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. For reference, his ability scalings at max rank are as follows:

• Q (Mystic Shot) — 1.0 AD + 0.2 AP ratio
• W (Essence Flux) — 0.8 AP ratio
• E (Arcane Shift) — 0.75 AP ratio
• R (Trueshot Barrage) — 1.0 AD + 0.9 AP ratio

Having gone through his abilities, the next logical step is to take a look at the item builds in question.

Items

Trinity Force provides: 30 AD, 30 AP, 30% Attack Speed, 15% Critical Strike Chance, 12% Movement Speed, 250 HP, 250 Mana. Its key passives are:

• Unique: Your basic attacks have a 25% chance to slow your target's movement speed by 35% for 2.5 seconds.
• Unique (Spellblade): 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.

Infinity Edge provides: 80 AD, 25% Critical Strike Chance. Its key passive is:

• Unique: Critical strikes now deal 250% damage instead of 200%.

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

Assumptions

In producing the contour plots (rainbow graphs) on the damage output increase that Infinity Edge gives over Trinity Force — a set ability combo is required. Having the advantage of foresight, 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 an 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 — referred to as Sheen proc from here on), number of critical strikes and the various AD/AP ratios according to each item.

5 Sheen procs out of 5 basic attacks:

You effectively have to crit at least two times in this scenario to outdamage 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.

0 Sheen procs:

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.

3 Sheen procs out of 5 basic attacks (realistic):

In this situation, you will never outdamage someone with an Infinity Edge. The trend is very clear — Infinity Edge outdoes Trinity Force in terms of damage output very significantly.

3 Sheen procs, 5 basic attacks, 2 Mystic Shots:

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 outdamage 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 movement speed 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,

DiffTheEnder