How tall is New Berk, really? (Revisited)

Not too long ago, I made a reddit post in which I calculated how tall is New Berk in How To Train Your Dragon: The Hidden World. Certain discord server ended up expressing some doubts that I misidentified the location of the lift on the full view of New Berk. While it turns out that I didn’t misidentify the location … let’s be completely fair and honest, 5 kilometers is a ridiculous height estimate. With 4K BluRay in my hands, the time has come to do a second round of calculations.

Seeing the forest for the trees

Another way to eyeball the size of Berk is to take a look at the trees. There’s spruces growing all over the island. Setting-wise, it would be reasonable to expect that these are the European kind of spruces — and these can grow up to 35-55 meters tall. Do spruces on New Berk grow this tall? Quick sanity check says they do:

Now that we know that our assumptions aren’t too out of the line, we can start counting pixels. Time to get our hands on some 4K footage for best results. The scene where vikings arrive to the New Berk seems to be a nice candidate for that:

That doesn’t seem to unreasonable, although it may not bode well for our initial “Berk is 5 km above the sea” estimate. Yikes. Let’s find a complete shot of New Berk.

Well. There’s not much to say, really. While the original estimate was within the same order of magnitude, the 4K screenshots seem to suggest that the height of the new island is between 3,3 and 5,6 km total, with settlement being between 1.7 and 2.8 kilometers above the sea.

But we aren’t done yet.

Autism intensifies

There’s still further confirmations to be had, and chapter title like that certainly doesn’t inspire much confidence in what’s to come.

Let’s try to guess the FoV of this shot.

Can we do it? Turns out yes … but we’re gonna need math. For starters, we need four points on the image that we know real-world coordinates of. Since we know how tall the boathouse is, we can quickly pull four points out of thin air.

Now, I haven’t watched enough Rick&Morty to understand the sort of math needed to calculate the FoV and direction in which the camera is pointing, but someone over at stackoverflow has — and they were nice enough to write their answer in Python — a language that I know a very tiny bit. That answer doesn’t give us everything, though, but let’s not worry. There’s also this math stackexchange post, which pretty much seems to cover the rest.

I know what you’re thinking. That’s a lot of math, and I really want to avoid having to do said math. Especially since I haven’t watched enough Rick&Morty to even figure out how my datapoints translate to solutions described in that post. That, and I’m also lazy.

There must be an easier way, right?

There is. Behind Berk, there’s a lake. Lakes have one very nice property: unless the lake is particularly big, its surface is going to be flat. With that in mind, the New Berk shot was picked very carefully: when moving through the scene frame-by-frame, I found a frame where the lake that surrounds New Berk is aligned with horizon. With horizon known, we only need another point, for which we know the between horizon and the direction in which the camera faces towards said point.

Fortunately for us, that will turn out to be easy. We can trust that support pillars for the lifts (as well as the walkways and terraces around the lifthouse are roughly perpendicular. This means we can easily determine the directions in which the axes of the 3D coordinate system point.

And not only we have that: we also have Minecraft.

Turns out that in Minecraft, pressing F3 gets you some nifty developer tools. Among said tools is a reticle that’s shows the direction of the main axes of coordinate system. The tools also tell us which direction we’re facing, with second number telling us the angle relative to the x-z (horizontal) plane.

Through the magic of Linux, we can make GIMP transparent and ensure it’ll always stay on top of any other windows:

That allows us to overlay GIMP over Minecraft. We carefully align the point where all axes cross with the corner of Minecraft’s reticle and wiggle the mouse around until the minecraft cursor covers the lines we drew in GIMP:

Let’s see whether the reticle truly fully overlaps our lines — it does. We also get to read out the angle that we need:

And now we can finally calculate the FoV of the camera.

The results are in: the shot seems to have vertical fov of 38.1°and a horizontal FoV just shy of 90°. This feels somewhere between just about right and too wide. Angle isn’t not unreasonably wide, though: assuming this were shot on 35mm film, the lens equivalent for that would be a 18mm lens.

Now that we know the FoV, we can try verifying the island height — and not only that: we can also calculate other dimensions.

Mathrix: reloaded

Back when I was making the initial calculations, I only had a shit (low-bitrate) 1080p version of the movie, so the details weren’t really clear while pixel-licking. Fortunately, though, I now have 4K bluray, which has all the bitrate and all the detail. And on 4K version, we are actually able to make out some impressive detail. What used to be a small blob of pixels is now indeed a viking:

What is more, we can reasonably guess where the floor ends, his legs begin and where his head ends. Your garmin is about to start making noises.

Would you look at that. Our optimistic estimates from earlier seem to be roughly confirmed: it’s safe to say that the island is about 3-5 kilometers tall, with Berk being 1.8 to 2.8 kilometers above the sea.

The Final Interpolation

Up until this point, all calculations have been done by estimating the island size based on some assumptions regarding tree sizes. But now that we know the distance between horizon and the bottom floor of the lifthouse, we can actually get a bit more accurate measurement.

So we know that this bit of rock is level with the lake, and we know that the bottom of the lifthouse is roughly at the top of the rock (maybe sunk a bit deeper thanks to foundations). In theory, this means we should be quickly able to determine the height of that arch, right?

Of course not.

It’s apparent that the shape of this rock has been at least somewhat changed when building the lift. The top is much flatter with the lift on. However, by eyeballing some things we quickly get how tall the part of the arch located under lake level is: anywhere between 17.5 and 20 meters.

As the last step, we can now pull up the screenshot that shows the Berk in its full flory, from the lift rock to the sea and re-calculate our heights:

So that’s just over three kilometers for the settlement and six to six and a half for the entire island.

This seems to be the definitive, most accurate answer possible.

Follow-up maths: The Lift and transfer stations

Since the New Berk is just over three kilometers above the sea, there’s one more thing that we need to keep in mind: the ropes have a maximum possible length before they break under their own weight. Let’s see if the lift we see at the end of the movie is even possible.

I will use this as a baseline when calculating my loads. 2″ (48mm) rope would weigh about a kilo and a half per meter, and it’s gonna break when you hit 120 kN (about 12 tons). This means that once the rope is 8 kilometers long, it won’t be able to support itself. But while this number is significantly over the 3.35 kilometer worst case scenario, it includes only the weight of the rope. It doesn’t account for the force of the ship being lifted, additional friction created by the wheels — both on top of the lift, as well as wheels that keep ships on tracks (you can see guiding tracks on the screenshot) — the lateral force that the wind exerts on the rope (as rope gets longer, those forces can get very significant), and any additional stress/force exerted by wheel/rope slipping and then quickly tightening.

If you don’t want to run the risk of rope snapping halfway up, you really want to treat maximum safe load as the maximum weight you can hang from that rope. The site suggests that maximum safe load is 1/12 of the breaking point, so we’ll take that as a gospel. 1/12 out of 120 kN is 10 kN or about 1 ton, which means you’ll need to plant a lift station once every 650-700 meters along the way if you want to lift nothing at all, and 300-500 meters (depending on how much rope can you afford) if you actually want your lift to be useful.

This means we’re probably looking at 8-10 transfer stations along the way. One could argue that, vikings being vikings, they had no OSHA and other pesky safety bureaus and organizations. On the other hand, though, they probably knew their ropes and how much they can carry, and knew their limits. Even if they initially felt a bit more adventurous and cut corners on the transfer stations (having as few as three or two, maybe none) they’d eventually come to the conclusions that transfer stations are not optional and built a proper number of them.