[Mapping Issues] World Dimensions

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[Mapping Issues] World Dimensions

Postby Gawain_VIII » Fri Mar 16, 2012 8:09 am

After discussion with Bruce on FB, I double-checked all my math and came up with the following:


All measurements in miles (or square miles for area), numbers are rounded and may not add up precisely.

*The HW Land area is calculated from the total surface area WITH the "missing" poles (since PWA3 didn't specify where the 40% came from). If calculated without poles, the land area is 42M square miles.

**Help me out with the Real-world surface areas. I took the figures from Earth's Wikipedia article which lists the measurements in metric. Converting km to miles of the circumference nearly matches. Calculating the surface area in km without converting, and my calculations match Wikipedia's article (within reasonable rounding error--I calculated the area given both radius and circumference, seperately, for comparisson). If I convert to miles before calculating area, I match the large-Mystara figures. If I convert AFTER calculating surface area, the numbers don't match up. The conversion rate between km and mi. didn't change. The forumla for calculating the area of a sphere (regardless of unit) doesn't change. How come I'm getting different numbers depending on when I convert the units? Rounding errors doesn't account for THAT big of a difference.



--Roger
Last edited by Thorf on Mon Jan 25, 2016 9:30 am, edited 3 times in total.
Reason: Added Mapping Issues to the thread title.
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Re: World Dimensions

Postby Birchbeer » Fri Mar 16, 2012 1:12 pm

I encountered weird issues converting square miles into square kilometers before (for a kingdom generator in km). The normal conversion is 1 mile = 1.609344 kilometer. For squared miles the conversion works like this: 1 sq Mile = 2.589988110336 Square Kilometers... basically you just square the kilometer. I hope this helps :)
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Re: World Dimensions

Postby Big Mac » Fri Mar 16, 2012 1:49 pm

There is an earlier thread called: Size of Mystara, which may be of interest.
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Re: Mapping Issues: World Dimensions

Postby Thorf » Mon Mar 26, 2012 10:29 am

Thanks for starting this thread, Roger. We've needed a place to discuss this for a while now, and the topic of sorting out the world maps is complicated enough that I think we can benefit from splitting it up a bit. I'd like to post the official stats for the world here for reference (there are various numbers depending on source). Do you mind if I add them to your original post at some point later to keep them accessible?

Getting down to business, today I did a preliminary check of the Hollow World map with the Iciria hex map overlaid. I counted 61.5 hexes from 90 degrees S to the equator, but 62 from there to 90 degrees N. The discrepancy will disappear when using the equirectangular map, so for now let's consider that a north pole to south pole measurement of 123 hexes. Treating the world as if it were a true sphere, and ignoring the fact that we have to wrap the last 15-25 degrees of land into the polar openings, this means the circumference of the world (again, assuming a perfect sphere) is 246 hexes. With 40 miles per hex, we can conclude that:

Hollow World circumference = 9,840 miles

There were three different official measurements: 11,908 (Hollow World Set, Poor Wizard's Almanacs II and III); 11,900 (Champions of Mystara); and 9,448 (Poor Wizard's Almanac). It's somewhat puzzling where these came from. The Poor Wizard's Almanac figure could well be from hex counting, as it's reasonably close. But the other two likely stem from the original poster map, and where that figure originated, we don't know.

So, the Hollow World may actually be a tad smaller (or bigger if you believed the first Almanac) than we thought.
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Re: Mapping Issues: World Dimensions

Postby Thorf » Mon Mar 26, 2012 1:51 pm

Okay, I had another go at lining up the maps, with this result:

Image

It's not perfect, but it's pretty good. The result is that the Hollow World is 125 hexes from north to south, resulting in the rather nice figure:

Hollow World circumference = 10,000 miles

Only an Immortal could be so precise. :D

In any case, while doing this it occurred to me that the hex maps for the Hollow World are going to change rather a lot when they are reprojected, because currently they are based on this pseudo-Robinson projection. While the Outer World maps are based on an equirectangular map which places north, south, east and west at top, bottom, left and right, the Hollow World's Robinson projection is less simple. I already showed this before with Suridal and Geoff's Selhomarr, which is one of the extreme cases, but it now looks as if it's going to affect Nithia, Azca, Milenia and Shahjapur too - in fact, all of the regions that have been mapped will change quite a bit. A good example of this is that the very flow of the River Nithia will change quite significantly.
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Re: Mapping Issues: World Dimensions

Postby Ashtagon » Mon Mar 26, 2012 2:00 pm

Just to throw a spanner in the works, is there anything to indicate that the HW's equatorial circumference must be the same 9or even vaguely similar to) its polar circumference? Certainly, the two numbers are different for our own Earth.
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Re: Mapping Issues: World Dimensions

Postby Thorf » Mon Mar 26, 2012 2:14 pm

For reference, here's what happens when you try to align it with the equirectangular map:

Image

But this is not accurate, because in order to get it to line up this much I had to squish the map from its 2:1 proportions.

Here's the map at its proper aspect ratio behind the Iciria map borders. Note how it has expanded out horizontally quite a lot:

Image

Of course the reprojection process will sort out all of this, so it's not really worth worrying about. It's just interesting to see how much distortion there really is in the hex maps.
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Re: Mapping Issues: World Dimensions

Postby Thorf » Mon Mar 26, 2012 2:25 pm

Ashtagon wrote:Just to throw a spanner in the works, is there anything to indicate that the HW's equatorial circumference must be the same 9or even vaguely similar to) its polar circumference? Certainly, the two numbers are different for our own Earth.


This is just something that we have to assume if we want to tackle these issues at all, because the tools are necessarily all made for the real world. Besides, the difference between equatorial and meridional circumference is only 21 miles for the earth. That small a difference really doesn't seem to be worth worrying about.

http://en.wikipedia.org/wiki/Earth_phys ... ics_tables
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Re: Mapping Issues: World Dimensions

Postby Birchbeer » Mon Mar 26, 2012 4:41 pm

Over the weekend, for fun I made a map derived off of the hex map and Thorf's fixed rectangular map. The amount of difference between the hex map and the rectangular map is a interesting and the distortion pops up in weird place. If so does that then mean we need to take the rectangular map, squish the tops and bottoms into triangles and then map it into Icosahedronal Net?

Whatever we do do, I'm hoping for a 2 versions of maps. One version with the civilizations mapped on it, and another without cities and civilizations on it; just plain terrain. (With just plain terrain I could envision a collective world Piazza game where peeps start with 1 city from each Civ, and grow from there;)
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Re: Mapping Issues: World Dimensions

Postby Hugin » Mon Mar 26, 2012 6:02 pm

Thorf wrote:Hollow World circumference = 10,000 miles

Only an Immortal could be so precise. :D

I don't think we can use that calculation though. The issue is with latitudes. A circumference of 10,000 miles gives us ~27.75 miles per degree of latitude(25.5 hexes times 40 miles/hex divided by 27.75). That puts Chitlacan, the capital of the Azcan lands, at ~36.75 degrees instead of the ~29 degrees it suggests on the world map.

The interpretation I've been working on gives me a circumference of 11,523 miles and 32 miles/hex. It isn't perfect but it then suggests a latitude of ~31.89 which is much closer to the world map's location.

Using the southern most tip of the Bay of Gam, at 41.5 hexes from the equator (1660 miles), gives us a latitude of 60 degrees using 27.75 miles/degree, a latitude of 52 degrees using 32 miles/degree, and the world map indicating 50 degrees.

Thoughts?
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Re: Mapping Issues: World Dimensions

Postby Thorf » Mon Mar 26, 2012 7:25 pm

I'm not entirely sure about your calculations here, Sheldon.

This is what I'm proposing:

Image

Chitlacan is exactly where you want it to be, right? What I'm saying is that placing the hex maps in this way allows us to extrapolate the actual size of the Hollow World, and according to these measurements it is 5,000 miles from north pole to south pole, and therefore 10,000 miles circumference.

Am I missing something?
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Re: Mapping Issues: World Dimensions

Postby Cthulhudrew » Mon Mar 26, 2012 8:40 pm

I'm not sure it is factoring into anyone's consideration or not, but if there is concern that certain latitudes need to be kept within the Hollow World in order to justify geographic constraints, don't forget that the magic of the Immortals provides microclimates for each of the rescued cultures that gives them a climatological niche similar to their Outer World origins, and they don't necessarily need correspond to actual physical geography.
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Re: Mapping Issues: World Dimensions

Postby Cthulhudrew » Mon Mar 26, 2012 8:48 pm

Also, just out of curiosity, has anyone ever tried aligning (re-aligning?) the HW's interior along either of the known direct points of access to the OW- Aegos/Neatharum and/or Quanahuac/Atruaghin's Conveyor? (Or even the Broken Lands/Atacalpa?). It seems to me that I've checked them briefly before, and they didn't seem to quite line up with the map as provided, but I wonder what it would do to the HW orientation vis a vis the OW if we were to use them as fixed points of reference instead of the obverse?
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Re: Mapping Issues: World Dimensions

Postby Culture20 » Mon Mar 26, 2012 9:42 pm

Thorf wrote:according to these measurements it is 5,000 miles from north pole to south pole, and therefore 10,000 miles circumference.

Am I missing something?

The "poles" aren't really poles, so judging circumference based on north/south measurement is problematic. Only east/west will work.
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Re: Mapping Issues: World Dimensions

Postby Hugin » Tue Mar 27, 2012 12:42 am

Thorf wrote:I'm not entirely sure about your calculations here, Sheldon.

This is what I'm proposing:

Image

Chitlacan is exactly where you want it to be, right? What I'm saying is that placing the hex maps in this way allows us to extrapolate the actual size of the Hollow World, and according to these measurements it is 5,000 miles from north pole to south pole, and therefore 10,000 miles circumference.

Am I missing something?

If I calculate circumference using the map I can get 11,304 miles around. 22.5 hexes from the equator to Chitlacan x 40 miles per hex = 900 miles. Chitlacan at ~28.7° gives us 900 / 28.7 = 31.4 miles per degree latitude. (Note: I am assuming that Chitlacan is right at the center of its hex.)

Now, if I calculate miles per degree of latitude by dividing 10,000 by 360 degrees, I get 27.78 miles / degree. When I take that to the map and see Chitlacan at 28.7° N, multiply that by 27.78 miles, I get a distance from the equator of 797.3 miles, or 19.7 hexes. But I can count out 22.5 on the map.

Does that help you see where I'm coming from?

[By the way, earlier when I counted the hexes to Chitlacan using the map from the Hollow World set (and assuming the equator was right along the bottom edge of the map) I counted out 25.5 hexes. However, I notice you placed the equator up 3 hexes from there. But either way, the math works roughly the same. Just thought I'd point that out in case someone noticed that I used different numbers in different posts.]
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Re: Mapping Issues: World Dimensions

Postby Hugin » Tue Mar 27, 2012 12:49 am

Culture20 wrote:The "poles" aren't really poles, so judging circumference based on north/south measurement is problematic. Only east/west will work.

Technically, you don't need the entire circle to determine what it's size would be. If I could only see a portion of the circle that I know is exactly 10° of its entirety, and it measures exactly 10 feet along that section. I can determine that 1° of the circle is exactly 1 foot in length, and then calculate that all 360° equal 360 feet in circumference.

The same thing works for a sphere and in any direction.
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Re: Mapping Issues: World Dimensions

Postby Thorf » Tue Mar 27, 2012 12:59 am

Culture20 wrote:The "poles" aren't really poles, so judging circumference based on north/south measurement is problematic. Only east/west will work.


The short answer to this is that the map shows 90 degrees N to 90 degrees S. For the longer answer, see the Sorting out the Outer World thread.
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Re: Mapping Issues: World Dimensions

Postby Thorf » Tue Mar 27, 2012 1:17 am

Hugin wrote:If I calculate circumference using the map I can get 11,304 miles around. 22.5 hexes from the equator to Chitlacan x 40 miles per hex = 900 miles. Chitlacan at ~28.7° gives us 900 / 28.7 = 31.4 miles per degree latitude. (Note: I am assuming that Chitlacan is right at the center of its hex.)

Now, if I calculate miles per degree of latitude by dividing 10,000 by 360 degrees, I get 27.78 miles / degree. When I take that to the map and see Chitlacan at 28.7° N, multiply that by 27.78 miles, I get a distance from the equator of 797.3 miles, or 19.7 hexes. But I can count out 22.5 on the map.

Does that help you see where I'm coming from?

[By the way, earlier when I counted the hexes to Chitlacan using the map from the Hollow World set (and assuming the equator was right along the bottom edge of the map) I counted out 25.5 hexes. However, I notice you placed the equator up 3 hexes from there. But either way, the math works roughly the same. Just thought I'd point that out in case someone noticed that I used different numbers in different posts.]


Answering your last point first, the reason for that discrepancy is simply that the Iciria poster maps do not show equal areas of the northern and southern hemispheres. Check out my overlay again and you can see it goes further south than north:

Image

Now onto the main topic. I think I've worked out where the discrepancy comes from: the world map shows the 60-90 degree regions as being smaller north-south than the 0-60 regions. But the hex map takes it all as if it were the same scale, which means that obviously counting by hexes it's going to be too short. In other words, if we go by the hex map (which is the usual policy, hex map trumps world map), we're going to get different latitudes from those marked on the world map.

I'm glad we discovered this now, before I started work on updating the world map. You're a lifesaver, Sheldon. :mrgreen:

So the question is, do we want to go with the hex map, and revise the altitudes to match it, or do we want to do what I have already done with the equirectangular projection and expand the 60-90 area north-south, adapting the hex map to fit? The north and south portions of the hex map are already warped east-west quite severely, so it may not be too big a deal warping them north-south too. But we'll be squashing them east-west while expanding them north-south in the process.

Another solution, which actually we touched on years ago in the Curvature thread, is to add some ocean to the bottom of the map in order to bring the 60-90 degree areas to the same size as the 0-60 areas. This has the advantage of preventing land from falling into the polar openings, but I think we might end up with too much water at the poles instead.
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Re: Mapping Issues: World Dimensions

Postby Hugin » Tue Mar 27, 2012 2:54 am

Thorf wrote:Answering your last point first, the reason for that discrepancy is simply that the Iciria poster maps do not show equal areas of the northern and southern hemispheres. Check out my overlay again and you can see it goes further south than north:

I figured there was a reason you did this. But as I said, it didn't have any effect on miles per degree calculations I was doing (thankfully).

Now onto the main topic. I think I've worked out where the discrepancy comes from: the world map shows the 60-90 degree regions as being smaller north-south than the 0-60 regions. But the hex map takes it all as if it were the same scale, which means that obviously counting by hexes it's going to be too short. In other words, if we go by the hex map (which is the usual policy, hex map trumps world map), we're going to get different latitudes from those marked on the world map.

I'm glad we discovered this now, before I started work on updating the world map. You're a lifesaver, Sheldon. :mrgreen:

This is interesting. I just compared the equator-30° distance on the world map and the 30°-60° distance, and they are exactly the same. Only the 60°-90° is different.

This says to me that there is a consistent distance of land per degree of latitude between the equator and 60° (at least). Somewhere beyond 60° we either see less distance per degree, AND/OR, 90° does not mean what we'd assume it to mean on a normal map. Now, the only way to get LESS distance per degree is to arc closer to the center of the circle (making the circle smaller). However, with the polar openings, we go farther away from the center, meaning there should actually be MORE distance per degree.

This also tells me that my latitude calculations are trustworthy as long as I keep my samples with the 60°N-60°S range. Outside of that, the latitudes can't be trusted.

So the question is, do we want to go with the hex map, and revise the altitudes to match it, or do we want to do what I have already done with the equirectangular projection and expand the 60-90 area north-south, adapting the hex map to fit? The north and south portions of the hex map are already warped east-west quite severely, so it may not be too big a deal warping them north-south too. But we'll be squashing them east-west while expanding them north-south in the process.

Another solution, which actually we touched on years ago in the Curvature thread, is to add some ocean to the bottom of the map in order to bring the 60-90 degree areas to the same size as the 0-60 areas. This has the advantage of preventing land from falling into the polar openings, but I think we might end up with too much water at the poles instead.

I'd go with the hex maps, using the world map as guidance for latitude (to determine where the equator, 30°, and 60° lies).

The issue really boils down to the higher than 60° latitudes. On a hex map there is really no problem since it doesn't make any claims on latitude anyway, simply distance, and we have plenty of distance to go into the polar openings. The latitudes, as they appear on a hex map, will no longer be equidistant from each other once we get to the start of the polar opening. Instead, they will begin to appear farther and farther apart from each other as the surface of the Hollow World slopes away from center inner sun.

As an added note to this, the opposite is true for the outer world. The latitudes on a hex map will begin to appear closer together once you pass the start of the polar opening. On my cross-section of Mystara, the greatest latitude on any map would be 80°. At this point you'd continue to travel through the opening directly along the 80th latitude until you started exiting to the other side and 'down' the latitudes again.

That make sense to you. (It might help to follow along on the sketch for a visual - I did!)
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Re: Mapping Issues: World Dimensions

Postby Thorf » Tue Mar 27, 2012 3:02 am

Okay, I used the graticule overlay to georegister the Iciria hex map, and reprojected it to an equirectangular projection. This means that north is now consistently up, south is down, east is left and west is right. Obviously the map has been stretched at the top and bottom.

It's very interesting to see how the Robinson projection affected the hex maps. We can see here for the first time just how strange it makes things, as the hexes become less and less accurate in both area and direction.

Note that the top, left and right edges of the map are inaccurate due to the lack of a graticule line to georegister. The bottom is pretty accurate, and you can see the result of the squashing of the 60-90 degree area crystal clear: whereas all other areas have 7.5 hexes to 10 degrees of latitude, the very bottom area, 60-70 degrees south, has only 5.

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Re: Mapping Issues: World Dimensions

Postby Thorf » Tue Mar 27, 2012 3:34 am

Hugin wrote:This is interesting. I just compared the equator-30° distance on the world map and the 30°-60° distance, and they are exactly the same. Only the 60°-90° is different.


By the way, we actually already knew this - it came up in the Curvature thread. A few years ago. I should have remembered. :oops:

This says to me that there is a consistent distance of land per degree of latitude between the equator and 60° (at least). Somewhere beyond 60° we either see less distance per degree, AND/OR, 90° does not mean what we'd assume it to mean on a normal map. Now, the only way to get LESS distance per degree is to arc closer to the center of the circle (making the circle smaller). However, with the polar openings, we go farther away from the center, meaning there should actually be MORE distance per degree.

This also tells me that my latitude calculations are trustworthy as long as I keep my samples with the 60°N-60°S range. Outside of that, the latitudes can't be trusted.


The definition of latitude also came up in our last discussions, but frankly I don't think it's useful to talk about anything other than regular spherical latitudes. Doing otherwise really doesn't help with mapping, because (as I keep saying ;) ) our tools for mapping are based on the real world. Besides, these numbers are more for our reference than for any in-game purpose.

I am willing to entertain the idea that 90 degrees was the start of the lip - the official maps explicitly mark it as such. It will involve some recalibration (groan), but it's doable.

The other option is to expand the 60-90 area to the same size as the rest of the map, and fold the extra bits into the polar opening. This should work, but it may make the expanded areas look a little stretched. Here's an example using an orthographic (view from space) projection centred on the south polar area:

Image

If we reject this, it will mean that there is no overlap into the polar opening, which does leave more space to fold the Outer World into the opening.

I'd go with the hex maps, using the world map as guidance for latitude (to determine where the equator, 30°, and 60° lies).


Based on that, the 0-60 degree areas have 7.5 hexes to 10 degrees, while the 60-90 areas have 5 hexes to 10 degrees. Rounded to the nearest half hex. This locks the position of the start of the lip in place at 80 degrees, does it not? Because "60-90 degrees" is 15 hexes, which is the equivalent of 20 degrees in the scale of the rest of the map. On the other hand, there's nothing to stop us from backing up the lip of the polar opening and folding some land in if we really need to.
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Re: Mapping Issues: World Dimensions

Postby Thorf » Tue Mar 27, 2012 3:54 am

I've been examining the Iciria hex map with graticule, and it seems that the world map's latitude lines aren't equally spaced even for 0-30 and 30-60. 0-30 N is 23.5 hexes, but 30-60 N is only 22.4 (ish). Looking south, 0-30 is about 23, perhaps 22.9 hexes, 30-60 S is about 22.7. Basically, none of the latitude lines are evenly placed.

If we are going to use the hex maps as the main reference, assuming no north to south deformation, I think we're going to have to discard the world map latitudes and use the world map only to complete the hex map to the north and south. That will give us a fixed north-south figure. However, since that figure is assumed to be the distance from the start of the northern lip to the start of the southern lip, this alone will not give us a final circumference. We could work that out by arbitrarily deciding some reference points (for example Quauhnhuac, which should be as close to Outer World Atruaghin as possible) and fixing the latitudes based on them, which will in turn fix the latitude for the start of the polar lip.

Or, we can leave it fluid and decide the altitude of the start of the lip based on necessity to make the inner and outer worlds fit together better.

Yet another option would be to pick one definition for hexes per degree of latitude, since they are not all that different - say 23 hexes for every 10 degrees - and draw new lines of latitude throughout the hex map to reflect the new standard.

We haven't been talking about east-west much yet, but I don't think it's going to help us, because not only does it suffer from the same problems as north-south in regards to not having equal spacing between lines, but it is also compressed due to the projection.

Edit: Of course, we could just go with one of the official figures and fit the hex-map-determined Hollow World inside that number. That would likely mean that 90 N to 90 S was my original 5,000 mile figure (probably not precisely, but very close), leaving 1,008 miles to split between the polar openings. I don't think that's going to be enough, because 504 miles along the circumference would lead to some pretty small holes.
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Re: Mapping Issues: World Dimensions

Postby Thorf » Tue Mar 27, 2012 4:11 pm

I'm leaning toward accepting the Hollow World maps at face value, statistics and all, and just ironing out the few little inconsistencies rather than revising the whole thing. The more I look at it, the more I think that the designers intended for 60-90 degrees to be smaller, but 0-60 to be regular.

Evidence
  1. The Hollow World Set lists the circumference as 11,908 miles. (Hollow World Set: Hollow World projection map)
  2. The region shown on the Hollow World world map between 60 degrees and N and 60 degrees S is roughly consistent. The regions marked as 60 N to 90 N and 60 S to 90 S are smaller north to south. This is because the lip of the polar openings begins at the place marked 90 on the maps. (Hollow World Set: polar openings poster maps, Dragon 161 map)
  3. The Poor Wizard's Almanac gives dimensions of 1030 miles diameter for the polar openings in the Hollow World. This narrows down to 774 miles at the narrowest point. (Poor Wizard's Almanac)

Conclusions
  1. Overlaying the hex map on the world map and counting hexes gives a north-south value of approximately 124 hexes. 124 x 40 = 4960 miles. Multiply by 2 to get the circumference minus the two polar openings: 4960 x 2 = 9920. (Hollow World Set: Hollow World projection map, Iciria poster maps)
  2. 11,908 - 9,920 = 1,988. This is the "leftover" circumference for the polar openings. (Hollow World Set maps)
  3. Between 60 N and 60 S there are 23 hexes for each 30 degrees of latitude. 60 N to 90 N and 60 S to 90 S only consist of 16 hexes for 30 degrees of latitude. (Hollow World Set: Hollow World projection map, Iciria poster maps)
  4. With the dimensions of the Outer World not yet fully determined, but likely bigger than the figures given in official sources, it probably will not be possible to maintain the official dimensions of the polar openings. Hex maps have general priority, and in this case the polar regions are not terribly important; if necessary, the lip can begin as early as 60 degrees without any negative consequences.

There are likely lots more things to add to these lists, but I'm too tired to think. More tomorrow.
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Re: Mapping Issues: World Dimensions

Postby Hugin » Tue Mar 27, 2012 5:10 pm

[Edit: This doesn't comment at all on Thorf's post directly above that was posted while I was drafting this one. I will read it and see if that changes anything.]

Excellent work, Thorf!

Thorf wrote:The definition of latitude also came up in our last discussions, but frankly I don't think it's useful to talk about anything other than regular spherical latitudes. Doing otherwise really doesn't help with mapping, because (as I keep saying ;) ) our tools for mapping are based on the real world. Besides, these numbers are more for our reference than for any in-game purpose.

Absolutely agreed. I have only ever used the standard latitude as meaning the degree of the angle at the center of the circle (planet) that is formed between the equator and another point along the surface. It's the only one that seems relevant to me as well. I guess I should have stated that earlier, but this is precisely the reason why a mentioned that you could only ever travel to 80° on Mystara; the openings result in there being no land at higher latitudes.

I am willing to entertain the idea that 90 degrees was the start of the lip - the official maps explicitly mark it as such. It will involve some recalibration (groan), but it's doable.

The other option is to expand the 60-90 area to the same size as the rest of the map, and fold the extra bits into the polar opening. This should work, but it may make the expanded areas look a little stretched. Here's an example using an orthographic (view from space) projection centred on the south polar area:

Here's my opinion and approach: ignore the higher latitudes markings of the world map as erroneous. The higher latitudes simply do not exist on Mystara because of the polar openings. Latitude markings only exist where a line going out from the center of the planet intersects a surface of the planet. As such (on my maps) an angle of greater than 80° will not intersect any surface since it just goes out a polar opening. Therefore, the highest latitude that would appear on a map is 80°.

So my approach here is to map land by 'vertical' (north-south) distance and just mark the latitudes as they fall. I have already determined the size of the planet by comparing the hex maps and the latitude markings on some maps to give me a satisfactory figure (to me at least). On the Outer World it worked out to 72 miles per degree of latitude along the 'standard' surface (i.e. not applying to any portion of a polar opening). Once I'm dealing with land within a polar opening, the relationship between 'vertical' distance and latitude is not consistent or standard. They'd have to be determined mathematically, but in all honestly have little meaning anyway.

To map out what appears in a polar opening, my first concern is to determine the distance between the two standard spheres/surfaces of the planet (i.e. from the point a polar opening begins on the outer world to the point it begins on the inner world). I believe this can be calculated relatively well. Also, we can calculate the "width" of a polar opening map at the various "height" using some relatively simple math. It will end up looking like an hour glass shape(ish).

If we reject this, it will mean that there is no overlap into the polar opening, which does leave more space to fold the Outer World into the opening.

Again, using my interpretation Mystara's measurements, there is plenty of room in the polar openings for land to go. And I mean PLENTY! So for the Hollow World I've worked out 32 miles per degree of latitude along the standard sphere. Using that I can see what hex the polar opening begins and any hexes beyond that are simply "that many miles into the opening", regardless of whatever latitudes a world map says it is; I ignore that part as a meaningless complication because they aren't using the same definition of latitude that I am. In fact, I have no idea what definition they are using.

Based on that, the 0-60 degree areas have 7.5 hexes to 10 degrees, while the 60-90 areas have 5 hexes to 10 degrees. Rounded to the nearest half hex. This locks the position of the start of the lip in place at 80 degrees, does it not? Because "60-90 degrees" is 15 hexes, which is the equivalent of 20 degrees in the scale of the rest of the map. On the other hand, there's nothing to stop us from backing up the lip of the polar opening and folding some land in if we really need to.

What I did is count 23.5 hexes from the equator to the 30° line on your Iciria 40-mile hex map. 23.5 hexes x 40 miles per hex gives me 940 miles, divide that by 30° gives me 31.33 miles per degree. I used 32 miles for simplicity. (Again, this only applies along the standard sphere, not to any portion of the polar opening.)

If I say that the Hollow World's polar opening begins at 64°, and starts a very gradual curve into the opening from there, and using 32 miles per degree, all the land between the equator and the top or bottom edge of the map, is all on the standard sphere. The very next hex above or below the map begins the curve into the openings. (Note: 64 degrees at 32 miles equals 2048 miles, or 51.2 hexes and there are 51 hexes from the equator to the top of the map.)

The next thing for me to do is determine how much 'vertical' distance is between the 'standard' spheres (i.e. in the polar openings).

Did I present this clear enough? Any questions/issues, etc.?
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Re: Mapping Issues: World Dimensions

Postby Hugin » Tue Mar 27, 2012 8:16 pm

Ok. Some comments on this.

Thorf wrote:The Poor Wizard's Almanac gives dimensions of 1030 miles diameter for the polar openings in the Hollow World. This narrows down to 774 miles at the narrowest point. (Poor Wizard's Almanac)

Here's how this would work using:
- my outer world circumference of 25,920 miles (close to Earth's),
- a Hollow World one of 11,908 miles (which is fairly close to my suggestion of 11,520),
- and the polar opening as given above (and using a spherical polar opening).
The openings begin at ~60.1°; that's just about 94 miles north of Alpha and well before the arctic circle. The openings would have a distance of 3503 miles between the outer and inner worlds. Put another way, if you started at the equator of the outer world and traveled north to the rim of the polar opening, you would have traveled 4327 miles. If you continued to travel through the opening for another 3503 miles you'd reach the hollow world's sphere, and would have to travel another 1989 miles to reach its equator - a total of 9819 miles.

It's doable, but not as ideal as it can be, in my humble opinion. The only things I'd change here is the Hollow World's size to 32 miles per degree (instead of 33.0778) and thus 11,520 miles in circumference (instead of 11,908), and the opening's shape to a funnel-like one (think of having a ball and just flattening one side a bit). It could even keep the same length in miles. Back to the Hollow World's 32 miles per degree, I like this for simplicity's sake; 4 hexes equal 5 degrees. So while 33 miles would be more 'canon', it is just more awkward as well. These two changes would allow us to pull the openings closer to the poles and into the (ant)arctic circles.

With the dimensions of the Outer World not yet fully determined, but likely bigger than the figures given in official sources, it probably will not be possible to maintain the official dimensions of the polar openings.

I know people are probably thinking I'm harping this too much, but honestly, I think the 72 miles per degree (25,920 mile circumference) works fantastically.

The whole point of my suggestions is their simplicity of use as well as compatibility with existing maps.
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