brumbarr

The maths of aiming mechanics explored.

15 posts in this topic

After a thread on the forums about  aimtime and how it works. I decided I wanted to figure it out exactly. All we know for now is that aim time is the time it take to reduce the aimcircle to 40% its size, dispersion is something that  say show much the circle gets bigger and accuracy is the size of the aimingcircle when fully aimed. However, we do not know  the exact relations between these 2 and how exactly they  influence the size of the aiming circle at all times.

So thats what I set out to do, finding a mathmatical description of the size of the aimingcircle. The method is simple: measure the size of the aimingcircle for different tanks and speeds in a trainingroom.   Thanks for uglycousin for giving me a second person to set up the trainingroom.

To measure the size of the circle, first I  did the test driving in the room. Then I watched the replays and paused at certain moments. I then took a screenshot of  my whole screen, makign sure I was always in 8x zoom. Then I took those screenshots into paint and measured the circle diameter in pixels.

I will now describe the process and results of my investigation. But if you dont want to read that, scroll down tot he conclusion on the bottom.

Disclaimer: the following formulas are NOT what WG uses, I made a linear  model that describes the size of the aimingcircle as close as possible. 

Aiming circle bloom

I assumed there where 3 variables that had an influence on bloom: speed, dispersion and accuray. I tried to do test in which I held 2 variables constant to  see the influence of 1.

 I started  with gathering data of 2 different dispersion numbers for which I picked 4 tanks with different accuracy and measured the size at each speedincrease of 10 untill 50 kph.

These are the raw results:

lucGfBq.png

 

From that I made a graph of the dispersion in function of speed , and calculated the gradient  of the graph assuming linear increase.  Then obviously  the  aiming circle size = C*v+accuracy.

With V=speed and C being the gradient, which consist of unknown  factors. To check the linear approach was decent I plotted the model and experiment:

ry3mSBe.png

 

As you can see the linear approach to the speed factor isnt perfect but not massivly different, only in the middle it differs.  I am happy enough with this.

Now we need to determine what the C factor consist off. Since there are only 2 variables left, it has to consist dispersion or/and accuracy components.

 

As you can see in the data, with the same dispersion numbers, the aiming circle fort he same speed is bigger when the accuracy is bigger.  So there has to be an accuracy factor in C,  which is proportional to accuracy.   

Here you can see accuracy vs circle size:

5ZITQUm.png

 

As you can see, the increase isnt marginal. We can now rewrite our formula as:

Size=Acc(D*v+1)

With D an unknown factor containing dispersion in some form.  As we can see, size of aiming circle is directly proportional to accuracy. So an increase in accuracy of 25% will results in 25% better gun handling.  This is why the E50/E50M have such amazing gun handling , their dispersion isnt great , but good, but due to the very good accuracy their gun handling is much better than at first glance.  The WZ-132-1 has the exact same dispersion values, so you would think the gun handling would bet he same, but no, since it has 33% worse accuracy is will have 33% worse gun handling, which is massive! Thats more than a vstab! 

Next task is determining the factor D. The only  variable left is dispersion, so I tested different tanks with differnt dispersion at the same speed, their accuracy was different, but thats fine, sicne we can normalise for that.    These numbers showed that the factor D was proportional to the dispersion values, so D=c*dispersion, with c an unknow constant.

Now the formula looks like this:

S=Acc(c*d*v+1)

Determining c was done by plottign the experminetal result and  trying some numbers until the model best fits the experiment. I took c=0.68.

The influence of dispersion can be see in this graph:

xK8nmDx.png

Now we have a formula that gives a perfect description of aiming circle size in function of all variables.  Next up is determing a  the time it takes fort he circle to shrink, or the actuall aiming time for the tank.

 

Aiming time:

We know aiming time is the time it takes fort he circle to shrink by 60% its startign size.

 So we can write:

S1=S2*(4/10)^(t/T) with T=aiming time, S1 size after time t, S2= starting size.

Solving this for t we get: t=T*(log(S1/S2)/log(4/10)).

 

We can now determine the time it take from any speed to reach any size we want.

To determe the time it takes to fully aim, jsut replace S1 by the accuracy of the gun. Note this time is independant of accuracy! ( which is logical, since it needs to go to  a smaller circle but also does it faster, these 2 cancel out)

Plotting this for 3 different tank in fucntion of time comming to a stop from a speed of 50 (40 for conway) we get:

kJEdGx6.png

 

Influence of equipment/skills etc.

Now that we have every formula we need we can quantify the influence of equipment/skills/directives/modules. To do so simply multiply the variable that gets influenced by (1-0,01*improvement in %). Dispersion values only get influenced by vstabs and the smooth ride skill.Other equipment only influences the accuracy value.

 Note that the same improvement to acc or dispersion results in a bigger improvement in size for what improves acc than what improves dispersion. Vstabs for example do not make the size of the circle shrink by 20%, they make the increase in size  less by 20%.

Lets take a look at a common dillema:vstabs vs gun laying drive, lets try this on 2 different tanks:

Uz6R9iQ.png

zfC2SPp.png

 

We can clearly see what the difference in vstab and gld is, vstab makes the circle smaller, so you start smaller but the decrease is still the same, gld starts at a bigger size but then starts to decrease faster, catching up tot the vstabs. In the BCs case, the time to fully aim is actually lower when equiping gld than when equiping vstabs.

Mathmaticly, gld decrease the total time to aim by 10%,  whereas vstabs decrease the total time to aim by subtracting 20% *initial size. To know wether vstabs or gld is better depends on the tank and how much you want to aim, you can determine this by pluggin in the numbers and plotting it for each vehicle, sicne i twill be different for each.

As general rules however, these apply:

·      - Bad dispersion + bad aimtime:

 Vstab better, unless you  fully aim from full speed.

·     -  Bad dispersion + good aimtime

:Vstab better, unless at high speed when fully aiming.

·       -Good dispersion + bad aimtime

:Vstabs always superior

·       -Good aimtime + good dispersion

:Vstabs always superior

 

 

Conclusion and TLDR:

·      - Accuracy has a massive  influence on aiming circle size on the move, they are proportional.

·      - Aiming circle size is proportional to speed/dispersion.

·       -To determine what gun has better actual gun handling: multply accuracy with dispersion, the lower the numbers the better the gun handling.

·     -  Size of aiming circle= Acc(0.68*d*v+1)

·      - Time to fully aim = Aiming time*(-log(0.68*d*v+1)/log(4/10))

·      - Vstabs is superior to gld in most situations.

·     -  Influencing accuracy gives a better boost than influencing dispersion values.

 

Whats next?

Next up I need to investigate how turning the turret and hull   effect dipsersion and work with the above formulas.

 I wil also try to combine this with  my previous thread where I determined shot distribution in the aiming circle, then I can plot change to hit a target vs time and determine the optimal time to shoot.

 

I hope you enjoyed the read and that i twill help you  determinign how a tank will perform. I hope that youtubers do become aware that accuracy has a massive influence over dispersion, as  currently reviews are misleading since they dont know what actually effects gun handling. Spread the word!

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Great job, I think I have read something similar before at wotlabs, but it is more easy to read than previous version. :)

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Good to see you here. I'm on the pubs under different nick though but I'm happy you are finally here. Good job. When I have some free time I really need to calculate those for the 30b and see how the rebalance really influenced it.

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Check out the Discrediting Smooth Ride Once and for all thread for the accuracy and dispersion calculations from the game client.  Also check out the How does aiming time and turret traverse work thread and the  How it Works : Accuracy, Aimtime, Dispersion... as impacted by skills/equipment/modules thread for some background info that may assist you moving forward.

Please continue the good work but also please define your terminology.

17 hours ago, brumbarr said:

All we know for now is that aim time is the time it take to reduce the aimcircle to 40% its size

It is commonly understood that "aim time" is is the time to reduce the aim circle to one third (33%) its size... do you have a reference for 40%?

 

I found your TL;DR is a little mid-leading (maybe because of the terminology definitions.)

 

Conclusion and TLDR:

·      - Accuracy speed and dispersion has have a massive  influence on aiming circle size on the move, they are proportional.

·      - Aiming circle size is proportional a function of accuracy, speed/dispersion and time.

·       -To determine what gun has better actual gun handling: multply accuracy with dispersion do the calculations, the lower the numbers the better the gun handling.

(if by dispersion you mean you are automatically including the speed component)

·     -  Size of aiming circle= Acc(0.68*d*v+1)

·      - Time to fully aim = Aiming time*(-log(0.68*d*v+1)/log(4/10))

·      - Vstabs is superior to gld in most situations.

·     -  Influencing accuracy gives a better boost than influencing dispersion values. Lowering dispersion values (equipment) and controlling speed (play styles) generally give a much better boost than influencing accuracy (BIA, equipment, gunner skill).

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8 hours ago, 8_Hussars said:

Check out the Discrediting Smooth Ride Once and for all thread for the accuracy and dispersion calculations from the game client.  Also check out the How does aiming time and turret traverse work thread and the  How it Works : Accuracy, Aimtime, Dispersion... as impacted by skills/equipment/modules thread for some background info that may assist you moving forward.

Please continue the good work but also please define your terminology.

It is commonly understood that "aim time" is is the time to reduce the aim circle to one third (33%) its size... do you have a reference for 40%?

 

For the aimtime figure I used to game client as source, movign over the aimtime the popup says:  the time it takes for the aiming circle to shrink by 40%. But I will try to do some test to confirm this figure.

8 hours ago, 8_Hussars said:

I found your TL;DR is a little mid-leading (maybe because of the terminology definitions.)

 

Conclusion and TLDR:

·      - Accuracy speed and dispersion has have a massive  influence on aiming circle size on the move, they are proportional.

   This is correct as it was, I mean accuracy as the final accuracy of the gun as given in the client or tanks.gg. 

8 hours ago, 8_Hussars said:

      

·      - Aiming circle size is proportional a function of accuracy, speed/dispersion and time.

 

        No native english speaker, sometimes hard to convert mathmatical terms in english from dutch. I meant 'evenredig' in dutch if that helps :P

8 hours ago, 8_Hussars said:

 

·       -To determine what gun has better actual gun handling: multply accuracy with dispersion do the calculations, the lower the numbers the better the gun handling.

  This is also correct as I originaly wrote it, multiplying accuracy with dispersion gives an excelletn single number figure for how  much the circle gets bigger and allow tanks to be compared without doing any compliacted calculations. Its a number for comparison, not for determining absolute size.

8 hours ago, 8_Hussars said:

 

·     -  Size of aiming circle= Acc(0.68*d*v+1)

·      - Time to fully aim = Aiming time*(-log(0.68*d*v+1)/log(4/10))

·      - Vstabs is superior to gld in most situations.

·     -  Influencing accuracy gives a better boost than influencing dispersion values. Lowering dispersion values (equipment) and controlling speed (play styles) generally give a much better boost than influencing accuracy (BIA, equipment, gunner skill).

Euhm? No? Did you read my thread actually? If you look at the formula accuracy has a bigger influence on aimign circle size than the dispersion values. Improving accuracy by 5% has a bigger improvement in gun handling than improving   dispersion by 5%.  Easy to see why if you look at the formulas.

 

 

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3 hours ago, brumbarr said:

Commets

A lot of this groundwork has been covered and vetted thoroughly already.  So I was just trying to illustrate that; so you are prepared to either agree with that work or explain why you propose it works differently and clarify what you actually mean with your mathematical modeling.
 

Quote

 

ComputedDispersion = GunAccuracy * SQRT (1 + DispMovingTraverse^2 + DispHullTraverse^2 + DispTurretTraverse^2)

where:

GunAccuracy = ReportedGunDispersion

DispMovingTraverse = DispersionMoving*HullSpeed

DispHullTraverse = DispersionHull*HullTraverseSpeed

DispTurretTraverse = DispersionTurret*TurretTraverseSpeed 

 

These calculations has been verified in the game files and it seems obvious that the proposed "accuracy * dispersion" is overly simplistic.  In fact it seems obvious that limiting speed and lowering dispersion affects final Gun Accuracy (ComputedDispersion) significantly more than nominally reducing gun accuracy.  The movement and dispersion is the reason that the aim circle grows so big so fast.

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On 1-12-2017 at 2:59 AM, 8_Hussars said:

A lot of this groundwork has been covered and vetted thoroughly already.  So I was just trying to illustrate that; so you are prepared to either agree with that work or explain why you propose it works differently and clarify what you actually mean with your mathematical modeling.
 

These calculations has been verified in the game files and it seems obvious that the proposed "accuracy * dispersion" is overly simplistic.  In fact it seems obvious that limiting speed and lowering dispersion affects final Gun Accuracy (ComputedDispersion) significantly more than nominally reducing gun accuracy.  The movement and dispersion is the reason that the aim circle grows so big so fast.

 

I come to the same conclusion as the formulas you showed, I think we are just having a misunderstanding here.

So the formula you have is:

ComputedDispersion = GunAccuracy * SQRT (1 + DispMovingTraverse^2 + DispHullTraverse^2 + DispTurretTraverse^2)

 

My first point was: 

- Accuracy has a massive  influence on aiming circle size on the move, they are proportional.

Which is entirely correct, if you look at the formula , everything is multiplied by  gun accuracy, so it has a massive influence, improve accuracy by 20% and your aiming circle at all times gets better by 20%. I am not sure why you think this is wrong, its simply looking at the formula.

·      - Aiming circle size is proportional to speed/dispersion.

Obviously, we agree on this.

·       -To determine what gun has better actual gun handling: multply accuracy with dispersion, the lower the numbers the better the gun handling.

·     -  Influencing accuracy gives a better boost than influencing dispersion values.

So, if we take the formula, and we improve the accuracy by lets say 5% we get:

ComputedDispersionImproved =0.95* GunAccuracy * SQRT (1 + DispMovingTraverse^2 + DispHullTraverse^2 + DispTurretTraverse^2)

So:  ComputedDispersionImproved = 95%*ComputedDispersion

A 5% improvement in accuracy is a 5% improvement in ComputedDispersion.

Now lets improve dispersion values by 5%:

We get :

ComputedDispersionImproved =GunAccuracy * SQRT (1 + (0.95*DispMovingTraverse)^2 + (0.95*DispHullTraverse)^2 +(0.95* DispTurretTraverse)^2)

=GunAccuracy * SQRT (1 + (0.95)^2*(DispMovingTraverse^2 + DispHullTraverse^2 +DispTurretTraverse^2))

Which is ALWAYS bigger than: 0.95* GunAccuracy * SQRT (1 + DispMovingTraverse^2 + DispHullTraverse^2 + DispTurretTraverse^2)  

So improving accuracy by a certain percent will always have a bigger effect than improving dipsersion by the same percent. Sipmly because of the fact you have that 1+ in the square root.

 

Now why I think acc*dispersion is a good measure of gun handling:

We take a look at  only movign dispersion, so the rest is 0, I use D=DispMovingTraverse, A=GunAccuracy. S=ComputedDispersion.

We can write that as :

S=A*sqrt(1+D^2)

S^2=A^2 *(1+D^2)

S^2=A^2 + (A*D)^2

The first term in the sum is the size of the circle when the gun is fully aimed, when the tank is stationary.

The second term tells how much the circle gets bigger. How much the circle gets bigger on movement is what I call gun handling. Just like  effective traverse isntead of nominal one, this is effective dispersion instead of the displayed one. With this value you can compare tanks with different accuracy to eachother and tell which one will have a bigger size increase when moving.

 

If you tell me I am wrong, atleast tell me exactly in what I am wrong. 

 

 

 

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It's a good stab in the dark for sure but this is literally from the game files:
 

Quote

ComputedDispersion = GunAccuracy * SQRT (1 + DispMovingTraverse^2 + DispHullTraverse^2 + DispTurretTraverse^2)

where:

GunAccuracy = ReportedGunDispersion

DispMovingTraverse = DispersionMoving*HullSpeed (kmh)

DispHullTraverse = DispersionHull*HullTraverseSpeed (Deg/s independent of turret movement)

DispTurretTraverse = DispersionTurret*TurretTraverseSpeed (deg/s independent of hull movement)

Then Smooth Ride seems to be better than Snap Shot in many cases and the benefits of both are minor at best and misleading/useless at worst.

now with formatting for easy viewing!
disp = Gun accy*sqrt(1+forward moving^2 + hull turning^2 + turret turn^2)
                   ^                        ^                                ^                         ^
              base value          SR skill                        nada                Snap skill    <-- skills that help. BIA helps all

and the files you can verify it in is: 
res/scripts/avatar.pyc
it's unencrypted but you do need a tool reader to view it. Also the lang is python if its not obvious.

My blog on the matter:

8_Hussars and my work on SR/Snap/Clutch:
https://docs.google.com/spreadsheets/d/1TqvlcnQIdPpAzy3X0OK4lxdNjWoqo9ejcdRqu_HqIuc/edit?usp=sharing 

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14 hours ago, Wanderjar said:

It's a good stab in the dark for sure but this is literally from the game files:
 

and the files you can verify it in is: 
res/scripts/avatar.pyc
it's unencrypted but you do need a tool reader to view it. Also the lang is python if its not obvious.

My blog on the matter:

8_Hussars and my work on SR/Snap/Clutch:
https://docs.google.com/spreadsheets/d/1TqvlcnQIdPpAzy3X0OK4lxdNjWoqo9ejcdRqu_HqIuc/edit?usp=sharing 

My result pretty much agrees with that, I only did it for moving. So then that square root and power go away and have have the exact same formula I found. 

Only problem is I have a 0.68 coefficient because I found that was needed to fit the data, not using that like in the game files results in a worse fit to my data. The theoretical circle is much bigger than what I measured. So I must have done something wrong when measuring or there is something else  going on I dont know about. 

 

Not that much of a PC guy, so I rather gather data and get resutls that way than lookign at files.

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On 11/30/2017 at 4:42 PM, brumbarr said:

For the aimtime figure I used to game client as source, movign over the aimtime the popup says:  the time it takes for the aiming circle to shrink by 40%. But I will try to do some test to confirm this figure.

I verified that the client does indeed say "reduced by 60%" (or 40% of the value).  The wiki says 1/3 (or 33% of the value) however I also recall RichardNixon determining the value was actually e (which I thought incorrectly was actually 1/e or 36.78 %) ... so something to think about.

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8 hours ago, brumbarr said:

Only problem is I have a 0.68 coefficient because I found that was needed to fit the data, not using that like in the game files results in a worse fit to my data. The theoretical circle is much bigger than what I measured. So I must have done something wrong when measuring or there is something else  going on I dont know about.

In that reference above  RicharNixon also says:
 

Quote

- There's a visual limit to the size of the aim circle that depends on the mod used, but has no effect on the true accuracy. This was very annoying. When using a mod with a low cap (eg Jimbo's crosshair), shots will frequently land outside the visible aim circle.

Even running vanilla, I seem to recall that others have indicated a feeling that the aim circle may be mis-displayed in sniper mode (will look for references or maybe I am miss-remembering).  Where the actual circle size was not representative of the actual "Gun Accuracy" especially when very bad or at different zoom levels.  For example the T49 dispersion on turret rotation may not even be representable in sniper mode due to its calculated size.  But the game has to present something to the player...  That may be an issue if counting pixels...

Perhaps, a test in training room (no movement) with a tank 100m away and see if the aim circle (fully aimed) seems reasonable (same as advertised) as sized against the other tank at 0 and other zoom levels. i.e. at 100m and fully aimed do we get a correct measure of the advertised accuracy (aim circle size) by counting pixels and relative to the size of the target enemy tank...  

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On 12/2/2017 at 11:00 AM, brumbarr said:

If you tell me I am wrong, atleast tell me exactly in what I am wrong.

I am just trying to help and share some work that others have done along the same lines.  My disagreement with the assertion that accuracy is more important than dispersion (maybe still terminology issues) stems from previous analysis and the following document I prepared when the Tier 10 Light Tanks were introduced.

Using the equations noted above it was shown for  T9 Light Tank Dispersions that even when accuracy was nerfed the resultant buffs to dispersion improved accuracy on the move. from the previous iterations.

Now on absolute grounds yes a 5% reduction in the accuracy does not correlate with a 5% reduction in dispersion... but a non-commander crew member digressive stat can only be reduced by limited and finite values (-4.1, -6.2, -8.3, -10.2. and -12.1 %) where dispersion can be controlled and reduced from max to 0%

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12 hours ago, brumbarr said:

My result pretty much agrees with that, I only did it for moving. So then that square root and power go away and have have the exact same formula I found. 

It's more accurate to say that the variables become minimized when fully stationary. That's still the formula, it just doesn't have anything to chew on. It also explains why turret rotation is one of the biggest factors in the accuracy formula, followed by tank hull traverse.
 

4 hours ago, 8_Hussars said:

Now on absolute grounds yes a 5% reduction in the accuracy does not correlate with a 5% reduction in dispersion

Yeah it only winds up as a 2.5 or some such. same as BIA/vents, but every little bit helps in the fight against LCM rng shitcode.
 

12 hours ago, brumbarr said:

Only problem is I have a 0.68 coefficient because I found that was needed to fit the data, not using that like in the game files results in a worse fit to my data. The theoretical circle is much bigger than what I measured. So I must have done something wrong when measuring or there is something else  going on I dont know about. 

which would be true if the visual matched the actual numbers instead of just bouncing off the upper bound. Not your fault and as we said, you aren't far off the mark. It's just been talked to death already so we're just catching you up.
 

4 hours ago, 8_Hussars said:

Where the actual circle size was not representative of the actual "Gun Accuracy" especially when very bad or at different zoom levels.  For example the T49 dispersion on turret rotation may not even be representable in sniper mode due to its calculated size


This is most obvious in Derp tanks (the KV2 and T49 famously) It maxes out fast visually but you may notice that after you stop moving everything it still takes a bit for the ret to start shrinking. That's the theoretical bound finally syncing with the visual model. With the Sigma rework it's actually a fairly accurate model of dispersion. Nothing in the game has less than a .3 disp (and that's to give the 25%RNG space to play in.... thanks WG)

It's also worth remembering that the listed aim time(s) aren't until the Ret stops shrinking, it's to 1/3 of max.

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On 5-12-2017 at 2:25 AM, 8_Hussars said:

I verified that the client does indeed say "reduced by 60%" (or 40% of the value).  The wiki says 1/3 (or 33% of the value) however I also recall RichardNixon determining the value was actually e (which I thought incorrectly was actually 1/e or 36.78 %) ... so something to think about.

I was trying to test it in a trainingroom aswell. Basicly just recording a tank aiming, measuring initial size. And then measuring the size after 1 aiming time has past.  Didnt really work out because I recorded in the wrong format to wathc frame by frame. But ill try it again.

But then again, if I cant trust the client side reticle. Can we even test what the real percentage is supposed to be or do we trust WG?

On 5-12-2017 at 3:05 AM, 8_Hussars said:

In that reference above  RicharNixon also says:
 

Even running vanilla, I seem to recall that others have indicated a feeling that the aim circle may be mis-displayed in sniper mode (will look for references or maybe I am miss-remembering).  Where the actual circle size was not representative of the actual "Gun Accuracy" especially when very bad or at different zoom levels.  For example the T49 dispersion on turret rotation may not even be representable in sniper mode due to its calculated size.  But the game has to present something to the player...  That may be an issue if counting pixels...

Perhaps, a test in training room (no movement) with a tank 100m away and see if the aim circle (fully aimed) seems reasonable (same as advertised) as sized against the other tank at 0 and other zoom levels. i.e. at 100m and fully aimed do we get a correct measure of the advertised accuracy (aim circle size) by counting pixels and relative to the size of the target enemy tank...  

hmm, thats kinda annoying.

But I wil try the following:

First try what you suggested , so against a stationary target we know the  size of. See how bgi the aimign circle actually is.

Second, drive at a constant speed ( so double or triple R), measure that, and how it correlates with the forumula.

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42 minutes ago, brumbarr said:

I was trying to test it in a trainingroom aswell. Basicly just recording a tank aiming, measuring initial size. And then measuring the size after 1 aiming time has past.  Didnt really work out because I recorded in the wrong format to wathc frame by frame. But ill try it again.

But then again, if I cant trust the client side reticle. Can we even test what the real percentage is supposed to be or do we trust WG?

hmm, thats kinda annoying.

But I wil try the following:

First try what you suggested , so against a stationary target we know the  size of. See how bgi the aimign circle actually is.

Second, drive at a constant speed ( so double or triple R), measure that, and how it correlates with the forumula.

Good luck!  It's never as simple as it should seem with WG...

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