Calculating a weapon's projectile accuracy.

[Gen.jamesWolfe]

So, this is the second tutorial I've posted here, which I've decided to release today after much delay, as I have managed to verify all the details on it's use. As has been mentioned before, I have been working on a mod for the SYW period, and have in the progress discovered new techniques and methods, which I hope to Publish in a series of tutorials, of varying sizes.

anyways, some background on this one: in making the SYW mod, I often have a nagging question: what is the accuracy of a missil type? how accurate is a cannon? a Howitzer? a company of fusiliers? what number should be given to represent the respective performances?

now for context, here is an example of a projectile from my mod:

projectile brown_bess                    ; * USED BY UK,HANOVER

destroy_after_max_range_percent_and_variation   30   90 
effect            musket
;effect_offset        -3
damage            0
radius            0.01 
accuracy_vs_units    0.39
min_angle        -35 
max_angle        80 
velocity        137 
bounce            0.5 0.6 0.5 0.5
display            aimed
model            SYW/data/models_missile/bullet.cas,  max
;effect_only

for a detailed look at these values, I would recommend this, with the note that damage in RTW is directly proportional (as far as I can tell), unlike what the author claims to be the case for M2TW. (affected_by_rain only exists in BI and M2TW).

For now, let us focus on the accuracy bit:the first thing that needs to be known, is what this value represents. now until relatively recently, this was not a well understood value, However, after myself and Paedric did some research on the topic a while back, we were able to figure out that the accuracy represents the maximum angle a missile will deviate from where it is meant to go, in radians (assuming the projectile is flat in trajectory). further, contrary to popular believe, there is no maximum: you could make the soldiers shoot in all directions in fact! in addition, the shots will fall into the resulting area (which is circular for a single shooter), in what is either a random, or normally distributed pattern with a low peak (hard to differentiate the two with what I know). One final thing: the accuracy of a given round is determined as it leaves the shooter. so say if the maxium deviation is ~0.3 rad, but to hit a target only needs 0.1, then the shot will be launched at an angle of 0.4 rad off the target. this is why when the shooting animation is triggered, the soldiers--particularly in low accuracy units--will be bent forward or backwards a bit more or less then what you would expect from the animation. So put another way, the number isn't so much a reflection of the missile's performance strictu sensu, but rather the aim of the soldier shooting the missile.

now this is fine and dandy--if you knew in advance what the angle of scatter is (which I did for muskets). However, most of the time, you don't really find many sources that conveniently and politely give you the maximum angle of scatter for historical projectiles . Instead, you will often find tables giving you a percentage of hits on a target of a giving size, the dimensions of the target, and the distance of the target from the shooter. a good example would be Napoleonystika's page on artillery accuracy. This is where this tutorial comes in: I will show you a possible method to derive an accuracy figure using this data.

you will need:

1-The distance to the target.
2-The size of the target. (the dimensions and shape, or at least the area of the target)
3-The probability of the hit (Directly taken from a percentage. so if 2% of the projectiles hit the target, then the number is p=0.02)
4-a good calculator. and if necessary the ability to make integrals.
5-patience. lots of it. and plenty of free time (my chronic unemployment has that covered for me).

bear in mind we are, when using just one data point, assuming accuracy decreases in a linear manner, rather than in an exponential manner. I'll get to the needed steps should the decrease be non-linear later in the guide.

anyways, what to do:

1-first, find the area of the target. how you do it will depend on the shape of the target, but either way, the area is a must.
2-divide the resulting area of the target, on the probability of the hit. this will give you a normalized area of scatter where the bullets could have potentially gone: note this does not mean that in real life the scatter was as (potentially) dramatic. so if the area of the target was 100 meters squared, and the probability of hits was 0.02, then the normalized area of scatter is 100/0.02= 5,000 meters squared.
3-now this part is a bit weird, but necessary to get the accuracy value: divide the above area by pi, then take the square root of the resulting number. This will give you a radius (naturally) of a circle. you want this, because the projectile will in the game, scatter and deviate from the target in a circular pattern--kind of like a shotgun blast. a series of these guys in a line, blasting off all at once, will produce the original rectangular shape (of the ideal area we just calculated: 5000 meters squared).
4-take radius that you calculated, and divide it by the distance from the shooter to the target. this will give you the tangent of the angle of scatter (i.e. the value of the accuracy). take the arctan of the tangent, and you'll get the angle (making sure to convert to radian, if you use degrees*), and therefore, the desired accuracy value!

using this technique, for example, I took this data (from Napoleonystika):

In 1782 the Prussian artillery conducted series of tests
for infantry muskets and artillery canister at 300 paces.
The target was a line 1.9m tall and 30 m long.
- 25 % accuracy for muskets
- 14 % accuracy for 12pdr cannon
- 13 % accuracy for 6pdr cannon

and arrived at a value of ~0.053 for the 6 pdr, and ~0.052 for the 12 pdr. (the pace is ~2 Prussian feet, in case you are curious)

this technique is useful for both direct fire, and indirect fire (from what I have seen), and as mentioned before, works best when assuming a linear decrease in accuracy over distance.

now, should the accuracy decrease in a non-linear manner, then do this:

-do the same steps as above, only for multiple points
-plot these points on a graph (calculator or by hand, up to you), so that the resulting scatter is on the y-axis, and distance is on the x-axis. you should get a function of some sort, which you may or may not have to normalize. it will often be exponential too.
-derive an equation to describe the line. as the decrease in non-linear, you must never have it intersect with zero on the y-axis. this is important for the next step.
-calculate the integral of the function, from where the closest distance was to the furthest. obtain the resulting area from that, and divide by two. this will give you the mean (IIRC). then use it to figure out (using the integral equation), at which point will half the total area be encompassed. This is the mean, which you will be using for the game. it won't be perfect, but it is better than just what it would be if it was linear.


Now looking at these, one question remains: does it work? does it approximate (even remotely) the accuracy of actual projectiles the data is meant to represent? well, having done the calculations for certain known performances of musket fire, I can positively say: yes. they do work, though obviously the match to real life will not be perfect, especially as it ignores the human element (the effect of panic for example). as such, the data will be an idealized version of the combat performance (where possible to discover).

for example, using this technique, I got an accuracy value of British foot guards at Fontenoy, of ~0.36--pretty close to the mean scatter of musket fire, going by the observations of Clauswitz and others (~0.39). in fact it is a bit better--which makes sense, as these are guards! at least in theory anyhow.... (this is assuming normal French spacing from the time, a distance of 30 English yards, and the estimated hit rates.).

the uses this has are of course, pretty obvious: no more does one have to "eyeball" the accuracy of a given projectile, but instead, could put some science into this particular matter. At least, that is the hope of this particular guide. One final thing about this particular tutorial that has to be noted, is that the trajectory of the projectile will be straight (i.e. there is not parabolic arc or similar). As a result, the angle of scatter is not the only factor that in the game will determine how many will hit a target. the velocity of the round, the size of the target--even the size of the round--can all potentially affect the results a bit, though the exact level is near impossible to determine.

In the future, I intend to post more guides, where applicable, on other things I've done, but until then, happy editing!

(final note: I decided to post this in the first place, seeing that as far as I can tell, no one else has thought this up. if anyone else has, I will acknowledge this here).


*to convert from degrees to radians, divide by 180, then multiply by pi.

Finally, I would like to thanks Paedric (once again), Napoleonystika, scotswars, etc, for their contributions to this particular quest.