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Old 05-28-2019, 05:58 PM   #21
Agemegos
 
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Default Re: [Space] Mapping Large Flat Areas

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Originally Posted by RustedKitsune View Post
Thank you. Thrash, your formula gives me about 1.5 hexes between stars at 50% density. Rupert, yours gives about (CORRECTED) 1.13 hexes in the same situation. I think Thrash has provided the answer here.
It seems unlikely. With a 50% chance of a star in each adjacent hex there is 98.4375% chance of a nearest-neighbour at distance one, and if there isn't one at distance 1, then 99.9755859% chance of one at distance 2 (i.e. an overall probability of 1.56% chance that the nearest neighbour is at 2). The chance that the nearest neighbour is further than that is negligible (0.00038%), so the expected nearest-neighbour distance is 1.016 hexes (for stars constrained to be in the centres of hexes).
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Old 05-28-2019, 06:08 PM   #22
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Default Re: [Space] Mapping Large Flat Areas

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Originally Posted by Agemegos View Post
It seems unlikely. With a 50% chance of a star in each adjacent hex there is 98.4375% chance of a nearest-neighbour at distance one, and if there isn't one at distance 1, then 99.9755859% chance of one at distance 2 (i.e. an overall probability of 1.56% chance that the nearest neighbour is at 2). The chance that the nearest neighbour is further than that is negligible (0.00038%), so the expected nearest-neighbour distance is 1.016 hexes (for stars constrained to be in the centres of hexes).
That was done to check the formulas. What I am doing right now is writing up the statistical distribution of various world types in Traveller (grouped into classes so I didn’t have to solve for the 1200+ possible combinations of the three physical stats Traveller uses for worlds), so determining the average distance between all worlds preferred for colonisation in an area is actually fairly necessary. Especially on really big maps.
Edit: I like all three formulas, but I’m using Thrash’s to make it obvious that a neighbouring star could be 1 or 2 hexes away. Close enough for vaguely mapping an area.
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Old 05-28-2019, 06:22 PM   #23
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Default Re: [Space] Mapping Large Flat Areas

As an example, consider a map of 36x36 hexes, for an area of 1,296 hexes. Average stellar density is 50%, for 648 systems. Weíre interested in the habitable ones, which average 14.8% of all systems (these worlds are far more common in Traveller than GURPS), for 95.9 worlds. Eh, 96 worlds.
For all three formula, average distance between them is:
Thrash: 3.93 hexes. (4 hexes)
Rupert: 2.94 hexes (3 hexes)
Agemegos: 0.55 hexes
Umm... Age? I think this needs to be checked. It is 0.5/(sqrt(N/A)), right?
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Old 05-28-2019, 06:45 PM   #24
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Default Re: [Space] Mapping Large Flat Areas

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Originally Posted by RustedKitsune View Post
As an example, consider a map of 36x36 hexes, for an area of 1,296 hexes. Average stellar density is 50%, for 648 systems. We’re interested in the habitable ones, which average 14.8% of all systems (these worlds are far more common in Traveller than GURPS), for 95.9 worlds. Eh, 96 worlds.
For all three formula, average distance between them is:
Thrash: 3.93 hexes. (4 hexes)
Rupert: 2.94 hexes (3 hexes)
Agemegos: 0.55 hexes
Umm... Age? I think this needs to be checked. It is 0.5/(sqrt(N/A)), right?
0.5/sqrt(96/1,296) = 0.5/sqrt(0.0741) = 0.5/0.2722 = 1.83

Three things to note.

1. The formula is for the case of a uniform distribution where any point can have a star at it, not for the queer case in which stars are constrained to be at the centres of hexes.

2. You are using "hex" as both a unit of distance and a unit of area, where your hex-area is not the square of your hex-distance. That is an error. The formula I gave you is for distance and area in consistent units: used the way you used it it gives distances in terms of the square root of the area of a hex.

3. The formula I gave is for actual Euclidean distance, not for the measure produced by counting hexes.
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Last edited by Agemegos; 05-28-2019 at 06:57 PM.
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Old 05-28-2019, 06:55 PM   #25
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Default Re: [Space] Mapping Large Flat Areas

Youíre right, I was inputting it wrong. Now I just need to figure out how to convert itís value into hexes if each hex face to opposing face =1
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Old 05-28-2019, 07:13 PM   #26
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Default Re: [Space] Mapping Large Flat Areas

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Originally Posted by RustedKitsune View Post
Youíre right, I was inputting it wrong. Now I just need to figure out how to convert itís value into hexes if each hex face to opposing face =1
Well the area of a hexagon is ≈0.6495 times the square of the long diagonal, and the long diagonal is 2/sqrt(3) times the width across the flats, so the area of a hex is 0.866 square hexes, and a hex-distance is therefore 0.931 times the square root of the area of a hex, which makes a square-root-area-hex 1.0746 times a centre-to-centre-distance-hex, which means that 1.83 hexes = 1.97 hexes, and don't you wish you had used units of distance and area?

There are eighteen other hexes within two hexes of a hex, and if each has 50% of a star with 14.8% of a habitable planet that means 0.926 chance that each is uninhabitable and 25% chance that they are all uninhabitable. There is 75% of a neighbour at distance one hex or two hexes, which makes it seem to me that an average nearest-neighbour distance of 2.94 or 3.93 hexes seems implausibly large.
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Old 05-28-2019, 07:46 PM   #27
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Default Re: [Space] Mapping Large Flat Areas

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Originally Posted by Agemegos View Post
Well the area of a hexagon is ≈0.6495 times the square of the long diagonal, and the long diagonal is 2/sqrt(3) times the width across the flats, so the area of a hex is 0.866 square hexes, and a hex-distance is therefore 0.931 times the square root of the area of a hex, which makes a square-root-area-hex 1.0746 times a centre-to-centre-distance-hex, which means that 1.83 hexes = 1.97 hexes, and don't you wish you had used units of distance and area?

There are eighteen other hexes within two hexes of a hex, and if each has 50% of a star with 14.8% of a habitable planet that means 0.926 chance that each is uninhabitable and 25% chance that they are all uninhabitable. There is 75% of a neighbour at distance one hex or two hexes, which makes it seem to me that an average nearest-neighbour distance of 2.94 or 3.93 hexes seems implausibly large.
Yeah, I made some interesting choices. But I’m definitely paying attention.
Although, if you have an circle of hexes, r2.5 hexes, that’s 19 hexes, for 9.5 systems, and 1.406 inhabitable planets. Going out to 3.5, that’s 31 hexes, 15 systems, and 2.22 inhabitable planets.
Although that distance is just between easily inhabitable worlds, not all worlds.
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Old 05-28-2019, 10:06 PM   #28
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Default Re: [Space] Mapping Large Flat Areas

To have a nearest neighbour at distance x you have to have a neighbour at distance x and no neighbours at any distance less than x.

A hex has 6x other hexes at distance x for integer x, so the probability of a neighbour there is 1-(1-p)^(6x).

A hex has 3x(x-1) other hexes closer than x, and the probability that they are all empty is (1-p)^(3x(x-1)).

So the probability of a nearest neighbour at distance x is (1-(1-p)^(6x))((1-p)^(3x(x-1))).

So the expected nearest-neighbour distance is the limit of an infinite sequence with the nth term n(1-(1-p)^(6n))((1-p)^(3n(n-1))). That's a pig of a thing, but it converges pretty sharply for moderate p, so we can solve it numerically.

I summed the first hundred terms for each of a selection of probability values up to 95%, which will put the errors in the 12th decimal place even for the lowest P tabulated below. I hope the results will be useful.

Probability
of a feature

in each hexE(hex count to nearest feature)
0.1% — 16.19
0.2% — 11.45
0.3% — 9.36
0.4% — 8.11
0.5% — 7.25
0.6% — 6.63
0.7% — 6.14
0.8% — 5.74
0.9% — 5.42
1.0% — 5.14
1.1% — 4.91
1.2% — 4.70
1.3% — 4.52
1.4% — 4.35
1.5% — 4.21
1.6% — 4.08
1.7% — 3.96
1.8% — 3.85
1.9% — 3.75
2.0% — 3.65
2.3% — 3.45
2.5% — 3.28
2.8% — 3.13
3.0% — 3.00
3.3% — 2.89
3.5% — 2.78
3.8% — 2.69
4.0% — 2.61
4.3% — 2.54
4.5% — 2.47
4.8% — 2.41
5.0% — 2.35
5.5% — 2.24
6.0% — 2.15
6.5% — 2.08
7.0% — 2.01
7.5% — 1.94
8.0% — 1.89
8.5% — 1.83
9.0% — 1.79
9.5% — 1.75
10.0% — 1.71
10.5% — 1.67
11.0% — 1.64
11.5% — 1.60
12.0% — 1.58
12.5% — 1.55
13.0% — 1.52
13.5% — 1.50
14.0% — 1.48
14.5% — 1.45
15.0% — 1.43
16.0% — 1.40
17.0% — 1.36
18.0% — 1.33
19.0% — 1.31
20.0% — 1.28
21.0% — 1.26
22.0% — 1.24
23.0% — 1.22
24.0% — 1.20
25.0% — 1.18
27.5% — 1.15
30.0% — 1.12
32.5% — 1.10
35.0% — 1.08
37.5% — 1.06
40.0% — 1.05
42.5% — 1.036
45.0% — 1.028
47.5% — 1.021
50.0% — 1.016
55.0% — 1.008
60.0% — 1.004
65.0% — 1.002
70.0% — 1.000 7
75.0% — 1.000 2
80.0% — 1.000 06
85.0% — 1.000 011
90.0% — 1.000 001 00
95.0% — 1.000 000 02
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Last edited by Agemegos; 05-29-2019 at 06:57 PM. Reason: extra precision
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Old 05-29-2019, 10:17 AM   #29
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Default Re: [Space] Mapping Large Flat Areas

One problem that I see is that you can have locations overlapping in hyperspace that are hundreds of ly apart in normal space. For example, star A and star B are 141 ly from Earth and are 200 ly apart from each other, but star A is directly above star B. On a two dimensional system, they overlap, meaning that it should be easier to travel from Star A to Star B than from Earth to either (just going by real world distance is still a 3D system).
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Old 06-01-2019, 01:39 AM   #30
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Default Re: [Space] Mapping Large Flat Areas

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The p.d.f. for nearest neighbour at distance x, for a uniform random distribution (i.e a Poisson process) is given here as 1-e^-(density * volume). For the case of a Poisson process on a 2-D plane that ends up implying that the average nearest-neighbour distance is 0.5/sqrt(number/area).
Ah! It generalises to n dimensions with the appropriate generalisation of volume and density. The expected distance to the nearest neighbour is the radius of an n-ball with generalised volume equal to π/4λ, where λ is the generalised density (i.e. number of features per unit of generalised volume.
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