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September 16

Geographical almost-centres and Croatia

Given the country's unusual shape, Croatia's geographical centre is rather awkwardly placed; if you Google <croatia geographical centre> you get lots of forums and similar content discussing the idea that its centre is actually outside the country, in western Bosnia and Herzegovina. From this I'm left wondering: (1) Are there any reliable sources for this claim? I couldn't find any. If so, it would be a good addition to geography of Croatia. (2) The geographical centre article discusses different methods of calculating the geographical centre of a region and the potential problems resulting therefrom. If Croatia's centre really can be defined to be in Bosnia, do all definitions put it there, or do some definitions put it in Croatia? (3) Is there any concept of most-centred-within-boundary? [This is the biggest reason I came to the Maths desk; I'm wondering if topologists would care about it?] Let us assume that Croatia's centre is outside Croatia: is there a term that refers to the Croatian location that is least-off-centre? It wouldn't necessarily be the Croatian location closest to the centre (imagine a narrow salient that would be closest but severely off-centre), but I suppose it could. I'm thinking of the point where, if you balanced a flat map of Croatia on it, the map would topple most slowly. Obviously some points are better than others — a point at the country's southern tip would be worse than places farther northwest — so I wonder if it's reasonable to define the best place when no place is ideal. Nyttend (talk) 07:51, 16 September 2024 (UTC)[reply]

PS, imagine Croatia like a balancing bird toy. If you broke off the bird's head, you probably couldn't balance it at all, but you'd do a lot better balancing on the body (or even the tail) than at the wingtips. Croatia lacks the "bird head", but you're probably better-off balancing in the northwest than anywhere else. Nyttend (talk) 07:55, 16 September 2024 (UTC)[reply]

There was a discussion not that long ago here: Wikipedia:Reference desk/Archives/Miscellaneous/2024 July 18 § Lake Lats and Longs. It contains, at least implicitly, answers to some of the questions. Since the notion of centre is not well-defined, neither is that of "least off-centre". The location nearest to a given point outside the area is on its boundary. The interior point furthest from the boundary works for most actual country shapes, including Croatia, for which this is a point roughly 20 km east of zagreb. — Preceding unsigned comment added by Lambiam (talk • contribs) 16:06, 16 September 2024 (UTC)‎[reply]

The visual center and algorithms to approximate the pole of inaccessibility are discussed in this 2016 Mapbox post. I thought that if you'd include just a spherical geometry of the Earth you'd get even more interesting questions, but it appears at a glance (? not sure?) the algorithm in the link already generalizes nicely to noneuclidean geometries and higher dimensions. SamuelRiv (talk) 17:46, 16 September 2024 (UTC)[reply]

Through any point on Earth there is (by the mean value theorem) a great circle that bisects the population of Croatia. Among such circles, consider the segments that cut the territory of Croatia into exactly two pieces. I propose the midpoint of the shortest such segment as a centre. (shamelessly OR) —Tamfang (talk) 01:29, 18 September 2024 (UTC)[reply]

If the border of Croatia is sufficiently fractal-like, there may be no great circle that cuts the area into just two connected pieces. --Lambiam 09:19, 18 September 2024 (UTC)[reply]

Doesn't need to be fractal; a C-shape with overlapping ends (ie a spiral with just over a full turn) can't be halved (by area) into only two pieces by an infinite straight line. There may be a coral atoll somewhere that approaches this? 213.143.143.69 (talk) 12:37, 18 September 2024 (UTC)[reply]

There's quite a few simply connected shapes that create uncomfortable solutions for this algorithm (so far splitting 3+ pieces, you can also just have a wide lobe with two long thin tails side-by-side; the shortest area or even-population bisector straight line would have to cut through both tails; the longer unbroken line that cuts the lobe can be stopped by making the lobe an S-turn; which can only be resolved (to retain contiguity) by removing the requirement of a great circle.

(The spiral atoll example need not be a problem, since the great circle need not be required to cut through the entire atoll, but just one segment.)

It's a good idea, but it needs a bit more refinement to get a sensible-and-unique definition in all cases. SamuelRiv (talk) 16:32, 18 September 2024 (UTC)[reply]

Seems to me your solution to the spiral problem (for which thanks) applies to any simply connected shape. —Tamfang (talk) 05:46, 23 September 2024 (UTC)[reply]

Could it be meaningful to measure the density of such circles passing through a given neighborhood? I think I see an approach or two, but it needs more thought. The maximum by this measure is my new favorite center. —Tamfang (talk) 05:43, 23 September 2024 (UTC)[reply]

Consider any point within the boundary. Calculate the maximum distance from that point to any other point within the boundary. Then choose the point where that maximum distance is least. This will give a 'centre' that is in some way the closest possible to the rest of the area. For convex shapes it is equivalent to the circumcentre (centre of the bounding circumcircle). Would this be unique for concave shapes? -- Verbarson ^talk_edits 11:38, 19 September 2024 (UTC)[reply]

Not necessarily; a symmetric C-shape where the ends curl back nearly to the centre would have two possible solutions, one on each tip. -- Verbarson ^talk_edits 13:27, 19 September 2024 (UTC)[reply]

1) For a practical problem like this, a finite set of 'diverse' solutions (or a continuous and symmetric set) is not really an issue -- just choose any one point; 2) you can also include the shortest path between two points within the concave polygon. A central consideration above is the calculation time.

Another possibility is to get the approximate convex skull, the largest enclosed convex polygon, which reduces you to an easy incenter calculation and you're done. SamuelRiv (talk) 18:24, 19 September 2024 (UTC)[reply]

Also, for cases like long bulbous C-shapes, we can adapt if we prefer the narrow midway 'neck' as opposed to the widest of the 'bulbs'. (To illustrate the scenario: consider North and South American -- what the cartographer calls the natural center of the supercontinent is either around the Isthmus of Panama, or it's at the geographic center of their choice of the gigantic North or South continent (or they can put the same label on both). One could find all the local extrema of the largest enclosed convex polygon at any point (assuming they are discrete sets, so the shape is not a perfect circular atoll for example, and if the approximate algorithm for the convex skull can be so adapted) and then choose the median polygon incenter (however you want to resolve it for even numbers). SamuelRiv (talk) 18:40, 19 September 2024 (UTC)[reply]

I assume the distance is measured on an interior path. —Tamfang (talk) 05:28, 23 September 2024 (UTC)[reply]

Either interior or straight-line. It may depend on whether you want to site a power station or an airport? -- Verbarson ^talk_edits 07:47, 23 September 2024 (UTC)[reply]