# 2519: Sloped Border

 Sloped Border Title text: "The slope will be 74° at ground level." "Okay, I think we can hack together a ... wait, why did they specify ground level? It's 74° everywhere, right? ... Oh no, there's a whole section in the treaty labeled 'curvature.'"

## Explanation

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Every single country demarcates an international border with other countries. The borders are established through law, treaty or consensus. Establishing an international border is very necessary for customs, immigration and security checks. Some countries (like Cyprus) have established a buffer zone in place of an international border to gain additional protection during a conflict.

In this comic, Cueball and Blondie established a "sloped" international border through a treaty. Usually borders are perpendicular to the ground so that all the air(space) above the ground belongs to the same country. This is called Air sovereignty. Thus it suffices to define the border on the earth surface, as 2D lines. The precise definition is that a line from the center of the Earth through the point of the border is drawn. Thus if the terrain is sloped then the border of the air sovereignty does form a sloped line with the terrain.

If the borders were sloped (with respect to the horizontal ground level) an airplane would need to know its precise height to decide which country's rules currently apply. With the help of the Global Positioning System this would be in principle possible, although the height information of GPS is less reliable.

The mathematical computation for an angled air sovereignty seems relatively straight-forward and should be expressed with a single line of code. However, how the line is curved (as indicated by the mention of curvatures in the title text) may make the issue very complicated. Does it continue towards the boundary of space strictly linearly? Or does it curve to follow a spiricular 'plane' as the Earth's surface (above which a position dictates an altitude) falls away beneath it through the Earth's roundness?

"GIS" refers to Geographic Information Systems. People developing these systems would be inconvenienced by the border described in the comic.

It is possible this comic is inspired by such boundary disputes as the Beaufort Sea 'wedge' which, while in this case perpendicular to the surface, suffers from alternative interpretations of how to extend it from the shoreline out towards internatuonal waters.

## Transcript

[Cueball and Blondie are standing on a podium either for TV, or for the photo op when signing a treaty. They are holding a document together between them. In the background are two informational graphics.]
Cueball: With this treaty, we are proud to announce the creation of the world's first sloped international border!
[Caption below frame:]
If I'm ever put in charge of a country, I'm going to spend all my time trying to think of new ways to make life a nightmare for GIS people.

# Discussion

I might be old-fashioned, but I've always wanted to live in Mandelbrotistan 3D. 162.158.89.156 15:49, 23 September 2021 (UTC)

I would make the country's border an Alexander horned sphere. 108.162.221.245 03:21, 23 September 2021 (UTC)

Waaait ... how can we define the border to maximize area of both countries? I'm talking non-measurable sets invoking something like Banach–Tarski paradox here ... -- Hkmaly (talk) 00:01, 24 September 2021 (UTC)

At least this border doesn't have thickness. --Angel (talk) 04:48, 23 September 2021 (UTC) -yet 172.69.55.107 05:46, 23 September 2021 (UTC)

GIS: Geographic Information System, that are the systems where maps (and the borders) are defined. They won't care much though, because for them the ground information is the relevant one. Once you get into air, you'll get a problem, because if the border is very sloped, and not in average straight, then an airplane might still be in the airzone of a different country than where it's flying over. Which will cause all kinds of problems, security wise. Liechtenstein might loose all control over its airspace, yet their inhabitants want safety even from aircraft flying above them. Can't imagine that going well, but bureaucrazy is that: it creates paperwork when it is not busy enough with the procedures it already created. 172.69.55.107 05:46, 23 September 2021 (UTC)

Liechtenstein and air control is a bad example for the problems with sloped borders because it's quite often the case that the air space of one country is done by air space controllers of another country. In the case of Liechtenstein this is done by SKYGUIDE in Switzerland that is also doing it for southern parts of Germany (being responsible for the collision of two planes near Überlingen: https://en.wikipedia.org/wiki/2002_%C3%9Cberlingen_mid-air_collision) 162.158.89.156 06:34, 23 September 2021 (UTC)

Wouldn't sloped borders also have interesting consequences underground when mining, building tunnels etc. ? 162.158.88.239 08:39, 23 September 2021 (UTC)

That was my immediate concern as well: Residents of a single building could be divided between two jurisdictions, by residing one above the other. Mining & water rights & airspace might be similarly complicated. If airspace ended earlier than international waters, & undersea oil fields extended even beyond that... Oh, wait. 👀
ProphetZarquon (talk) 05:27, 25 September 2021 (UTC)

I introduced into the explanation a hint of the more precise problem with airborne geometry upon spherical (or, possibly, geodesic) coordinates. The shallower the angle, the more possible that the 'curves with the ground' altitude calculation is to actually wrap itself all the way round the Earth before (presumably), whatever altitude limit there is to make space the same upper edge as International Waters are to horizontal edges. Taking the Liechtenstein case, as above, you could easily enclose them in a 'pyramidal' (or wedged, if not applied from all around them) air-claim by angling over them - or greatly increase their air-claim over neighbours if the angle is away. With inverse issues for the Mineral Rights issue. You need to agree in advance what happens when angled boundaries hit perpendicular ones, and whether the 'rhumbs' projected from the border mash together when equidistant points on a crinkly border project their own air-distance line. And if it is from an agreed surface level datum or local ground level, with the complications that arise from both cases. (Yeah, I originally thought there were about four different bones of contention that need to be ironed out in the codicil on curvature, but I now think there's about six of them needing strict definition, not counting the compound cases which further may need specifying in advance or forever requirev adhoc arbitration.) And none of this even takes account of Relativity and curved space frame of reference that might very subtly shift whatever reference you just agreed upon, if you let it go high enough. 172.70.34.165 12:05, 23 September 2021 (UTC)

Should there be a part of the explanation talking about how GIS is already a nightmare? Because the hobby is "_new_ ways to make life a nightmare" 162.158.75.159 13:35, 23 September 2021 (UTC)

There should ; I expect it's because the borders are often defined by natural features, which may be very detailed AND changing in time. -- Hkmaly (talk) 00:06, 24 September 2021 (UTC)

It's not worth wondering: is the boarder sloped across cardinal directions, like East to West, or is it sloped inward or outwards from the country in question? If the latter, outwards will make it like a funnel, meaning country A has greater airspace than its surface area at "ground" level. (Which is another consideration: where does the initial angle begin?) If sloped inwards, well then that country loses a lot of advantage. If it's based on cardinal directions... I do not want to consider how many complications that would create along various sections of the border. 108.162.219.51 16:55, 23 September 2021 (UTC)

Also, NERD SNIPE! 108.162.219.51 16:56, 23 September 2021 (UTC)
My presumption would be that the line-on-the-ground, however it winds around (e.g. following the centre of a river, the apex of a ridge or the point-to-point (with or without Great Circle adjustment) between two defining nodes) is a sequence of presumed horizontal line segments of arbitrary length, normalised to be parallel to the horizontal at the whatever ground elevation they cross. That line and the perpendicular through that line from the centre of the Earth (the vertical, by all accounts) thus define the third mutually perpendicular line that is the 'slopeward'(/antislopward) baseline. The defined angle indicates the inclination from the vertical on the vertical/slopeward plane.
Where landforms complicate matters the border rises or falls across contours, or twists and turns with a convex and/or concave groundtrack, the dominant inclined border is that originating from the closest source-point wherever there is potentially conflict.
If Cueball's border was around an enclave otherwise within his area of control, this would result in a tent-like (but strangely irregular) territorial enclosure (assuming not truncated by the Karman Line 'air limit', or similar). But I think he's content to make this just any shared border (e.g. the mostly 'straight' US-Canada fifty-whatever-parallel one) which means probably all other territorial limits (in that case, maritime) remain vertical (certainly not similarly leaning, in non-right-prismatic form) and except in a very few edge cases would end up dominating the slope-vs-vertical intersections.
As to 'advantage' (except for the territory sloped away from), I don't think there really is one. At best, it makes true geofencing of drones a bit more complicated than saying "don't cross this line; don't go above/below these altitudes" for some doubtless functional reason. For the people in the RHS 'illustrative' sub-image, it seems to have no practical effect other than to identify limbs/other extremities as cross-border in rather more unusual slices of the body than a normal border-straddler would expect. 162.158.159.11 22:33, 23 September 2021 (UTC)

Hmm, defining the calculation of the slope is tricky indeed. The commenter above suggests that the slope is relative to "vertical". However, the interpretation of "ground level" could deal with "level" meaning the ground slope, not the ground height. In other words, consider the slope on the side of a mountain. Let's suppose that in a local area, with a section of border running north/south, the ground is sloped 30 degrees to the east. Does that mean the 74 degree border is 104 degrees to the east at that point? The ground changing shape (whether due to natural erosion or bulldozers) could change the borders significantly.108.162.246.220 00:01, 24 September 2021 (UTC)

Sloped borders wouldn't be so weird, if the 3D component of the borders would follow the same rules as most 2D sea borders would do, which is that the border lies at a line of points equidistant from the coastlines. If country A is flat and B has a mountain range and the border is at the foot of this mountain range which has a 30° slope, the border would be at a 15° slope inward from being perpendicular (to sea level) leaning in on A. 141.101.77.144 04:51, 24 September 2021 (UTC)

Borders are usually vertical; what definition of "vertical", though? Some geodetic system? (Which one?) Local plumb line? Something else? Sabik (talk) 06:48, 24 September 2021 (UTC)

Just had the additional thought that this is probably creating additional borders, which will need more treaties. If the sloping border is quite shallow, there's a chance that the country on the left (with more airspace than ground) will now have an airborne border with whatever country is on the far side of the country on the right. And conversely for the mineral rights issue, the country on the right may now have an underground border with whatever country is on the far side of the one on the left. How does international law deal with a border treaty between two countries causing a non-signatory to have an extra border with one of them? Would those countries need to be consulted? --Angel (talk) 09:21, 24 September 2021 (UTC)

This kind of thing can happen with tunnels and bridges. For example, the border between the City of London (north of the river) and the Borough of Southwark (south of the river) runs right down the middle of the Thames as you might expect. But the whole of London Bridge belongs to the City. So if you happen to be on a boat underneath the bridge and on the southern half of the river, you're in Southwark; while a passenger on a bus crossing the bridge directly above you is in the City. So the border really must be sloped - and curved - and, as you trace the arches of the bridge, in some places horizontal. 162.158.159.95 14:21, 24 September 2021 (UTC)

Note that the only data point we have is at ground level, and the knowledge that the border is at least somewhat vertically curved. For all we know, there may be specific plausible landmarks defining the border at different altitudes, and a from a long distance away, the border might appear to be essentially vertical.

If, say, planes define the border by the top of the mountain range, underground defines the border by the longest mines built by either country, and ground level is halfway up the mountain side, but angled to allow structural supports to be at a right angle to the slope.... And then you start defining curves to merge the various points into a uniform line plot..... This might almost make sense. And it probably returns to vertical once you get below the mines, or above the aircraft. 172.69.90.15 15:35, 24 September 2021 (UTC)

The Earth is not a sphere. In general the local vertical does not go through the center of the Earth. I would expect international law to be based on the local vertical (easily measured since antiquity by plumb bob or similar) rather than the line though the Earth's center (requiring surveying precision not widely available till the late 20th century). Or perhaps international law just defines the Earth as flat.172.68.65.107 20:17, 24 September 2021 (UTC) ...it probably does, alongside Pi being 4. ;) 172.70.42.129 13:15, 25 September 2021 (UTC)

I added the explanation with the line from the center of the earth, because that's how it is defined—independent of what you expect. See for instance "International Air Law and ICAO" from Micheal Milde p. 37. https://books.google.de/books?id=YqaYJ3R0nKQC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q=line%20earth&f=false. --162.158.90.23 09:45, 29 September 2021 (UTC)

If the border surface is tilted off vertical and also following a curved line on the surface, at sufficiently high altitude the border surface can self-intersect in a complex manner, producing enclaves of air/space/underground which are disconnected from the country that they belong to. That is, assuming you count the parity of border crossings to identify which country owns the space. You could also make a greedy border so the first time you cross is you are in the other country and stay there even in the detached enclaves. The enclaves would be even more exciting where the borders of three countries meet. I'm not even sure how the parity counting would work, when you are in country A and cross the B/C boundary. 108.162.241.143 03:31, 28 September 2021 (UTC)