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This blog post is "just for fun", assuming you find mathematics interesting. A little hypothetical formulation.
I mentioned in my previous post that aside from sometimes being faster, numerical approaches to modeling champion damage are more appropriate than algebraic ones. But what if we wanted to use algebra anyway, just to be awkward? How about this situation: Two Ashes, each with a BotRK and "simple" items (no
Statikk Shiv). We want to model a duel between the two and determine who dies first, accounting for regeneration from HP/5 (but no
Perseverance
) and lifesteal.
There are three variables here, the health of one ADC (x), the health of the other (y) and time (t). Damage and regeneration are simply components of derivatives of health with respect to time. The formulae I came up with look like this:
So what the heck do all the symbols mean? Subscript x and y indicate who the constant belongs to e.g. cx would be one of ADC X's stats.
d = Expected Average Damage (AD modified to account for crit chance and damage)
a = attack speed
h = health regeneration per second
M = all relevant physical damage modifiers (armor, Havoc
, etc) Technically attack speed could be incorporated into this, or else M could be separated out, especially for ease of modifying values.
L = life steal
From left right, the chunks of each equation are "rate of change of health"="lifesteal*yourDPS"+"health regen"-"enemyDPS".
Solving the equations to get x and y in terms of t (with known initial health or time of death) would then allow us to plot health against time on the same graph and see who hits zero first. So what is the point in this? None, really, it is purely academic. We could create a hypothetical program which determines each constant based on runes, masteries, items and champion level, takes your chosen starting health for X and Y and solves the equations in terms of t before plotting them onto a graph.
The only practical use I can see for this would be to determine the optimum itemisation to duel the enemy ADC based on their current build, however even that is pretty useless. There is a lot more going on in the Rift, and at the end of day, crits could throw things miles of prediction. Not to mention this is only for Ashe, a relatively simple champion to model and then we have ignored her passive. I'd also argue that we'd be better of with a simulation rather than a formulation, but there we go.
If you are wondering, no I probably cannot solve these equations. The x's and y's being multiplied by t's would necessitate product rule using my known method of solving simultaneous differential equations, producing something vastly more complicated than what was on the A-level syllabus and probably being too hard for me. I'm sure my best mathematician could do it, but I am more of a biologist.
I mentioned in my previous post that aside from sometimes being faster, numerical approaches to modeling champion damage are more appropriate than algebraic ones. But what if we wanted to use algebra anyway, just to be awkward? How about this situation: Two Ashes, each with a BotRK and "simple" items (no
Statikk Shiv). We want to model a duel between the two and determine who dies first, accounting for regeneration from HP/5 (but no
Perseverance
) and lifesteal.There are three variables here, the health of one ADC (x), the health of the other (y) and time (t). Damage and regeneration are simply components of derivatives of health with respect to time. The formulae I came up with look like this:
So what the heck do all the symbols mean? Subscript x and y indicate who the constant belongs to e.g. cx would be one of ADC X's stats.
d = Expected Average Damage (AD modified to account for crit chance and damage)
a = attack speed
h = health regeneration per second
M = all relevant physical damage modifiers (armor,
Havoc
, etc) Technically attack speed could be incorporated into this, or else M could be separated out, especially for ease of modifying values.L = life steal
From left right, the chunks of each equation are "rate of change of health"="lifesteal*yourDPS"+"health regen"-"enemyDPS".
Solving the equations to get x and y in terms of t (with known initial health or time of death) would then allow us to plot health against time on the same graph and see who hits zero first. So what is the point in this? None, really, it is purely academic. We could create a hypothetical program which determines each constant based on runes, masteries, items and champion level, takes your chosen starting health for X and Y and solves the equations in terms of t before plotting them onto a graph.
The only practical use I can see for this would be to determine the optimum itemisation to duel the enemy ADC based on their current build, however even that is pretty useless. There is a lot more going on in the Rift, and at the end of day, crits could throw things miles of prediction. Not to mention this is only for Ashe, a relatively simple champion to model and then we have ignored her passive. I'd also argue that we'd be better of with a simulation rather than a formulation, but there we go.
If you are wondering, no I probably cannot solve these equations. The x's and y's being multiplied by t's would necessitate product rule using my known method of solving simultaneous differential equations, producing something vastly more complicated than what was on the A-level syllabus and probably being too hard for me. I'm sure my best mathematician could do it, but I am more of a biologist.
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