what meridian you want to focus on.

Edit: Actually it looks like I put Sundsvall at -30... Perhaps because the map looks even more off-centre if it's at 0...? I'm not sure.

Image

The latitudes are set using the borders of the map as 90 N and 90 S and equally spacing the rest in between. It is possible for us to adjust this by stretching the map slightly, and we will almost certainly do this at least a little when we update the map to reflect the detail shown on the official hex maps.

Next, the icosahedral net. Earlier in this thread I introduced this image:

Image

Now here's another variation on the same thing, splitting the edge part in two so that it lines up with the map better.

Image

Unfortunately, as I was saying yesterday, it doesn't line up perfectly:

Image

The north and south regions will be squashed to fit in the triangle, so they are not a problem. What is a problem is the extra space at the far right. It seems that an icosahedral net is not quite the same aspect ratio as a flattened out sphere. It's not a great distance, but squashing it into the net will break the aspect ratio.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by nerik » Tue Mar 31, 2009 9:44 am

One thing that might help, if I can figure out how to explain it!

Your cylindrical map (the 2:1 rectangle) represents the surface of a sphere, which is curved, while the Icosahedral map is made from plane triangles. Consiuder a 'vertical' section through both, that of the sphere is a circle, while that of the icosahedron is a hexagon.

Essentially, both maps intoduce some distortion, just the Icosahedral map (I think) has less overall distortion. A better way of looking at it might be to scale your icosahedral map to the same length as the rectangular map, this will make it too tall, but will ensure that equatorial distances match up.

It occurs to me that there may exist software to do things like this.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Gawain_VIII » Tue Mar 31, 2009 12:50 pm

I don't know if this will work, but it sounds like a simple fix:

What if you kept the 2:1 ratio, but scaled the image to fit the longitudal distance--don't squish the map (that's what caused the problem in the first place). The "empty space" at the tip of each triangle will correspond to the polar openings.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Tue Mar 31, 2009 1:00 pm

Gawain_VIII wrote:I don't know if this will work, but it sounds like a simple fix:

What if you kept the 2:1 ratio, but scaled the image to fit the longitudal distance--don't squish the map (that's what caused the problem in the first place). The "empty space" at the tip of each triangle will correspond to the polar openings.
That would certainly work, but at the same time it would warp the latitudes, pushing everything towards the equator.
nerik wrote:Your cylindrical map (the 2:1 rectangle) represents the surface of a sphere, which is curved, while the Icosahedral map is made from plane triangles. Consiuder a 'vertical' section through both, that of the sphere is a circle, while that of the icosahedron is a hexagon.

Essentially, both maps intoduce some distortion, just the Icosahedral map (I think) has less overall distortion. A better way of looking at it might be to scale your icosahedral map to the same length as the rectangular map, this will make it too tall, but will ensure that equatorial distances match up.
Yes, I've been thinking this too - the discrepancy comes from the fact that an icosahedron is not a sphere, and it doesn't have the same properties as a sphere, even if it is less distorted than a flat unprojected map.

The problem with scaling it like that is that it will affect the scale of hexes within the grid, so that they also become squashed.

I don't think there's any way around adding distortion, we just have to decide what sort of distortion we dislike the least.
It occurs to me that there may exist software to do things like this.
Yes... In fact I am currently e-mailing someone who may be able to help with this. He's an expert on Dymaxion maps, and he sounds interested by our project, so hopefully he can help guide us through the process. :D

This problem has made me doubt the icosahedral net a little, but I find my faith restored when I look at this image Michele posted back on page one of this thread:

Image

We have to get to this point! It just looks right. :D

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Wed Apr 01, 2009 9:20 am

Now that we appear to be getting closer and closer to a definitive solution to the problem of taking curvature into account in our mapping, I am starting to wonder how we are going to go about adapting the official maps to fit this picture.

There are lots of options, and I'm not entirely sure which is the best. The key issues are whether or not to correct official hex maps; how to correct official hex maps; how to update/correct world maps (projected and unprojected).
  1. Existing hex maps are correct and already take projection into account; place their outlines into the projected world map (warping the overall shape of the world and especially undeveloped areas in the process) and work backwards from there to recreate a new unprojected map. Hex maps for new areas can be developed from the updated projected map.
  2. Existing hex maps do not take projection into account, based as they are on the unprojected maps; update unprojected maps with detailed info from the hex maps, then project the map and finally create hex maps that take projection into account. Official hex maps will be redeveloped while staying true to the originals. Areas above the 50 degrees North line will likely need significant squashing, while areas south of that will likely stay largely the same.
There may be other options besides these. They're a bit complicated so I'll try to explain the issues involved more clearly.

Essentially, we are working with two sets of maps: maps that take projection into account and map the world as if it were a globe (projected maps), and maps that don't adjust for projection and map the world as if it were flat (unprojected maps).

The official projected maps are the Hollow World Set world maps. They show the world in a Robinson-like projection. (Unfortunately the Outer World projected map does not fit with any other Outer World map.)

All other official maps, including the Master Set world map and all of the hex maps (which were based on the Master Set map), are unprojected maps. This means that they are made using a simple regular grid of latitude and longitude, where north is always up and east is always right, but distance is only true along the central meridian and the equator.

There is an additional problem with Hollow World hex maps, because they were based on the projected world map, but they treated it as if it were unprojected. This becomes more of a problem the further from the central meridian you go, so that Shahjapur is likely more warped than for example Azca or Milenia. Geoff's maps of Suridal and outer world Davania suffer from the same problem.

Early on in this thread, we came to a consensus that it is preferable to take curvature into account and create projected maps wherever possible. Unfortunately this means that most of our existing maps are at best problematic, at worst completely wrong. So how do we fix the problem?

This is where my choices above come in. Choice number 1 favours the shapes and areas of official hex maps, choosing to treat them as if they were actually based on a projected map and altering the rest of the world to accommodate their shapes. It might be possible to pull this off if we work creatively, and we have lots of room to spare on the Outer World which we could use to help keep the shapes we're familiar with. But invariably the world map would change in shape, with the likely result being the enlargement of the Known World area at the expense of other areas. Norwold will be particularly problematic.

Choice number 2 favours the shapes of the projected world maps, assuming them to be the new standard, and working outwards from them to fix the distortions present in other maps. This option is potentially the more difficult of the two, but it also seems like the more "correct" option. Without doing tests it's not yet known how much or in what ways the hex maps would need to be changed, so it's also a bit of an unknown quantity.

I can't commit myself to one or the other until we've seen what they entail, but right now I'm leaning towards option 2 - as long as the revisions to the hex maps aren't too jarring.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by cab » Wed Apr 01, 2009 9:27 am

Thorf wrote: I can't commit myself to one or the other until we've seen what they entail, but right now I'm leaning towards option 2 - as long as the revisions to the hex maps aren't too jarring.
Your reasoning seems sound... But because so many of us have been using the official hex maps in our campaigns for so long, you run the risk of losing compatibility with Mystara as it is actually played. As you say, it'll have a big impact on Norwold especially.

Can you instead disproportionately morph the sea areas, leaving the land less affected?

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by nerik » Wed Apr 01, 2009 9:52 am

I've done some math: assume Mystara is the same size as earth, each triangular face of the propsed icosahedral map ends up being 49.92 x 72 mile hexes high...

...seems convenient?...

...unfortunately, to fit the hex map onto the icosahedral net, we need a hex size such that the height of the triangle is a whole number of hexes plus a half.

So we could fit 49.5 x 72.6 mile hexes - which is quite close.

(or we shrink the planet by 0.8 % :shock: ).

Assuming, of course, I did my sums right!

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Wed Apr 01, 2009 2:06 pm

cab wrote:Your reasoning seems sound... But because so many of us have been using the official hex maps in our campaigns for so long, you run the risk of losing compatibility with Mystara as it is actually played. As you say, it'll have a big impact on Norwold especially.
I agree that's an issue - and frankly no one is more attached to the official maps than I am. :D But if the changes turn out to be non-extreme, it might work out okay. That's why I'm saying that we need to test it out and see how it looks before making a decision.

A compromise solution is also possible, involving a little bit of squashing and a little bit of rescaling on the world map. It does get a bit messy, though.
Can you instead disproportionately morph the sea areas, leaving the land less affected?
Well in the case of Norwold that might change the distances between coasts a bit too much. But we can certainly try - and note that this is precisely what I was referring to with option 1: keep the developed areas while adapting the world around them to fit in.

I think if anything's clear about this, it's that we're going to have to do some testing and make some examples before we can make any decisions.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Wed Apr 01, 2009 2:10 pm

nerik wrote:I've done some math: assume Mystara is the same size as earth, each triangular face of the propsed icosahedral map ends up being 49.92 x 72 mile hexes high...

...seems convenient?...

...unfortunately, to fit the hex map onto the icosahedral net, we need a hex size such that the height of the triangle is a whole number of hexes plus a half.

So we could fit 49.5 x 72.6 mile hexes - which is quite close.

(or we shrink the planet by 0.8 % :shock: ).

Assuming, of course, I did my sums right!
I don't think we need to worry about making it absolutely identical to Earth in size. Well it would be nice, but only if it really works out that way.

Incidentally, it would be great if you could post your workings as well as the result. :) And thanks for doing the calculations - it's something I'm avoiding until I absolutely have to! ;)

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by nerik » Wed Apr 01, 2009 8:03 pm

Ok, I messed up the maths in my post above :oops: , so lets try that again:

For simplicity, assume the planet is a sphere, with a mean radius of 3,959 miles (from Wikipedia).

This gives a circumference of 2 x PI x 3,959 = 24,875 miles.

Now the simplest way to determine the length of the icosahedron's edges after it is projected onto the sphere is to consider a horizontal slice through the centre of the icosahedron - this is a regular decagon, and will be equivalent to the circumference of the planet, thus each side will have a length of one-tenth of the circumference, or 24,875 / 10 = 2,487.5 miles.

As the equator cuts through the equatorial triangular faces half way up, the length of the sides of each triangle must be twice the length of the side of the decagon, or 2,487.5 x 2 = 4,975 miles.

There are almost certainly subtleties I'm ignoring just now, but I should get a reasonable estimate. Also, I would probably do better with diagrams!

Anyway, we have each edge of the Icosahedron having a length of 4,975 miles. This makes each triangle 4,975 x sqrt(3) / 2 = 4,308.5 miles. So each triangular face is 4,308.5 / 72 = 59.84 hexes high.

Rounding to the nearest odd number of half-hexes for reasons stated earlier, that would be:

59.5 x 72.4 mile hexes, or we shrink the planet by 0.6 % to 3,935 miles in radius.

The whole planet would have 47,191 hexes, and 12 pentagons! (I think).

Anyway, I hope that was useful.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Sat Apr 04, 2009 2:13 pm

How's this?

Image

Currently the projection is centred on 0 longitude. It seems to split Iciria quite nicely, but Suridal is more problematic. I'm afraid that sorting out Suridal's position would have a bad effect on Iciria.

Any thoughts? Can anyone see a better place to centre it on?

Edit: Here's the same image again, split slightly differently:

Image

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by metal » Sat Apr 04, 2009 4:07 pm

Thorf wrote:Edit: Here's the same image again, split slightly differently:
I like this view best. To me Suridal is easier to look at in that view than in the first one. ;)
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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Sat Apr 04, 2009 4:43 pm

metal wrote:
Thorf wrote:Edit: Here's the same image again, split slightly differently:
I like this view best. To me Suridal is easier to look at in that view than in the first one. ;)
Yes, that was my intention.

You know, I'm beginning to realise that part of the reason we're having such problems with curvature is that we are trying to map the whole world at once - or rather we're assuming that each map must be able to link up with the one next to it. If we were to discard this requirement, we wouldn't have to worry about curvature problems nearly so much, but instead concentrate on making nice maps of each area.

Just look at this map for a good example. The middle row of the icosahedral net is at a pretty uniform scale, directions are all true to the compass (up is north, left is west, etc.), and the land is not interrupted by divisions in the map. It's fair to say that we could easily create a continuous hex map of the region between 30 degrees north and 30 degrees south.

It's only when we introduce the idea of mapping the whole world that we start running into problems.

To put things another way, think about our own world. In an atlas and elsewhere you can see lots of different types of maps, including world maps, regional maps, local maps, and so on. They all have their own purposes - their own information to convey. You can't link up the regional maps to make a world map; the larger scale maps don't fit together like that. Moreover, most people wouldn't even think to try.

I'm really starting to think that of course we need a world map, and of course it needs to be nice and accurate, but it doesn't need to be able to show the whole world in 8 or even 24 mile hexes. 72 mile hexes on the icosahedral projection we can probably manage quite well, but look at the official maps; maybe there's a reason why TSR's cartographers decided against using hexes on their world maps.

The fact is, now that I have the data and the programs to manipulate it, we can easily create accurate projections on which to base maps of any region of Mystara, inside and out. If we take away the requirement of being able to fit all those maps together, we can instead concentrate on making each map the best it can be, showing its own region in the best possible detail. Things like the problem with Suridal in the icosahedral projection - and the fact that north and south are not always up and down - will be non-issues, because we would instead base the map on an accurate projection of a smaller area.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by OldDawg » Sun Apr 05, 2009 1:57 am

Thorf wrote: That's a very fair question. Does the curvature change if we're talking about the inside of a sphere rather than the outside? The official cartographers seem to have taken the position that it doesn't. I can't think how it would be any different, other than that the surface is interior rather than exterior - but it's still the same surface. (Well, for Mystara it's not, but that's only a matter of distance, not proportion.) The polar regions are another matter entirely, but for now I haven't been taking them into account.
[Blasted repost :x ]
Doc Math here, but yes curvature does change when you are on the inside vs. outside of a spherical shell of non-trivial thickness [I'm using the formal geometric defintion of curvature here. For perfect spheres, K=1/radius]. I get the sense, though, that you are using the term a bit more loosely to indicate the transformation effect on the relative spatial relationship of tagged points, in which case your statement is correct.

I had other technical comments on the EW distortion and a suggestion for shifting the scale of reference away from the equator to a higher latitude, but the refresh ate my homework :?

When you finally get around to the lip, you are still going to need to define what latitude means for the purposes of your project. You have two main choices
  • latitude = the angle of rise from the equatorial plane on the line segment from the planetary center to the surface point, or
  • latitude = angle of rise from the local horizon to view a fixed polar reference like the North Star [equivalently, the angle between the planetary axis and the local horizon tangent]
these definitions are functionally identical in the special case of a perfect sphere, but they differ in other circumstances (elliptical, or the lip). Case 1 will not have a 90N OW, but case 2 will. For my own work, I favor definition 2 as it is precisely how human explorers would have marked things, and it is mathematically more useful when you get into the degree of smoothness in going from OW to Lip to HW.

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