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[wierd]

I agree that either a cylendar or the inside of a torus will translate with less distortion.

However, each has a significant problem.

The cylendar world has no thickness! There are Z levels to the world, so either Z has to be truncated/distorted, or the world isn't a cylendar.

A torus still has distortion from projection, but not as pronounced-- but it then has gravitational anomalies, and now lacks a proper day and night. (Either the sun is in the center of the torus, making it into a niven ring world, or it is a free floating torus like a halo ringworld. In either case, differences in centrepetal forces will make the effective gravity at the poles vary wildy from that at the equator, and now plotting how a moon appears in the sky becomes a lesson on cat herding.)

it's easier just to say that the DF cartographers just suck, and have inaccurate maps.

[Snaake]

... Since the DF planetary system appears to lack other planets besides just the planet the dwarves and such live on, its orbit can be essentially circular, if plotted through the barycenter of the planet/moon system, and not the center of mass of the planet.  Incidentally, this would give an explanation for seasonal insolation differences without needing axial tilt at all, which would explain the curious lack of reversed seasons below the equator.
...

Sure, they're not mentioned, but I don't see a particular reason that there wouldn't be other planets. In fact, current knowledge of solar system formation dynamics seems to suggest that a gas giant has a large stabilizing effect in helping habitable-zone rocky planets like the Earth to form (by protecting them from comets etc), or at least in protecting any fledgling life on them. You may be onto something with the axial tilt, but I've had a few beers and don't trust myself to think that through completely (I have studied meteorology, but very little astronomy, so...).

Disregarding the previous disclaimer, since this is too interesting not to comment on right away...  On the other hand, no axial tilt and a (nearly) completely circular orbit around the sun should lead to negligible seasonal differences, especially in the temperate zones, but actually anywhere. Especially if the mass of the planet is much larger than the mass of the moon, the eccentricity that the moon causes to the planet's orbit around the sun (since it's the combined center of mass of the planet+moon that actually orbits) just isn't big enough, if the moon:planet mass ratio is small. It might even require a pretty crazy mass ratio for that eccentricity to have any effect on seasons, to be honest. So it just seems to me that if you have seasons, you need to have them reversed on different hemispheres. Plain hot-cold maps (as opposed to cold-hot-cold) could be explained by the planet having it's rotational axis more or less pointed towards the sun, but with axial tilt/wobble causing seasons as per usual. Or just having a huge ocean on the other hemisphere that hasn't been discovered at all in the pseudo-medieval world. Think of the Americas, or Australia/the huge Pacific Ocean, and how late those were mapped.

As a final note, the moon missing planet-core materials like slade is actually a nice analogue to our moon being mostly composed of what used to be part of Earth's mantle.

[wierd]

Yay legends mode!

Legends mode can output a "timeless" heat map of the generated world map that is a nice greyscale image.  Sadly, it has lots of noise in it, but that's actually what you would come to expect from there being atmospheric mixing, and other climactic events going on.

Interestingly, the "large island" sample world I have selected appears to be squarely in the norther hemisphere, as the high heat energy band is at the very bottom of the world map.

I may need to do worldgens of the various types. They may generate features very differently.


*edit

Ok, after some hair pulling on the temp map spat out by legends mode, I have been able to derive a flat illumination falloff.

It looks like the 30 degree incident angle happens "exactly" 143 "large world map tiles" up from the bottom of the map. I am in the process of attempting to derive some distance units for each large world map tile. Vertical distances shouldn't be distorted, if we presume it is your typical worldmap projection style.

I am going to embark on a super tiny embark that is exactly 1x1 embark, and count the discrete tile dimensions of it, then attempt an extrapolation for scale of the big worldmap tiles.

Edit again*

The smallest embark you can make is 2x2 embark tiles. I just counted the number of tiles from edge to edge of this tiny embark, and it was 96 tiles.  This means each embark tile is 48 play tiles wide. Each world map tile is 16x16 embark tiles in size.

I will give an approximate real world distance value for 1 play tile of 1 meter. (That's about 3 feet, the width of a normal doorway.)

This means each embark tile is 48 meters tall, and each world map tile is 768 meters tall.
Our 30 degree line was drawn exactly 143 world tiles north of the equator, which places it approximately 109824 meters north of that position, as an arclength.

It is important to stress that my CAD software cannot generate a sphere that size, as it will crash my workstation. (Yes, I've tried before.....)  instead, I will scale down the sphere by exactly 1000:1 scale, so that I can generate a perfectly scaled sphere instead. This gives a 109.824 meter arclength, at 30 degrees, for a 1:1000 scale of the planet. This may actually still be too big, and I may have to scale it even smaller. We'll see.

Edit once more**

The planet has an approximate sphere radius of 104874.19494 meters.
(Earth has an approximate sphere radius of 6371000.00 meters, according to google.)

That's a whopping 1/61 size of the earth, or there abouts.

Next, we need surface gravity.

I think I will do that tomorrow.

[wierd]

Mapping the region onto the resulting sphere shows what I though I would see. The world map shows approximately 1/8 of the total global surface for this worldgen.

As expected the map gets smooshed what good when mapped onto the sphere's surface.

It however looks pretty sexy otherwise.

Spoiler: woah, really big image file here! (click to show/hide)

I need to find the best way to project this map

The 45 degree measure is from the centerline of the map, to the two extrema, as mapped from the equatorial line.  The faint red line is the measured 30 degree light angle line, where solar energy delivered to the surface is 50% of what is delivered directly to the equator. The white elipse, is the equator projected around the whole sphere.

A possibly better projection would be a cubic projection, but that wouldn't have the 30 degree inclination line in the correct location.

[nanomage]

I am really sorry, but I fail to see (or I just miss) how you prove that upper and lower lines of the region are the the actual equator and pole. Why can't it start at arbitraty latitude and end at another arbitrary one?
Also, regarding your suggestion to derive eccentricity from the varying length of month: I was under the impression that the length of synodic month is constant and it's only the length of calendar month that varies. Though it's quite hard to me to say for sure without drawing charts and calculations, I believe that effect of eccentricity on the respective length of waxing/waning moon periods should be greater than it's effect on the overall synodic month length.

[wierd]

Insolation

I took a DF large island, went to legends mode, and explorted its temperature map.
I then averaged out the falloff in temperature from hottest latitude to coldest latitude.
The hottest latitude has the strongest insolation, and I treated it as a 90 deg angle of incident to the surface. (Straight down.) I then measured the temprature values of the resulting gradient until I found the 50% point between the extremes. As the article points out, this is the 30 degree angle of incident to the surface point, and has a sine value of 1/2. Using the measured arc-length, and the angles of incident, I generated a circle that matched that geometry. From the circle, I derived the sphere's volume.

The numbers all work out. The resulting map is exactly a 1/8th pie wedge of the computed volume of the planet, as derived from its heat map and surface distance.

If it were wrong, then the 30 degree line drawn from the heat distribution would not align at the 30 degree point of the sphere.

It does align.

The equator is indeed the bottom of this map. If you want to suppose that the equator is further south, off the genned map, then you are supposing an undefined system, where I have no methodology of getting that information. (This was the biggest size of region DF will gen) With the information I DO have, this is a valid solution, and it works out cleanly as far as surface topology is concerned.

We could do a multiple worldgen comparison, if the world generator makes worlds with a centrally located equator, and compare the heatmap gradient against the one I used, to see if the extrema change, but since the world I genned did not have that feature, I have placed it appropriately according to the math and data availale.

[nanomage]

Thank you wierd. I think I understand what you did, and I was just trying to understand if you already have any strict means of ruling out the hypothesis that equator is "somewhere further south". I of course agree that yours is a valid solution, and it is better than my vague assumption about arbitrary latitudes.

There are still issues with it however. One you have derived yourself and that's extremely small planet size (it's twice as big if we take Toady's assumed 2m per tile, but still tiny), and the second is that in your model, the entire top line of the map should be one actual point (the pole) of the sphere, with all distances in it's immediate vicinity being scaled down tremendously. So embarking anywhere near the top line of the map should place us roughly in the same area, and this is not the case with DF regions.

[Gavakis]

Jeez.. That's cool.

[Snaake]

Toady has said that for minecart physics purposes, a tile was 2x2 meters, not 1x2. I don't think we have definitive dimensions in terms of SI units otherwise. I think you could safely make some other assumptions too, e.g. that the mantle and core of the planet are below hell. I.e. Dwarves can only dig about in the crust, which is only 5 (ocean) to 50km (mountains) thick on earth as well.

Each embark tile being 48x48 was known, although I'm not sure how well it's documented e.g. on the wiki.

I think it should be possible to calculate the radius (/circumference) of the planet even if you take the latitude of the hot side of the map as an unkown, as you still have a temperature gradient over a known distance. May involve calculus though, so is of course more complicated, and said latitude being 0 will indeed probably be a valid answer. Or at least a possible one.

edit: What did you assume/measure for the solar constant/how did you, in general, get from the temperature gradient to the radius of the planet? For a solar constant S₀ (skipping the subscript from here on), the amount of radiation reaching the surface at any given location is (at solar noon, averaged over the year, or during the equinoxes) is Q = S*cos(y), where y = latitude, north is positive. For 2 places on the map, Q₁=S*cos(y₁) and Q₂=S*cos(y₂) must be true, and thus you can get

Q₁/cos(y₁) = Q₂/cos(y₂), or
Q₁/Q₂= cos(y₁)/cos(y₂)

So hm. You can get ratios of the cosines of latitudes. Which you have done for the case of y₁ = 0 degrees and y₂ = 60 degrees (insolation half of the equator's). Note that I'm using latitudes, you were using angles of incidence. With no axial tilt, the sun's height above the horizon is 30 degrees at a latitude of 60, and sin=cos(Pi/2-a), or vice versa.But what if the bottom of the map is instead e.g. y₁ = 20 degrees? Then with Q₁/Q₂=2 as before, y = arccos(0,5*cos(20 degrees)) = 62.0 degrees (to 3 S.F.). Huh. So apparently y₂ doesn't react very quickly to changes in y₁. Further testing shows that eg. y₁ = 60 degrees only gives y₂ = 75.5 degrees.

There's still the option of what if the hot end of the map is in the other hemisphere, but I don't feel like getting into that, and it does seem unlikely.

There's still lots of stuff that we're just assuming/can be artifacts of the time/space compression, especially in fort mode. I was about to remark that we don't really know if there even is a sun (like you commented about the moon earlier), but we do at least know that dwarves do get "irritated/nauseated by the sun" if cave adaptation has set in . One thing that bugs me is that fort mode has no day/night cycle, for instance, so are the surface temperatures supposed to be the ones for noon, average daytime temperatures, or averaged over 24 hours?

Just hoping I got the math more or less right, it's been a couple of years since the climatology course where we did these kind of calculations.

[wierd]

Since the heat map is more or less consistent in the x direction of the plot, and varies only with the y axis, i'd say it is probably "average daily".

It has a LOT of random noise, but averages out more or less cleanly. (It has a bumpy histogram.)

For reference, the final averaged extrema were 256 shade greyscale values of 4 for the top of the map, and 194 for the bottom. This means that at most, the equator could be 33% further south than the plotted position before you simply run out of spectrum, and have a max value of 255.

As for a game mode with day an night, Adv mode fits the bill.  IIRC, temperature, weather and pals are active in adv mode, and you can probe a tile for temp with dfhack.

[Snaake]

Since the heat map is more or less consistent in the x direction of the plot, and varies only with the y axis, i'd say it is probably "average daily".
...

Probably. But that would mean that realistic afternoon temperatures in scorching biomes would be crazy. And again, by "daily" do you mean daytime (when sun is up), or 24h? Then again, I don't think it's that much more unlikely that fortress mode just treats everywhere as being noon, or day, all the time, so the mostly constant temperature with respect to longitude doesn't really give any info on this.

[wierd]

Ok, I am quite confident in my calculations now.

I have genned many many many more worlds that are large islands for cross comparison. Some have been southern hemisphere regions. Not a single one has had the equator in the middle. When stitched together, these worlds all have the same banding patterns for biomes, with a conserved equator.

Spoiler: BIIIG file, but looks neat. See the well conserved equitorial band. (click to show/hide)

The planets produced really are just that small.

I am in the process of stitching 8 such regions together (4 northern, 4 southern) to map onto my sphere to get a sample planet.

[Gentlefish]

regions or islands?

[wierd]

Large islands.  I would have spent ALL NIGHT trying to gen large regions that match up cleanly.

[Gentlefish]

Aha, true. I can't wait to see the outcome, this is pretty darn cool!