2519: Sloped Border

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Sloped Border
"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.'"
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.'"


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Every country has land and sea international borders that demarcate the extent of their territory and their legal jurisdiction. These borders are established through law, treaty, or consensus. Establishing an international border is maintained by present-day customs, immigration, and security checks. Some countries (like Cyprus) have established a buffer zone outside of their international border in order to gain additional protection during a conflict, and most countries have an offshore Exclusive Economic Zone in order to preserve exclusive proprietorship of marine resources such as oilfields and fishing grounds.

In this comic, Cueball and Blondie have established a "sloped" international border through a treaty. Usually borders are perpendicular to the ground[citation needed] 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 1D lines across the curved 2D surface. The precise definition is that a line from the center of the Earth through the point of the border is drawn. Sloped terrain is immaterial to the border of the air sovereignty which is still vertical, even if not perpendicular to 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 if another country's jurisdiction currently applies. With the help of the Global Positioning System this would be in principle possible, although the height information of GPS is less reliable. (It might be possible to program a computer to use altitude data from the airplane's altimeter along with latitude and longitude data from the GPS and a relevant ground relief database to make an accurate determination.)

Most countries would not agree to a border that cuts into their airspace and shrinks their territory as the altitude increases; most cases of countries losing area have come about as a result of trying to avert, or losing, an armed conflict. It is entirely possible that Cueball's country has compelled Blondie's country to accept its demands, of which the redrawn border is one. Alternatively, Cubeall's country may be deliberately reducing its own airspace purely because it will cause problems.

There is at least one famous case of a border being affected by elevation: the Franco-Swiss border bisects the staircase of the Hotel Arbez. Hence, although part of the upper floor is geographically in France, the entire floor is Swiss territory, because it is only accessible through Switzerland.

The mathematical computation for an angled air sovereignty seems relatively straight-forward at low level and could be expressed with a single line of code or a single equation, although the people acting on the information are likely unfamiliar with code and equations and likely use tools with completely no support for sloped borders. The mention of curvatures in the title text may reveal some emergent problems that need accounting for.

A totally straight line drawn far enough upwards at an angle will find the surface of the Earth curving away beneath it (not even considering terrain undulations) and the angle to the local vertical will reduce as it continues, tending towards vertical as you head towards infinite altitude.

Alternately (although it seems this is not the case) the profile of the sloped border may be assumed to remain at a constant angle to the shifting vertical, in which case it describes a certain form of spiral (if shallow enough it could loop round the Earth).

A third option is that it gains altitude at a constant rate, with respect to the passage of land measured on its surface track, to form a different spiral, in which case it will still loop around the Earth but at an angle that increasingly tends towards horizontal.

While the comic doesn't mention this, such a boundary should probably also extend underground, in the opposite direction. (The straight-line version, if implemented, will eventually reach a depth at which it is tangential to the radius and then rise back through the surface an equal distance further around the planet.) This would then impact, at practical depths for such things, planning rights for property foundations and, at deeper levels, mining rights for minerals.

Practically an upper-limit to a nation's claim (somewhat below satellites, e.g. the Karman Line) and a lower limit (well before reaching the Earth's mantle) will prevent many of these complications, together with intersections with other (probably vertical) 'territorial volume' borders that will supercede in any compound claims to ownership. - However, it is still very important to specify exactly which curve (i.e. with respect to what) the boundary is designed to respecting.

"GIS" refers to geographic information system, a set of tools and methods for capturing, analyzing and presenting spatial and geographic data. While altitude is already an (optional) element in the blocks of information, people developing these systems would be inconvenienced by the additional requirements demanded 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 international waters.


[Cueball and Blondie are standing on a podium. They are holding a document together between them, filled with unreadable text. On either side of the podium are two informational graphics each on a stand. They are placed a bit behind the back side of the podium. The graphic to the left shows a cross-sectional view of a non-vertical border, shown as a dotted line going up between Cueball and Blondie, who both are standing on the ground. The angle is indicated and noted, and the line tilts towards Blondie's side. The graphic on the right shows a skewed perspective of a similar setup of the non vertical border, shaded so what is behind it becomes gray. There are also some lines on this plane to indicate where it is. It almost looks like a window, but people can move through it. There are also two more persons than on the left, Megan, who is on the same side of the border as Cueball, and another Cueball-like guy standing next to Blondie. Megan is entirely on Cueball's side of the plane, but the other three are positioned so they are intersected by the 'shaded plane' of the border, with the effect that some or most of their bodies are beyond the sloped boundary, in the gray area, but not all. Cueball and Blondie are postureed in a mutual greeting across this border, as the others look on.]
Cueball: With this treaty, we are proud to announce the creation of the world's first sloped international border!
Angle: 74°
[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.

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

I would make the country's border an Alexander horned sphere. 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 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. 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) 06:34, 23 September 2021 (UTC)

Wouldn't sloped borders also have interesting consequences underground when mining, building tunnels etc. ? 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. 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" 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. 16:55, 23 September 2021 (UTC)

Also, NERD SNIPE! 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. 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. 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. 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. 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. 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. 20:17, 24 September 2021 (UTC) ...it probably does, alongside Pi being 4. ;) 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. -- 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. 03:31, 28 September 2021 (UTC)