Calculating # of hits (Tornado Shot)

So I've been toying with how to calculate number of hits with tornado shot, and I think I've got a working model. I'm going to use this thread to process the math as I have time to work on it.

So here's a single shot (no extra projectiles):
ME ----- d1 ----- FOE ------ d2 ------- Explosion

If you draw a circle at the point of the explosion with a radius up to the FOE then the enemy's hit box acts as an arc on the circle, and the projectile hits if it passes through that arc. With three secondary projectiles you essentially have 3 chances to get hit that arc.

So chance to hit with a secondary projectile is:
3 * arc length / circumference of circle
= 3*(HitBox)/(2*π*d2)

A simple example would be firing at a monster 35 units away, with a hitbox 5 units wide, striking the ground 35 units behind the monster. The chance to hit with a secondary projectile wouild be:
3*5/(2*3.14*35)=7% of a second hit.
Not very high, tbh.
But if you aim right behind the monster(reducing d2 to about 5 units) you get:
3*5/(2*3.14*5)=48% of a second hit.

*** This highlights the importance of targetting close to your enemy! ***

If we use GMP the explosions occur in an arc, with a spread based on where you target the shot. It might be that the projectiles are always equidistant from each other, but I'll need to do some testing, and I'm still trying to work out the math to determine the distance from the foe for each additional projectile. I think I'll need the spread angle so I can do some trigonometry. I also need to find the hit box sizes of typical foes, because the size of the enemy will greatly impact the likelihood of additional hits. Then I'll just need to use some probability to check the odds odds of one hit, two hits, three hits, etc.

Some questions I could use some help with:
Q1a: Can a secondary projectile can hit a monster that was the direct target of the inital strike? If so, then a second hit would presumably be guaranteed since the explosion would occur inside the hit box of the enemy.
Q1b: Can all three secondary projectiles hit the enemy if it explodes inside them?? Or does the anti-shotgunning rule apply?
Q2: If the no-shotgunning system applies does that mean a monster can only hit by maximum two hits per attack (piercing first attack and one hit from secondary projectiles)?

We might need a developer to weigh in on some of that.

I'll come back later to do some of the GMP math.

Q1a: it can, and I imagine the hit would be guaranteed unless the mob is very fast moving or small.

Q1b: anti-shotgun applies - only one secondary projectile per burst of 3 can hit the same target.

Q2: without extra projectiles, yes. Each extra projectile can cause one extra hit (from its secondary projectiles).

I wonder now... is the answer to Q2 still true for enemies with huge hitboxes (cough kitava cough)? Though to be honest, this only really matters with additional secondary projectiles or lots of luck, since the normal spread makes it almost impossible anyways

If what you're saying is true then that brings up an interesting point: when you fire GMP Tornado Shot at very close range the explosions are almost colinear (virutally no spread), which would mean almost guaranteeing six full hits, or I suppose five if the explosions occur before the creature is pierced. That would then be immediately identifiable as the ideal use of the skill - walk up to the foe and shoot it direclty in the face point blank. Not to mention the clear benefit of using Point Blank keystone at this point.

That said, you might be mistaken. It's possible the explosions have a radius, and the projectiles fire out from the edges of that radius rather than a specific point (which would mean enemies inside the radius would not get hit). Do we have official confirmation anywhere?

Continuing with my ruminations and calculations:

The average humanoid player/monster has a hit box radius of '2', so their arc would be approximately 4 units.
(source: Mark_GGG)
Small monsters can be assume to have size '1', arc size 2 units.
Large monsters can be assumed to have size '3', arc size 9 units.

Later on I may try to calculate exact values but for now I think I can assume the arrows explode in a line perpendicular to the trajectory of the initial shot (there is actually a slight arc, but I think it will be numerically insignificant).

In that case d3, the distance from the foe to the next adjacent arrow after the first, is a simple calculation:
[(Distance from foe to first explosion)^2 + (Distance from first explosion to second explosion)^2]^1/2
And d4, distance from foe to the 2nd adjacent arrow, is similarly:
[(Distance from foe to first explosion)^2 + (Distance from first explosion to third explosion)^2]^1/2

The first variable is set by the player (d2), and the second variable is related to the distance of the player from the target spot (d1+d2), though we need more information about the spread to learn the specifics. For now I will work with a visual estimate using unit measurements from the wiki (https://pathofexile.gamepedia.com/Distance).
The edge of the screen is esimtated to be 85 units from the player.
A shot fired at half this distance (42.5 units) had a width of about two thirds that distance, or about 28 units. This puts the distance between the explosions at a half-the-screen shot at approximately 7 units.
Aside: the spread does not appear to be a linear progression, rather it jumps up a lot early on and then seems to expand slowly, so we cannot use this calculation for all distances.

The chance of being hit by a projectile from each explosion is thus as follows:
E1(hit)= 3*(HitBox)/(2*π*d2)
E2/E3(hit)= 3*(HitBox)/(2*π*d3)= 3*(HitBox)/(2*π*[(d2)^2 + (7)^2]^1/2)
E4/E5(hit)= 3*(HitBox)/(2*π*d4)= 3*(HitBox)/(2*π*[(d2)^2 + (14)^2]^1/2)

To determine average number of hits we then would need to use probability (we cannot simply sum the chances of hits), calculating the chance of not getting hit by any, then the chance of not getting hit by four, three, two, one, and zero. I'm a bit too tired to work out those formulas now so I'll come back to it in a bit.

Questions:
1. If anyone has some numbers on that spread please let me know! We need these values to build a function that can address all possibilities [which is ther first step to building an online calculator]
2. If anyone can confirm that the secondary projectiles come from a point, or from the edge of the explosion radii, that would be great.

Ok, I couldn't sleep without finishing my example :P

So now we just need to set of the instantiating variables.
Lets say the target is a normal size dude, radius 2 (arc 4).
Lets also say we fired our shot as close as we could get behind him without accidentally targetting him, or two humanoid widths behind him (d2 = 8).

E1(hit)= 3*(4)/(2*π*8) = 23.9%
E2/E3(hit)= 3*(4)/(2*π*[(8)^2 + (7)^2]^1/2) = 1.91/[64+49]^1/2 = 1.91/10.63 = 18.0% [x2]
E4/E5(hit)= 3*(4)/(2*π*[(8)^2 + (14)^2]^1/2) = 1.91/[64+196]^1/2= 1.91/16.12 = 11.8% [x2]

A quick glance suggests on average the dude will probably get hit by one extra projectile.
Chance of five additional hits is easy, just multiple all the five fractions together: 0.0108%
(about 1 in ten thousand)
On the other side of the scale the chance of no extra hits is: [chance of all others missing]=[0.761*0.82*0.82*0.882*0.882]=40%
So there's a 40% chance that no secondary arrows hit. Almost half the shots at half the screen away with a small distance between foe and arrow will result in no extra hits on a single target!

The middle values are a bit more complicated. There's probably a computational formula for it but I guess we'd have to calculate the odds of just E1 missing, then just E2, etc then sum all five to get the chance of four additional hits and so on. Any math experts want to jump in here? Otherwise I'll come back later, maybe tomorrow, and use an excel spreadsheet to work it out. Might be able to reuse the sheet for future calculations too (if I create all the formulae as functions and put variables somewhere on the table).

In any case once we have the chances of each possible event we can multiply each by tooltip dps multiplied again by the number of hits at that probability, sum the results, and we'll have the actual dps of the attack on a single target. I'm expecting a result somewhere near 2x.

"

FCK42 wrote:

I wonder now... is the answer to Q2 still true for enemies with huge hitboxes (cough kitava cough)? Though to be honest, this only really matters with additional secondary projectiles or lots of luck, since the normal spread makes it almost impossible anyways

Yes: only one arrow from each explosion can hit any individual enemy. This isn't "because they don't hit", this is because the skill is specifically built to only hit once that way by GGG.

Given that, more initial shots >>> most damage boosts, since each extra explosion generates a new "can hit the boss" arrow.

This is a pain in the ass without a decision tree calculator. If anyone knows of free one, would you let me know? As it stands I have to do it manually in excel and I'm at 10 calculations for two/three arrows hitting (with just five arrows via GMP). The process will get all the more painful if I introduce Additional arrow quivers and Dying Sun.

Ok, so here are the results using just GMP:

A: firing 25 units behind humanoid foe
B: firing 5 units behind humanoid foe
C: firing 5 units behind a large foe
D: firing 5 units behind an immense foe (max size, 5)

Scenario A B C D
No additional hits 70% 23% 8% 0%
One additional hit 26% 49% 44% 28%
Two additional hits 4% 27% 36% 20%
Three additional hits 0% 9% 21% 38%
Four additional hits 0% 1% 6% 29%
Five additional hits 0% 0% 1% 8%

% of DPS 35% 135% 206% 340%
^ from secondary projectiles.

If you are piercing with your first shot add +100% to get modified tooltip dps.

As you can see the dmg is considerably higher for larger enemies, and also the proximity of the click-point to the foe is significant. Just adding another 5 units to Scenario B (shooting '10' units behind the foe instead of '5') cuts the additional damage down to 79%

It's also worth noting that the additional two secondary projectile enchantment adds 66.6% to the probability of each hit, greatly increasing the dmg of secondary projectiles. In Scenario C, for example, the dmg goes from 206% to 340%. Including initial piercing hit in the calcutation (306% to 440%) this results in 44% MORE damage.

It's a good enchantment!

[the benefit caps out a bit on huge and immense foes where the first few tertiary shots are guaranteed to hit)]

Unfortunately because the decision tree isn't automated at this point it would be prohibitive to calculate scenarios for 6-8 projectiles. I may still do it eventually, but for now this satisfies my curiosity.

In comparison with barrage for single target, for example, a GMP Tornado shot targetted well on a large foe does 3x tooltip.
I'm not familiar enough with barrage to know if a well-targetted shot will hit with *every* projectile, or if spread will cause some to miss, but presuming they all hit then barrage is doing 8x tooltip, and a quick check shows tornado shot tooltip is roughly 85% higher than barrage.

1.85x3 = 5.55 vs 8.
Barrage still wins out vs large, and they tie for immense foes.
This is even true with enchantments, despite the Barrage enchantment being only 25% more dmg (8 vs 10 on large foes).

So keep that barrage gem in your inventory. :)

Unless of course we get confirmation that a tornado shot at point blank, where all the explosions overlap, hits the target with all secondary projectiles. If that is the case Tornado Shot blows Barrage away.

MAAAAAAAAATH!

[Unreal Voice]"MATHACRE!"[/Unreal voice]

...a few points that may be of use:

*Tornado Shot is guaranteed to fire a secondary projectile 'behind' itself, i.e. in the direction of the player, or was the last time I checked. GGG introduced this behavior specifically to make shot place for TS more pertinent for skilled players, since placing the shot immediately behind a target maximizes the chance of hitting it with multiple subprojectiles from a GMP'd Tornado Shotgun blast.

*Shotgunning applies to an individual Tornado Shot's subpros, and to all mother projectiles from the initial shot. A GMP'd Tornado Shot (5 Mothership arrows) is capable of hitting a single entity a theoretical maximum of six times - once with the parent/mother projectiles, which are counted as one cast/attack for the purposes of anti-shotgunning code, and once per subprojectile bundle. TS calculates each subpro burst as an independent attack, allowing multiple subpros from a GMP'd TS shot to impact the same creature, but only once per subpro burst. It's of not benefit to try and detonate a Tornado Shot 'inside' something's hitbox, it can still only be hit by one subpro per blast. 'Extra Secondary Projectile' Lab enchants increase the odds of hitting a specific enemy with one subpro per Tornado Shot click, but cannot shotgun with it.