Cannon Inaccuracy Rework

[restive sable]

Currently, the way inaccuracy works for all projectile-based guns, as well as lasers in FtD, functions based on a rectangular, or uniform, distribution. This means that the azimuth and elevation axes are assigned a uniform inaccuracy, with all shells guaranteed to fall within the listed value.
This is, of course, unrealistic. Real cannons and ballistic systems of all kinds will usually have a Gaussian distribution, where shots will group more tightly towards the center with a few more spurious outliers.

{MATH AHEAD, PROCEED AT YOUR OWN PERIL}
variables going forward:
x = azimuth or elevation. Given the distribution for azimuth f(x), and the same for elevation g(x), the full distribution function h(θ, φ) = f(φ)*g(θ).
A = nominal inaccuracy value
σ = standard deviation
We can consider the inaccuracy distribution in FtD to be
F(x) = rect(x/2A)/2A.
This probability distribution describes a region from +A to -A, which integrates to 1.
It can be determined that given this function, the standard deviation is equal to
σ = A/sqrt(3).

Standard deviation can be best described as how inaccurate any particular distribution "feels."

Now, a Gaussian distribution function that has this same standard deviation can be produced, and is equal to:
G(x) = sqrt(3)•exp(-3x^2/2A^2)/(A•sqrt(2•pi))

Now, the obvious problem with the Gaussian distribution is that its domain stretches from positive to negative infinity. This means that there is an astronomically low, but nonzero chance of a given shell firing with an inaccuracy of 180 degrees and hitting its own cannon. To address this, I propose that any shot that would otherwise fire with an inaccuracy greater than 2A on either axis will instead be forced to fire with perfect accuracy. This covers the edge cases (there is approximately a 0.053% chance of shells falling outside 2A this way, and so it will not appreciably change the standard deviation.)

[muted bobcat]

funny thing about inaccuracy, it used to be gaussian vertically, and uniform horizontally. but now it's just uniform in both axes

I suppose a rectangular shape is much easier to calculate, though it would be nice to have something more circular. but in practice it doesn't affect combat that much

[ebon edge]

It might be nice for shots to somewhat favor closer to the center (on both axis) but still be mostly uniform. But it's not going to matter a whole lot if it's not.

[restive sable]

I know it's a QoL feature more than a proper balance change, in fact i specifically meant for it mathematically to be neither a buff nor a nerf.
However I can definitely see why it would be computationally expensive, especially with craft firing batteries of dozens of shells in one physics tick.

[mighty lake]

from experience with using gaussians for this exact purpose in quantities greater than in ftd, it’s not noticeably expensive

[thin knoll]

Yeah and if you felt like it you wouldn't even need to use a real gaussian, just approximate it, which can be done cheaply by just adding like 12 random numbers (avoids expensive math functions)

Could also do something like a cached box-muller transform
I'd think that the real question is balance, given that you'd need a significantly wider normal distribution to get the same average inaccuracy, and even then you'd get a lot more shots hitting close to dead-center which favors drilling weapons

[restive sable]

Yes, you could alternatively add N uniform distributions scaled to A/sqrt(N) to achieve the same standard deviation

[restive sable]

If the standard deviation is equal between two distinct distributions, it should in theory be neither better nor worse at drilling, especially because of the fact that cannon accuracy is also only a small piece of the puzzle here, namely with detection error adding its own form of inaccuracy

I specifically defined a Gaussian which would have the exact same average inaccuracy (aka standard deviation) as the current form of the vanilla inaccuracy distribution

[fluid locust]

Useful in game examples for this topic

Something to consider, realistic accuracy would be a big buff to the accuracy of projectiles., which may not be intended by the devs

[restive sable]

Consider that I enumerated a Gaussian function that would yield the same standard deviation of inaccuracy, thereby meaning it would be equally accurate on average.

The tighter concentration of many shots closer to the center is mathematically perfectly offset by the smaller amount of very spurious shots

[fluid locust]

I am not sure how you would balance a tighter grouping of shots to be just as inaccurate as things already are, if the inaccuracy isn't changing than how does changing the spread pattern not effect accuracy

Unless your turning the current volume of the square shaped spread patterns into a circle while keeping the same volume, the result seems highly redundant

[restive sable]

Essentially, there exists a statistical parameter for any given probability density function called standard deviation. It measures the average extent to which any given instance is expected to err from the average position. (which in our case is the true aimpoint).
Essentially, I gave a mathematical function which does two things.
-It bakes in a more circular distribution rather than square, and
-It produces a gradually decreasing probability that any given shot will be a certain error from the aimpoint, which is probabilistically more true to how actual cannons of all kinds function.

Shown here is the 1-dimensional probability density functions I described (current FtD model in red, proposed Gaussian model in green). Notice that the Gaussian function has a higher peak probability density function in the center, but extends out past the nominal inaccuracy mark.

basically, there is a mathematically rigorous way to prove that a cannon with this specific Gaussian distribution and one with the existing uniform distribution have exactly the same amount of average error on shots, and so should score hits on a given target equally often

[muted bobcat]

well, not necessarily

the gaussian would hit more often against a small target and the uniform would hit more often against a large target

[willow crane]

ngl, instead of gaussian, you could just do a pyramid, with the very edges having a 30% chance of the shell going there, and then increases in the middle, so instead of pure maths you can just have like 3 preset numbers

like 10% 20% 40% 20% 10%

in a square, like the game already has

would be a lot more mathematically simple

tough the outer parts have more area so youd have to mess with the percentages to make it so the mid part gets more shells

[tropic plover]

This would significantly complicate calculations that an advanced player would have to do to determine their precision and optimal range. How would you deal with that?

[restive sable]

Define "precision" and "optimal range"

[tropic plover]

It also would completely require redoing all inaccuracies in the game

Since precision would become a lot more different for all weapons

[restive sable]

If by optimal range you mean the range at which shells have the highest likelihood to hit, this value is trivial at 0m for any cannon

[tropic plover]

Expected rate shots will hit within some distance of the aim position

[restive sable]

It sounds like you're referring to std. deviation

[tropic plover]

No.

I want to hit a boat one kilometer away.
Given inaccuracy X, what percentage of shots will hit the block I'm aiming at within 5 blocks?

Assume trivial planar geometry

If I want to create a vessel that only shoots when the stability, range and inaccuracy would lead to a shot landing within some specified radius of target block, rectangular inacc is much much easier to deal with than normal

[restive sable]

This is trivial in desmos

Set up an integral

[tropic plover]

I want to do that in game

No integrals in BB

no error function

unless you want to approximate

[restive sable]

You can use the CDF function of the Gaussian for this

[tropic plover]

CDF of gauss has an erf which is not a thing in breadboards

[restive sable]

Basically a CDF (cumulative density function) is the integral of a probability density function which is what I showed here

[tropic plover]

The CDF of gauss uses a non-algebraic function erf

[restive sable]

It's reasonably approximatable with arctan

Or even a piecewise linear function if you're feeling extra engineer

[tropic plover]

Even then, disregarding the BB issues, this would completely and utterly change the balance of the game.

Especially stuff like 0Q lasers which would become significantly more penetrative

[restive sable]

Not really, as they would spend non-zero amounts of ticks shooting way outside the nominal inaccuracy metric

[tropic plover]

This would look very strange, don't you think?

A ton of laser shots will seemingly randomly drift a trillion light years away for 1 frame

They hit 40 times a second after all

But most shots will hit much tighter than they do now

That's why I'm saying they would become more penetrative

More shots will land at intended targets

[restive sable]

~68% of shots will land within one inaccuracy metric, 95% within two, and nearly all within three

[tropic plover]

^ look at your own graph here.

68% of shots will land within the area where current rectangular inacc is, but notice that within that 1 inacc, more shots land near the center.

[restive sable]

I am aware

[tropic plover]

That's kind of the thing though.

[restive sable]

With rectangular distribution, 100% of all shots land within a rectangular prism in space. Given that all of FtD's parts are roughly rectangular, this actually makes rectangularly distributed lasers significantly better at boring a minimally sized hole if the the distance error at target is roughly on the order of 1m, assuming the target is not at an angle

There is an effective maximum sampling pixel density baked into combat in the form of the fact that boats are made of 1m voxels, which completely overpowers any changes to the distribution when error is less than this distance (this is very much often the case for lasers, at least out to 1.2km or so with 0.05° inacc)

[tropic plover]

Okay, I did some calculations

Say, our inaccuracy square is 5x5. We want to bore a hole. That means the laser will only be hitting the central block 4% [1/25] of times with rectangular distribution.
With gauss, however, it becomes roughly 2.46%

So things actually become more inaccurate

That's a negative change in other words

Still a huge balance change

Things become around 33% less accurate overall

[restive sable]

Maybe make a chart detailing the probability of hitting center square for any 2n+1 length and width?

[tropic plover]

I'm monte-carloing it rn, writing out the formula analytically would make me tear out my hair

since blocks are square

the uhhh

the double integral for 2D gaussian PDF over -1 to 1 for both x and y

[restive sable]

Ah so you're doing this in 2d

Cool stuff

[tropic plover]

well duh

obviously

you're rolling the dice twice

for azimuth and for elevation

Also, I fucked up

it's not 2.46%

I did not account for tan

this is azimuth/elevation

fuck

uno momento por favor

[restive sable]

I should also mention that the Gaussian function in 2d inherently has radial symmetry, this would mean that you are neither rewarded or punished when firing at a craft that is at a steep angle

[tropic plover]

btw, you also did not account for how inaccuracy happens over distance I think

The distribution for ele and azi change might be uniform

but the effect it has on where the shot lands is not

It's already non-uniform

i.e. when a laser bores a hole, over infinite time it will be a rectangular hole

but short-term it's not

it also depends on inaccuracy itself of course

[restive sable]

This is extremely computationally difficult to test, but I would imagine that a Gaussian, being radially symmetric, would also punish certain shaped craft who happen to just barely take up all the space of their inaccuracy distribution

consider this case

[tropic plover]

Okay, so here's a more concrete example.
0.2 inaccuracy, 1000 meters distance: uniform distribution: 8.25% hit rate
Gauss: 5.06%

1 inacc: 1000 meters, uniform: 3.28%, gauss: 2.13%

Finally

0.05 inacc, 1000 meters

Uniform: 100%
Gauss: 55.9%

This is a huge drop

[restive sable]

see this is part of my point

consider a cube-shaped craft

it will almost always get hit by incoming shells where the inaccuracy is on the order of its total size, just because of its shape

rotating it by 45 degrees suddenly causes some of these shots to miss

[tropic plover]

Nobody makes cube shaped craft though

For obvious reasons

and rotating with least exposed area is literally the meta so

[restive sable]

yes but i mean

rolling with respect to the direction of incoming fire

should do nothing to help or harm you

yet with rectangular distribution, it does

[tropic plover]

It still would because of the way inaccuracy is set up

It does two rolls for azimuth and elevation separately

This inherently makes it triangular distribution with uniform

If it instead changed angle directly with one roll, then yes

Gauss multiplied by gauss is still gauss, but with a different scale

[restive sable]

rolling for multiple independent events is analogous to multiplication

[tropic plover]

^

Ah, found it, bessel function governs it

[restive sable]

ah yes, my beloved sombrero function

[tropic plover]

Regardless, because it sets azi and ele separately, it probably doesn't have quite the same mapping

[restive sable]

it would not be a triangular distribution going into a 2d graph.

rect(x)*rect(y) produces a uniform 1x1 rectangle in cartesian space

[tropic plover]

Okay, so I went ahead and implemented your idea

This is the histogram of hits for rect distribution.
The x axis is manhattan distance from the origin

1000 meters, 0.2 inacc

Gauss. I also did the with an inaccuracy greater than 2A on either axis will instead be forced to fire with perfect accuracy

oh wait

either

Yeah, this then

lol

either axis changes meaning a lot

@restive sable

Well, 2A implies that nearly 10% of shots must be nearly dead on.

Unless you mean on either as in, both must be >2A

and not one of them should be 2A for a shot to become perfect

2 sigma on either axis -> perfect shot makes gauss ridiculously more accurate than rect

[restive sable]

I realize I made a slight mistake, it should probably have to be 3A or similar

[tropic plover]

3A is this

@restive sable

Apparently perfect hits are still way more likely in gauss in this case, but misses will be further away

0.05 inacc

with 3 sigma cutoff