Mapping Issues: Curvature, Latitude and Global Projection

A directory of geographical maps for the world of Mystara.

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Andaire
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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Andaire » Sun Apr 05, 2009 9:36 am

OldDawg wrote:these definitions are functionally identical in the special case of a perfect sphere, but they differ in other circumstances (elliptical, or the lip)
Earth is not a perfect sphere either; do you know which definition is used?
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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Thu Apr 09, 2009 8:56 am

Is this issue getting too complicated to keep up with? We are dealing with some pretty high level cartographic concepts here, so it might take some time to digest.

In any case I'd love to hear what you think about my most recent posts - starting with the summary of the issues on page 4 of this thread, but especially my very last one at the bottom of that page.

(Edit: I got the page wrong - it's page 4, not page 3. I fixed it.)

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Thu Apr 09, 2009 9:00 am

Andaire wrote:
OldDawg wrote:these definitions are functionally identical in the special case of a perfect sphere, but they differ in other circumstances (elliptical, or the lip)
Earth is not a perfect sphere either; do you know which definition is used?
From what I've read about real world mapping, the ellipsoid nature of the earth is often deemed irrelevant and ignored for world maps and other small scale maps. It's the more detailed large scale maps that need to reflect it precisely, because even a small distortion at that scale could throw everything out of whack.

In any case, I don't intend to deal with this issue of sphere versus ellipsoid; it's simply not within my ability to do so. That we can now properly deal with spherical worlds is enough for me. :D (And I've heard that working with an ellipsoid makes everything a lot more complicated! :lol: )

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Thu Apr 09, 2009 9:13 am

One more idea: I've been considering how to deal with distortion on official hex maps a bit more, and it seems to me that our problem is really the hex grid itself. Unlike a graticule (latitude/longitude gridlines), hexes are always uniform. This is what causes the deformation problems with Geoff's Davania map: it's not actually wrong, but the hexes give a misleading impression of distance and area. Add a graticule to the map and it becomes much clearer how the map is distorted.

This hex problem can be dealt with in a number of ways. For a start, we could always warp the hexes. But that would make the maps look weird since the hex art would also be warped.

We could mark a graticule over the hexes. That would be enough in itself to show the distortion of the hexes, but depending on how it's done it might make the map look a bit weird - too crowded, perhaps.

In any case, the solution that I am currently considering is this: first, we must accept that official hex maps are all made on an equirectangular projection. This means that close to the equator they are not bad, but east-west distortion gets progressively worse the further north or south you go. Accepting this fact allows us to plot the hex maps on a world map in the same projection, and add a graticule to the hex map. Doing this allows us to georegister the map, i.e. tell the program what latitude and longitudes it depicts, and consequently reproject the map. Assuming the map is one of a reasonably small area, we can then create a new, more appropriate (i.e. less distorted) projection centred on that area. The hexes will of course be distorted. Finally, this newly re-projected map can be used to make a new super-accurate hex map of the area. This will be the Atlas map.

The great thing is that the official maps do not become obsolete as such; they're just showing the world in a different projection. We're not throwing them away; all we're doing is accepting that they are distorted in a certain way (i.e. as equirectangular projections).

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Thorf » Thu Apr 09, 2009 9:43 am

In real terms what this means is that we will end up with two sets of maps:
  1. the base maps, in equirectangular projection, including both hex maps and non-hex maps, but the hex grids are not regular on these maps.
  2. the corrected maps, in various other projections, again including both hex and non-hex maps, but on these maps the hex grids are as regular as they can be (100% accuracy is simply not possible when mapping a sphere to a flat surface, but it should be possible to make the grids a lot more accurate than they currently are - especially for small areas).
For example, I'm eager to get the Hollow World sorted out properly. The number of maps involved is very manageable, so the whole project is a great deal simpler than mapping the Outer World. The process involved is as follows:
  1. Use the 8 mile per hex maps (Azca, Nithia, Milenia and Shahjapur) to update the coastlines and terrain of the 40 mile per hex Iciria map. (Mostly they fit quite well, I believe.)
  2. Use the 40 mile per hex Iciria map to update the coastline (and terrain?) of the equirectangular version of the Hollow World map. (Some rescaling/redrawing of Iciria on the world map may well be necessary.)
    At this point, the official equirectangular maps are effectively complete.
  3. Using the equirectangular map as a reference, work out the exact coordinates displayed on the hex maps. This allows them to be georegistered.
  4. Reproject the hex maps to the desired projections, choosing the most appropriate one to minimise distortion of the area shown on each map. (This will cause the hex grid to look warped.)
  5. Redraw the map with a new hex grid overlay. This is the finished corrected map.
Really what this process provides us with is truly an Atlas of Mystara! The resulting maps will not line up perfectly with each other when put side to side; for example if Northern and Southern Iciria were done separately you would not be able to fit them together. But this drawback can be avoided by providing maps at various scales. This collection of maps will look and feel just like a real world atlas does. :D

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Zendrolion » Fri Apr 10, 2009 4:48 pm

Thorf wrote:
  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.
IIRC, LoZompatore and another Mystaran fellow, Zaryck, calculated the non-spheric curvature to begin at about 67° OW latitude. Anyway, the real "lips" should start way beyond that; in fact, what is indicated as 90° OW and 90° HW in HW set's maps could represent the start of the torus region - i.e. the parts of the OW and of the HW warping one toward the other. Perhaps the diagram showed in Voyage of the Princess Ark in Dragon #161 could help; there, "real" torus seems in fact to be located between 90° HW and 90° OW.

About the 60-90 degree problem, as I said, to me the best option is the third: consider the OW a "normal" globe and set latitudes accordingly, ignoring the polar opening for this purpose.

Finally, about polar openings' size, I think it can be guessed knowing Mystara's circumference, from the map of the planet's section featured in HW set. I'll see if eventually something is also found in VotPA.

About this issue:
  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.
What lands are folded into the polar openings is shown in Dragon #161 (at least, for the southern polar opening). Diagrams featured in HW set, showing a view of Mystara from the OW's poles can also help determine this. It seems that lands beyond 90° latitude both on OW and HW are folded into the polar openings.

Regarding OW map management:
  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.
I'd go with option (2). As you said, it's far more correct than the first, and I definitely prefer to realize real shape of projected (and hexed) lands, rather than continue to work with wrong/unprojected maps only for the sake of nostalgy.
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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Mike » Sat Apr 18, 2009 5:05 pm

Thorf wrote: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.
Hi Thorf! I am thrilled with the direction you are taking on this -- earth sized, master set map as base, icosahedral projection. I am interested in helping with this if I can.

I do not think the HW being inside should affect projections; while it is smaller it is still a perfect sphere just like the outer world, and can be mapped to an icosahedron.

I used to create and assemble icosahedral models of fantasy planets from folded paper; these models had both an outer and an inner surface. Turned the paper one way up and it is the outer world, turn it the other way up and it is the hollow world. Of course since the crust has a thickness, to be accurate you'd need two icosahedrons, one within (and slightly smaller than) the other.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Mike » Sat Apr 18, 2009 6:02 pm

AARGH! I can't count hte number of times I've finished a post and then accidentally hit some key combination that reloads the page!! Grr..
Thorf wrote: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.
That is true. An equilaterial triangle with a base of length x has a height of 0.866x. An icosahedron has the equivalent of five triangle-edges around the equatorial circumference (5x); the polar circumference is six triangles -- two edges and four faces, which is 2x + 4(0.866)x, or 5.4641x.

Put in perspective, if we assume an equatorial circumference of 24840 miles, then the polar circumference is 27145 miles. That is a large distortion, bit it is entirely due to flattening faces of the sphere. Instead of taking a curved path, you are taking linear shortcuts ("tunneling") between points on the surface. If the icosahedron is "inflated" into a sphere such that the twelve vertices lie on the surface of the sphere, the distortion disappears.

For gaming purposes I see this as an acceptable tradeoff between curved geometry and flat maps. I take the equatorial circumference as "real" and push all the distortion into the poles, for several reasons:

(1) the adventurers are more likely to visit the equator than the poles, so I am more concerned with equatorial map accuracy.

(2) if by some chance they decided to circumnavigate the globe and *measure* it, they'll likely do so around the equator not the poles.

(3) on Mystara, the polar regions are actually holes so the distortion can disappear into the ether.

(You could rotate the icosahedral net such that the equatorial circumference is the larger. I haven't tried it but it strikes me as awkward.)

My solution then would be to enlarge the icosahedral net to match the equatorial circumference and ignore the mismatch at the poles, since there are holes there anyway. A side benefit is it would give a little more "breathing room" to fit the land masses in below the edge of the polar openings.

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Re: Mapping Issues: Curvature, Latitude and Global Projection

Post by Mike » Sat Apr 18, 2009 6:35 pm

Thorf wrote:In any case, the solution that I am currently considering is this: first, we must accept that official hex maps are all made on an equirectangular projection. This means that close to the equator they are not bad, but east-west distortion gets progressively worse the further north or south you go. Accepting this fact allows us to plot the hex maps on a world map in the same projection, and add a graticule to the hex map. Doing this allows us to georegister the map, i.e. tell the program what latitude and longitudes it depicts, and consequently reproject the map. Assuming the map is one of a reasonably small area, we can then create a new, more appropriate (i.e. less distorted) projection centred on that area. The hexes will of course be distorted. Finally, this newly re-projected map can be used to make a new super-accurate hex map of the area. This will be the Atlas map.
I think this will be inevitable to some degree, but it should be possible to preserve most of the mapped areas intact and without distortion. This includes the KW and most of the savage coast, as well as equatorial areas of the HW.

This can be done by "selective compression" -- choosing areas that are less developed (or undeveloped) and compressing them more severely, in order to avoid changes to well-trodden areas. Doing so would be more an art than a science; compression can be done most easily and least noticably in deep mountain ranges, open sea, and large features such as expansive plains and forests.

Another technique is rotation. The canonical hexmap of Alphatia could be preserved almost perfectly if it is slightly deformed and partially rotated with respect to north-south; instead of compressing it, let it "follow the curve of the earth" and spread eastwards. This would change the direction of "north" in some areas but would preserve the hexmap. Skothar would need to be adjusted slightly to make room, but then Skothar is largely unmapped and rarely visited, so it seems a good tradeoff.

Like Alphatia, Norwould could be "bent" the other direction, curving westwards in order to preserve the hexmaps. (Maybe magnetic north "flows" around the polar opening so that navigators become confused the further north they get.)

Also it seems to me that the "world maps" (such as the Master Set map) are far too large scale to define local details with accuracy. I suggest that some minor local distortion/modification is permissible. Possibly the original maps had inaccuracies, the cartographer relying on legend or even imagination in some instances? This may even be true of hexmaps of remote areas... medieval maps were notoriously distorted and inaccurate! After the world hexmap is complete, a new "re-mastered" Master Set map can be created showing the corrected mountain ranges and coastlines.

To summarize I think it is vital that the known world and other well-trodden regions remain intact as far as is possible, even if this means slight deformations to coastlines or more severe compression of unknown areas.

Some may see this as heretical but I am more interested in a map that is useful for play than perfect fidelity to (conflicting) canonical sources. And I do think with with a bit of artistry, the changes can be made to look acceptable. It's interesting that when people re-project the world everyone says "wow I never realized it looked like that". Since nobody really knows what the planet looks like (yet) this is a great opportunity to make needed changes.

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Re: Mapping Issues: Curvature, Latitude and Global Projectio

Post by Gecko » Tue Nov 16, 2010 12:23 am

Gawain_VIII wrote:According to cannon, the Known World (before it was labelled as Mystara) was the exactly like our R/W Earth, except that it was perfectly spherical instead of bulging at the equator, as our own planet does.
oh wow, I had missed this little nugget back when the discussion was going on. This is news to me.

trying to think it through here-

Mystara/Urt is slightly smaller and significantly less massive than earth AND spins slightly slower (right?) but presumably more rigid, so trying to conceptualise it and, if I understand the dynamics behind the phenomenom, then that does seem likely indeed. cool.

Ultimately It wouldn't matter anyhow, even on earth the difference is only something like a third of 1 percent.

I guess my post has nothing really new to contribute to the discussion, that just caught my interest for a moment.

cheers,

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