Ezreal: Trinity Force vs Infinity Edge

With the release of Pulsefire Ezreal, a lot of players have been picking up Ezreal and trying to learn him in time to prepare for the day they can play with the skin. I've come across a lot of these new-Ez players and their builds ranged from full AP to full AD to hybrid and everything in between. I was intrigued by two popular options however. It is widely known that Ezreal (along with Corki/Urgot) is one of the best users of Trinity Force among AD carries. It comes as no surprise then that a lot of these new Ez players rush a Trinity Force and forgo the traditional Infinity Edge rush. My curiosity was piqued and I had to get the math done behind this to see how much damage you stand to lose by rushing Trinity Force. So I present to you - Ezreal: Trinity Force vs. Infinity Edge.

Abilities

Ezreal has 4 abilities which scale off AP and 2 off AD. It stands to reason that the AP from Trinity Force would help him quite a bit in terms of damage output. Let's take a quick look at his ability set for reference.

Now having gone through his abilities. The next logical step is to take a look at the item builds in question.

Items

So cue the item descriptions:

The official LoL website fails to show all of Trinity's information in one box. They are missing the mana/health components in the above picture but more importantly Trinity Force also gives the following passives:

Unique: Your basic attacks have a 25% chance to slow your target's movement speed by 35% for 2.5 seconds. 
Unique: After using an ability, your next basic attack deals bonus physical damage equal to 150% of your base Attack Damage. 2 second cooldown. Does not stack with  Sheen or  Lich Bane.

 

That's a fairly straight forward description. An important note is the critical damage increase which will play a significant part in our calculations to come.

Assumptions

Now in producing the contour plot (rainbow graph), on the damage output increase that Infinity Edge gives over Trinity Force - a set ability combo is required. Having the advantage of foresight and being the author, I can tell you that Infinity Edge far outweighs Trinity Force in terms of damage output. So in the interest of showing data values where Trinity Force can hope to match up to Infinity Edge - these are the assumptions:

  • Ezreal has no runes and masteries equipped.
  • The only item he has built is a Trinity Force or a Infinity Edge.
  • His spell rotation consists of QWER, i.e. he uses every spell once.
  • He lands 5 basic attacks in one trade.
  • The bonus damage axis is any damage that is not IE/TF. 
Important Note: Since the bonus damage axis does not take into account any damage apart from the IE/TF, you would have to take into account Ezreal's base AD, runes and other items to read accurately. So at level 18, you would have to look at >101 bonus damage (assuming no extra items) and at level 1 >50 bonus damage due to your base AD as well as extra bonus damage due to runes/masteries. The graph has been designed this way so results can be read off for various points in the game. Bonus damage values from 0-50 are effectively impossible to hit values and should be ignored. They are in the graph simply to provide an axis starting from zero.
 
It's important to be clear that the more bonus AD (damage on top of the IE or TF) Ezreal has, the more it favours the Infinity Edge as the critical damage multiplier will boost the bonus AD. Many of you will be saying that Ezreal lands more than one Mystic Shot (Q) in one trade and you would be right. However, the more Q's he lands, the more it favours the Infinity Edge due to the 1.0 AD ratio. On with the graphs.
 

The Results

The results take into account sheen procs (the 150% damage bonus from Trinity proc - referred to as Sheen proc from here on), number of critical strikes and the various AD/AP ratios according to each item.
To help you understand the following graphs, I've made a video on how to read contour plots (aka rainbow graphs) to help prevent confusion. The video can be seen below:
 
 
 
 
 
 
Firstly let's take a look at the increased damage when every one of your five basic attacks procs Sheen.
 
 
As you can see - you effectively have to crit at least two times in this scenario to out do the damage of a Trinity Force since having 200 bonus damage on top of an Infinity Edge only happens at very late game when you have ~100 base AD and a fully stacked Bloodthirster. However this is an unrealistic situation as you will never proc Sheen five times out of five basic attacks.
 
What if you proc Sheen zero times?
 
 
Regardless of how many times you crit or how much AD you have, if you do not proc Sheen - an Ezreal with an Infinity Edge will always outdamage an Ezreal with a Trinity Force. Once again this is unrealistic as after you have used your QWER, you will at least proc sheen once. So let's take a look at a more realistic graph.
 
 
 
 Here we can see the damage output difference when you proc sheen 3/5 times - a much more realistic situation. In this situation, you will never outdamage someone with an Infinity Edge as well. As you can see, the trend is very clear, Infinity Edge outdoes Trinity Force in terms of damage output very very significantly. However for the readers who want to see one extra Mystic Shot in the mix, I'll include the following graph:
 
 
This is nearly identical to the graph above, with a narrower margin between Trinity Force and Infinity Edge. Once again, in a realistic situation, you will critical strike once out of five times and therefore still out damage Trinity Force.
 

Conclusions

Even after trying to give the edge to Trinity Force by limiting the study to a small number of basic attacks, including a full QWER rotation and numerous sheen procs. It is painfully obvious that the Infinity Edge is significantly better damage wise than a Trinity Force. It is important to note that an Infinity Edge also scales any other AD you might have due to the critical damage increase. Having said this, I would strongly recommend not rushing a Trinity Force first for damage purposes unless you absolutely require a bigger mana pool or the utility from the slows and MS increase. However that is my opinion and you are free to disregard it if you wish.

I hope you guys enjoyed a return to some serious mathcrafting after a period of absence of rainbow graphs. Please do leave a comment below on anything that might have passed through your mind while reading this :) Cheers :D

 
 
 
 
 
If you enjoy reading the article - I'd really appreciate you clicking the up arrow - thank you in advance!
 
 

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Cheers,
DiffTheEnder

Comments

  • #39 nornamor

    Second best item after IE, pure damage-vise is actually equal between fully stacked BT and PD.... Though unstacked, the PD wins outright (after IE + 1-3 dorans that is).... unless enemies have so much armor that LW is the best choice.... Also PD gives movespeed == utility and speeds up farming.

    Thats why  the standard build is (Standard, because it gives the most damage increase overall):

    1.1-3 Dorans + Berserkers greaves

    2. IE+ Vamp scepter(optional) 

    3. Zeal. (sell dorans at some point)

    4. Depending on enemy armor stacking, finish LW or PD, then finish the other one. 

    5.  Choose 2 of these items in most suited order (very game dependent): BT, QSS, second PD, Guardian Angel, Banshees Veil

  • #38 nornamor

    This post is nice to ppl who did not know this already, great article for the new Ezreal players, however every seasoned Ez player/or theorycrafter knows that bought BT and IE outdamage trinety BYYY FAAAR.... So this post becomes kinda stupid after that, since the reason for trinity, is not the damage, but the extremely nice buildup and power you gain in lane, mutch the same as the triple dorans+wriggels-->PD build for other ADs:  

    - Ezreal has a low mana pool early and very high damage burst damage..... for only 1250 gold you can increse that burst (sheen proc) and that mana pool, making you win trades in lane. 

    -  The Phage combined with 1-2 Dorans (1800-2200gold) gives you a Lot of damage and survivability for laning, making you win trades as well, and if you win trades the slow makes sure they have to blow summoners or you get a kill.

    --> These two items makes you win certain match-ups for sure, making you stronger and ahead of the enemy AD.... at this point you can snowball -->

    Depending on how long the game have lasted: 1. you build either straight for Trinity as you are still laning and this makes you stomp even harder in lane.

    2.  Build IE for maximum damage, and do a regular AD carry build, even though you have some nice options: phage+sheen --> Trinety after IE, or sell sheen+phage -->frozen mallet (makes you kite like a boaws).

     

  • #35 Pagezoso

    Now if IE is the best first item, what is the next best? Phantom Dancer or Trinity Force?

  • #34 alorama

    get an intrinity edge problem solved

  • #33 iceverfrost

    I'd be more interested in seeing a post on the Gold Efficiency/Value Added of going for Phantom Dancer vs Trinity Force as a second item for Ezreal assuming you rushed Infinity Edge first.

    Last edited by iceverfrost: 6/20/2012 11:47:08 AM
  • #32 alorama

    Now i might just happen to be a bit crazy in saying this because i personally love ezreal and play him loads and read up all the stuff on ezreal and tested all the stuff on him so im pretty sure that most of the time on ez unless u have mega sustain support or are destroying ur lane u rush a bloodthirster. NOT TO MENTION ezreal usually gets early game sheen unless when he b's he can buy straight up bf, the again after that unless he can get straight up IE/BT not trying to bash you thats just my personal opinion of what i like and believe is most common.

     

    Last edited by alorama: 6/20/2012 11:45:34 AM
  • #27 Purpose2

    I've never been sure what to pick up first on Ez, and even with this information I think I may still get Phage or Sheen first, but likely finish IE before finishing Trinity Force. - just going to be very dependant on the situation I guess! Thanks for this Diff The Ender! I've added the link to my weekly eSports news roundup. :)

  • #25 N4xu5

    would be way more interesting if its stronger to get pd or trinity on ez. Cause you cant rly compare utility against raw dmg. Its like Comparing the dmg of a Frozem mallet against Bt. There is no reason to do it. But as ezrael with full lategamebuild, u have to choose between pd and trinity, so it would be nice if some1 could compare them.

  • #26 Arvendragon

    You don't need utility as an AD Carry.

    You need to be able to pew pew.

  • #31 Wynden

    Utility = Positioning = Not dying = More damage?

  • #36 l3rowncow

    tell ashe that. because ashe is the number 1 ad carry due to number 1 utility and not a certain void carry that has 0 escapes and no real utility except for a small slow, but deals RETARDED amounts of damage with any items at all.

    tl;dr ->at end of day, ad just has to go pew pew.

  • #24 ElevationLoL

    I love reading these btw, keep it up :)

  • #23 ElevationLoL

    Could you do the math on when Infinity Edge + Last Whisper do more DPS than Infinity Edge + Phantom Dancer?

    As an AD player, I've been wondering about this for a long time. I tend to buy Last Whisper immediately after Infinity Edge if the enemy team has 1 Chain Vest on their team. Or does anyone else know?

  • #30 Wynden

    Look at the actual amount of armor that they have. If two to three are >100, then buy Last Whisper. Otherwise, Phantom Dancer is better. Generally, the PD is better because it offers better positioning and by late-mid game, you start relying on your auto-attacks more than your spells (except for using spells to proc your passive, letting you AA more).
    Note: Some of the enemy champs might have taken armor runes and masteries and built smaller items that give them ample armor. i.e. Wriggle's.

    This is what I remember from a while back, so it might be outdated. I'm not sure.

  • #22 OldFusion

    First of all you are looking at damage you get from Trinity Force rather then all the other stats for example Attackspeed and Health that TF offers that IE doesn't get. Secondly Trinity Force gives you more damage on your Q and more burst damage then TF offers.

  • #37 l3rowncow

    why would you drop 4k into an item that makes your laning stronger but falls off once the game actually starts? this post has convinced me that there is no reason to build a TF. UNLESS you build an IE immediately afterwards like you should on corki. that would actually make sense, because then you are boosting the damage of the TF by alot with the IE crits, and then relying on passive for some atk spd, while still have the advantage of phage in lane. Id have to look at his base atk speed to actually know if this would be optimal, but on corki it is optimal from the mouth of chaox. ill have to test on Ez bc no one good plays him really at all.

  • #21 Makumasa

    I think this article is a bit misleading since early game the only source of crit chance you would have would be from the Infinity Edge itself, meaning that only the left side of the graph is relevant since it's highly unlikely you'll be criting more than 2 out of 5 auto attacks. If we only look at the 1 crit column, the most realistic one, we can see that the damage increase is marginal at best (~150 damage). The calculations also disregard the attack speed gained from Trinity Force, which makes the difference even more marginal if even existent. 

    Weigh this marginal damage increase against:

    +250 health, making you less squishy and helping you live through fights.

    +250 mana, which is really useful on Ezreal early game. More spells = more damage.

    +12% movement speed, letting you escape, chase, and dodge skillshots more easily.

    On hit slow which works on your Q, letting you chase and kite.

    Composed of cheaper items that work well individuality meaning that you get the biggest bang for your buck each time you B.

    I'd much rather have all this than the marginal damage increase I would get from rushing Infinity Edge. Instead, I'd build Infinity Edge later since late game is where it really shines since it scales with bonus AD and crit chance.

    That's just my take on it, though.

  • #20 Sedim

    I'm interested in knowning whether or not you took into account the fact that Ezreal's Q procs Sheen/TF and then immediately applies the damage.  Its more important for AP Ez with a Lich Bane as every 3 seconds you get to hit someone with a Q that applies 100% AD and 120% AP.  For AD Ez with a TF it means that most Qs hit for 250% AD and 20% AP.

    I know you were just testing raw damage in a straight up fight, but Ez's main usefulness as a carry comes from the ability to move around a lot and continue doing damage with Q and high attack speed.

    Also you didn't seem to take attack speed into account.  Ez's passive increase his attack speed makes crit damage and chance important, and TF gives 30% attack speed.  I think you are better off looking at how many autoattacks you can get off in 5 seconds with both items to see if there is a difference.

    Last edited by Sedim: 6/20/2012 3:44:42 AM
  • #18 sCrapoloviC

    Actually I really prefer getting TriForce before Inf. Edge cause of the little but impressiv constant Sheen boost + the lane dominating Phage slow. This Phage is securing or even enabling me soooooo much kills on lane and it allows me so much constant harrass if its combined with Sheen so that I still would rate TriForce much much much higher on Ezreal then the Edge. Hitting someone with your Q - getting this Phage prog. - will nearly guarantee a Sheen proc AA on the target and this Poke really hurts! E+SheenAA+Q+AA+W+SheenAA = really huge burst with MS+AS slow its just amazing to see how scared your opponents are after this. And dont forget that Ezreal gets Bonus AS after using a ability which also synergizes quite comfortable with the % amount of AS increase through TriForce.

    For me Triforce is just THE Item for Ezreal, or metioned in the comments Corki. There is nearly no way to avoid building it, or at least parts of it in the early stage of the game.

    Pure mathematical DPS doesn't mean, that he really do more damage cause of the TriForce utility I would bet he can get more AA on his targets then with an Edge. And don't forget that the TriForce makes you atleast a little bit "tanky" which IE definitly doesn't.

    Last edited by sCrapoloviC: 6/20/2012 3:00:29 AM
  • #15 xepel

    TriForce can be built from more smaller pieces than Inf Edge, and being able to pick up _something_ each time you go back will help you out a lot more than waiting it out for the big stuff. Yes, in the ideal world you'd come back with 1650+ gold, but that doesn't always happen, especially if you are not dominating your lane.

    Also, who is going to let you get 5 auto attacks on them before mid-late game teamfights? Early game is all about poking, especially as Ezreal. With two AAs + Q/W + 1 sheen proc (a quick lane exchange), I bet TF outdoes IE. Not to mention the phage slow and TF move speed allows you to actually kill people, where with IE they can just run away. As well, that extra health is great on squishy Ez. And where's attack speed in this calculation? You should be comparing times instead of simply the number of hits.

    Anyway, I do get IE first if I'm dominating my lane, and there's no doubt that IE is a great item for outputting damage. However, if I am not doing so great, I'll pick up pieces of TF and combine them later for the cheap cheap cost of 300 gold.

    TLDR: TF is more than just AD/crit chance; of course ignoring 60% of what it does will make it seem like a terrible item compared to IE.

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