Properly calculate city borders based on culture

[TomWerner]

We now have nearly tile-for-tile accuracy with the real game for loaded save files! There is 1 tile that is different from the real game to C7 in the screenshot below, which I've circled.

The difference I suspect is due to rules 7/8 from https://forums.civfanatics.com/threads/the-eight-laws-of-border-dynamics.106882/:

• VII. If an unassigned tile finds a tile of a single civ directly adjacent to both the NW and SE, it is assigned to that civ.

• VII. If an unassigned tile finds a tile of a single civ directly adjacent to both the NW and SE, it is assigned to that civ.

I'm not sure the best way to implement this without doing a second pass over all the tiles, but that may be what we need to do.

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This change required changing how we track tile owners, from a player owner to a city owner, so that we can correctly check rank and culture comparisons.

#288

[KulkoBSW]

IF you use an exponent of 1,8 it is working for all Culture size up to 20K (Which is a victory condition) but it breaks again slightly at 101K, no idea what formula they are using there.
So the better formula should be Power( (Power(abs(X1-x2),1,8) + Power(abs(y1-y2),1,8)), 1/1,8)

[TomWerner]

@KulkoBSW nice! I've updated the code and PR description. How'd you figure that out?

[KulkoBSW]

@TomWerner Have you tested it out with your code. I realize you still have that additional Sqrt in DeltaX/DeltaY and I am not totally sure how that will interfere with my test data. So the exponnent might still be slightly off.

As for How I got this? Studying math in University. changing the exponent changes the roundness of the circle and there is basically no other valid Metric on the plane. So I just wrote an excel to try out different exponents until I found one that fit.

[TomWerner]

Yeah, I tested it with a quick hack to exponentially increase culture, and it looked good, and I also loaded the save game I was using for testing and updated the screenshot.

The Sqrt(2) factor is needed because this grid is rotated, that factor cancels out the rotation.

[KulkoBSW]

I also checked it out on 5 editor saves and with that addition it works even for Culture 100K, so its probably the correct formula.