It’s been a while since I’ve written a post for this project, but that’s not from lack of progress or activity — quite the opposite. In fact I’ve been repeatedly amending and revising a post for over a month now. Now it’s finally ready to share.
Planet Size
After the last article, I discovered that my calculations were a bit off. I’m not exactly sure why this is. I posted about this in the Facebook Mystara Cartographic Society (embedded below, if it’s working), and after discussions there (check the “Is my maths off in some way?” comment thread for details) decided to go with a round 36,000 miles for the circumference — primarily because chasing an exact 72 miles per hex was not working, and because this figure gives an easy 100 miles per degree.
Unfortunately, this didn’t work out either — it ended up giving far bigger hexes than I had expected, with 24 mile hexes coming out at 26 miles.
So I left it for a month or so. And then it hit me: I can reverse-engineer the circumference from measurements in Illustrator.
Basically, what I did was measure a hex, then divide the circumference by that number. Then I multiplied that by the desired hex size, which gave the actual circumference I needed to define the ellipsoid.
It turns out that Atlas Mystara has a circumference of 33,032 miles, which is quite a bit smaller than previous estimates. But this indeed made the measurements for hexes come out nearly perfectly in Illustrator, so I’m locking this in as our final circumference for the new 2018 model.
Phew! Now I just have to go back and re-georegister all those maps. Thankfully I already know how to do this without losing all the GIS data I’ve been inputting, and it’s not very hard.
As a consequence of all this, I have had to rebuild the model yet again — hopefully for the last time!
We are rapidly approaching the conclusion of this project. This may even turn out to be the last post, if community feedback is mostly positive. So let’s get into it.
The Atlas of Mystara World Model
Mystara 2016 Diagram 1
Diagram 1 shows the world model, as determined by the world dimensions we came to previously. All of the diagrams and images in this post are derived from Placement 5, which seemed to be the most popular of the placements I proposed. (There was also support for placement 4, with its tunnel-shaped polar openings, but if we can preserve the polar openings in their original shape, that would be preferable.)
I have split the world into four parts, as illustrated on the right of this diagram: the Hollow World, the Outer World and the two Polar Lips. The north-south measurements of each of these are also given; these are essential for fitting the world maps to the 3D model correctly.
The next three diagrams show the model itself, which is very simply a 3D extrusion of the shape in diagram 1.
Mystara 2016 Diagram 2
Mystara 2016 Diagram 3
Mystara 2016 Diagram 4
I made the rendered diagrams blue to differentiate from the 2012 versions of these same diagrams.
Finally, here’s the sunlight diagram:
Mystara 2016 Diagram 5
The areas of black marked in the northern polar opening show the extent of the Darklands — where neither sun shines. An area the same size exists in the southern polar opening. There’s also an area which experiences both day and night in summer, marked in grey, but most of the land at the poles gets constant day in midsummer and constant day in midwinter.
With that out of the way, it’s time to explore the model.
Atlas of Mystara 3D Model
This is it: hopefully the final model and placement. I put some clouds on this one for fun, but the rest of the mock-ups are cloud free for better clarity.
From straight on, Mystara looks like a ball with its top and bottom cut off:
The latitudes seem to be working rather well, don’t you think?
Here’s a better view of the Arm of the Immortals and the Savage Coast area:
Skothar:
And Davania:
When it’s on the model, Davania is far more visibly South America and Africa with Antarctica and Australia added on. It’s very clear to see in this image.
Next, let’s take a look at the polar openings. The north pole:
Frosthaven sits just south of absolute north. The composition of the polar opening seems close to the official maps. Note that Nentsun is actually quite small — it’s only marginally larger than Frosthaven in size.
Next, the south pole:
Just the very southern tip of the continent falls into the opening, but it goes about halfway in. As expected, southern Davania is far smaller than it looks on the world maps, due to the pinching effect of the poles, which is lessened on Mystara thanks to the polar openings, but it still significant.
Now let’s delve into the Hollow World. In these shots, you can see the polar openings as well as the Hollow World itself.
The extent of the expanded Grey Sea and Sea of Keleb is apparent in this last one.
Let’s take a closer look.
The fuzziness is because I haven’t rendered these shots, as it takes close to two hours to do so. That can wait for the finished Atlas versions of these maps, hopefully in a few years’ time.
It’s perhaps a little hard to make out how the curvature works here, so let’s try some perspective shots.
The placement of the Hollow World’s landmasses seems pretty good.
Conclusion
I’m quite happy with this model. What do you think?
If you have any feedback, please don’t hold back. Now is the time to iron out any issues.
If nothing major comes up, I will declare the project closed, and all my maps from that point on will use this as the model. Of course there’s a huge amount of work needed to turn these collages back into usable maps, but that is for the next project.
In which I slice and dice the Hollow World to get the straightened out map to fit the hex maps…
More Size Issues
There’s one more issue here that needs to be tackled: the size of the Hollow World. Back in 2012, when we were talking about World Dimensions at The Piazza, both Sheldon Morris and I came to roughly the same size for the Hollow World, based on expanding the hex grid from Iciria to the rest of the map, and counting hexes. My count was 124 or 125 hexes, as I have mentioned in previous articles. But let’s look again: what was I counting?
The Hollow World map has latitudes marked just as the Outer World did. There’s something crucial to notice here: between 60ºN and 60ºS there are 23 hexes for each 30º of latitude. 60ºN to 90ºN and 60ºS to 90ºS only consist of 16 hexes for 30 degrees of latitude.
But if we take the hex map at face value, we can at least conclude that the north-south dimensions of the central meridian provide an accurate measurement of the extents of the Hollow World — at least, the extents of the part up until the polar openings.
This is all getting a bit complicated, but here’s my conclusion: the vertical height of this map is the accurate measurement, while the latitudes are a red herring that we can and should ignore.
So what needs to be done? Well, we need to unwarp the map — straighten it out so that north is always straight up, south straight down. But we don’t need to map it to the grid of squares an Equirectangular projection will give us. Look again at my Equirectangular map above. See how all of the squares are regular throughout the map? But as we have seen, the hex map dictates that the areas from 60-90º are in fact smaller than the rest of the map.
Here’s what I propose: we can still use my reprojection to straighten out the map, but it needs to be cut up and rescaled vertically, so that the latitude lines match the pseudo-Robinson projection’s lines. This will result in an image like this:
Hollow World Set Hollow World map in Equirectangular projection, vertically scaled to match latitude lines with the original map
Here’s a comparison of the two. Note how the latitude lines match up.
This image is no longer 2:1. It’s now 1925:1000 — in other words, a little too short in the east-west dimension.
This makes me remember that when we fitted the world map to Iciria’s hex map, we needed to stretch it slightly east-west. Since we’re reprojecting based on the graticule, the stretching will be discarded in the reprojected image. Does this mean that we need to stretch the map out a bit to get it the right east-west dimensions again?
Well, the original hex map’s east-west dimensions should be true at least along the equator, so we can use this as a check.
In simple terms, we want the reprojected hex map to match the original (pseudo-Robinson) hex map along both the equator and the central meridian.
Let’s see with an overlay:
Equirectangular Iciria on top of pseudo-Robinson Iciria
It’s pretty obvious right away that they don’t match.
The latitudes will of course need to be rescaled manually as we did for the Equirectangular world map.
Equirectangular Iciria scaled to fit pseudo-Robinson Iciria’s latitudes
Note how the central column of hexes is quite regular now that the latitude lines have been matched up. Remember, we’re ignoring latitude and going with the hexes as an absolute measure of distance north-south.
But what of the longitudes? Shouldn’t they have been accurate along the equator?
Apparently not. Looking at other Robinson projections, they do seem to be taller than they are wide.
Conclusion: the Hollow World hex maps are probably all slightly wider areas than they appeared to be! This is because the Robinson projection is a compromise projection, which doesn’t present any single aspect (shape, area, distance, etc.) of a map completely accurately.
But wait: there’s more. It turns out that the pseudo-Robinson world map’s western (right) hemisphere is significantly wider on the map than the eastern (left) hemisphere. Check out the width of the grid squares on the left and right, and it’s quite obvious that the right is bigger. This all gets normalised when you reproject, but I propose that we match the longitude graticule in the same way we matched the latitude lines. Specifically, we should match the junctions at the equator, restoring the visual appearance of the map. It may be necessary to add some sea on the far side of the world in order to keep the central meridian where it is, but this is fine by me — I’ve always thought that the outlying continents are too close anyway.
Here’s what we end up with:
Equirectangular world map on top of pseudo-Robinson world map; note how the grid has been matched along the equator and central meridian
The resulting world is now 1967:1000. Closer to the 2:1 ratio than before, but not quite there. Let’s fix that by centring the central meridian and adding sea to the blank space:
Equirectangular world map with central meridian restored to centre, sea added to fill the resulting blank
This looks pretty good. It would be easy to add some islands to make sure the equatorial mountain range isn’t completely broken.
The ratio is now 2031:1000. So we’ve passed the important 2:1 ratio. More on this shortly.
Building an Equirectangular Hollow World
This may have been hard to follow, so I’ll sum up. We now have:
An Equirectangular projection world map, scaled to fit the visual dimensions of the pseudo-Robinson world map. The graticule grid is now totally wrong, but we can fix that in future maps.
An Equirectangular projection Iciria hex map, scaled to fit the north-south visual dimensions of the pseudo-Robinson Iciria hex map.
It’s time to put these together.
If you remember, this involved stretching of the world map last time. We’ll do the same this time, adapting the whole world map to the hex map’s dimensions. Here’s the resulting world:
Fully corrected Hollow World map, with hex maps overlaid
It’s reached a ratio of 2234:1000. Add a graticule and fill in the north and south extremes with sea, and here’s what we get:
Fully corrected Hollow World map, with hex maps overlaid and new graticule in red; 66º lines in yellow mark the beginning of the polar lips
It’s messy due to the old graticule being burned into the map, so I made the new graticule thick red. You can see that it’s mostly the eastern (left) hemisphere that had longitude problems, which makes sense as it’s the one that came up short.
Hollow World Placement
The polar lips starting at 66º look fine. The Blacklore Valley was always going to end up close to the polar openings, but its exact location may need some tweaking so that it doesn’t go too far into the opening; the Hollow World Set polar opening map seems to place it not quite inside, after all. This should be possible by adjusting the amount of sea at the edges of the map.
But the big question for placement is the size of the Hollow World. All of our tweaks have regularised the Hollow World so that it now appears to have a steady 7.5 hexes per 10º. With 40 mile hexes, this works out as:
7.5 x 40 = 300 miles per 10º latitude. 300 / 10 x 360º = 10,800 miles. Unsurprisingly, this is the figure we originally came up with. The official figure was 11,908 miles. Doing the formula backwards, that would mean 8.27 hexes per 10º.
We can easily add a little more sea at the edges to accommodate this size. This has the added benefit of bringing the Blacklore Valley up to right after 66ºS, which should mean it will be at just the right spot, on the gradual slope into the opening but still facing the inner sun.
Here’s what this looks like:
Fully corrected Hollow World map scaled to official 11,908 circumference
I would definitely add islands to the equator.
It took a surprising amount of fiddling to get to this point, but I’m pretty happy with the result.
Any questions?
One thing you may have realised is that even though I adjusted the Equirectangular world map vertically and horizontally to match the graticule, I only adjusted the hex map to match latitudes — not longitudes. As I noted, the result of this is that the area of the hex map will be slightly wider than it appeared on the original map. So why did I do this?
Because the projection we’re dealing with artificially expands distances east-west as you move away from the equator. To put it simply, the three HWR maps cover less area than they appear to, because they’re all away from the equator. Allowing this slight widening should counter that a little throughout the continent, allowing the shapes we know to remain closer to how we know them.
Also, I want to place as much of the Hollow World land as possible within the 66ºN to 66ºS span. This means we need more land from east to west, so this slight stretching seems to make a lot of sense here, too.
Next up: Tying up some loose ends before we go back to the 3D model and testing placements.
