There exists a constant related in some arcane way to the amount of AP the angel has immediately after investing mana in someone such that:
Maths
Which is nice because now the rate of change of the max AP can be calculated based only on a starting point for the current AP and itself, but is still I think pretty impossible to solve analytically :P
Given some values for the constants (maybe not necessarily ) I could probably draw some graphs showing the evolutions of these quantities in time. Maybe I'll do that later with imaginary values.
Last edited by pedromvilar on Thu Aug 11, 2016 10:36 am, edited 1 time in total.
Okay after thinking about it for thirty seconds I've figured out what value must have. If the angel has just invested an amount of their AP such that their current AP is c(0) and their max AP is m(0) then
This makes it easier for me to come up with values. I'm gonna plot graphs later/when I find a tool that does that numerically for me.
Okay so I just plotted this out a bit and apparently what actually happens is that in the short term an angel's current AP tends to quickly approach a value that's N/M of the maximum AP and then both of them grow approximately linearly, staying at that fraction.
I'm not totally sure this result is correct and will verify it on the weekend when I'm neither at work nor ridiculously tired due to having spent the day at work.
Here are some graphs of the growth and recharge of AP as a function of time (units agnostic). Let N be the maximum rate of recharge of current AP, M the maximum max AP growth rate, c(0) and m(0) how much AP and max AP an angel has immediately after spending it and c and m how much AP and max AP an angel has at some other point in time, and assuming the constant rate of growth/recharge K is small:
If the angel has not spent any AP (c(0) = m(0))
AP just grows linearly without hitches
If the current AP value recharges more slowly than the max AP value grows (N < M)
If the starting ratio of AP and max AP is greater than the ratio between recharge rates (c(0)/m(0) > N/M) c and m grow slowly and eventually settle at a rate of change that leaves c/m very close to N/M
If the starting ratio of AP and max AP is equal to the ratio between recharge rates (c(0)/m(0) = N/M)
the ratio c/m remains equal to N/M forever
If the starting ratio of AP and max AP is less than the ratio between recharge rates (c(0)/m(0) < N/M) c and m grow less slowly and eventually settle at a rate of change that leaves c/m very close to N/M
If the current AP value recharges as fast as or faster than the max AP value grows (N ≥ M)
the current AP approaches the max, getting arbitrarily close, but never reaches it
In all cases, the recharge and growth rates are greatest when the current AP is 5/12 (exactly halfway between 1/2 and 1/3 because I'm lazy :P) of the max. The rates N, M, and K can be tweaked arbitrarily to get the most desirable result.
A noteworthy thing about the above is that in the case where N/M = 1 the current AP value doesn't actually approach the max, it's just that both of them grow enough that the difference between them becomes negligible compared to the max.
related in some arcane way to the amount of AP the angel has immediately after investing mana in someone such that:
