Citrine guide - Meet The Math (Patreon)

Regarding Citrine herself there's not much to math out as it were - she's more effect and result based rather than interesting scaling or considerations.

Instead I want to focus on her 4th ability and the difference between headshotting vs going for the crystals. Both require you to aim for the right spot, and while the crystal has a larger hitbox and stationary enemy it also requires you to set up that situation first. Headshots need no ability support, though can be supported through Condemn and Ensnare, or damage boosted with yet other abilities.

So let's break this down on an equation basis.

For weapons which are able to headshot, against units which suffer critical headshots, the equation for average damage bonus on headshot is: 3 * (1 + CC * (CD * 2 - 1)) 

For weapons which target the crystals instead, forcing 300% critical chance, the equation is: 1 + 3 * (CD - 1)

Written as an inequality, the point at which you should aim for the crystals instead of headshot is:

3 * (1 + CC * (CD * 2 - 1)) < 1 + 3 * (CD - 1)

Divide both sides by 3 

1 * CC * (CD * 2 - 1) < 1/3 + 1 * (CD - 1)

Clean up the constants

CC * (CD * 2 - 1) < CD - 5/3

Divide by (CD * 2 - 1)

CC < (CD - 5/3) ÷ (CD * 2 - 1)

Here we've managed to single out the Critical Chance term, and have a couple terms for our weapon's Critical Damage. We know that Critical Chance can never be below 0, so that already gives us a limit to the equation. 

Because of that, we can see that if Critical Damage is equal to or less than 5/3 (1.67x) then it is ALWAYS better to headshot as that would otherwise require less than 0 critical chance.

On top of this, we can also see that the right side converges on Critical Damage divided by 2x Critical Damage. This will by asymptotic to 0.5 for higher levels of Critical Damage, showing that 50% or higher critical chance will ALWAYS be better to headshot than to shoot the crystals. Remember that this is still relying on the condition of a headshot capable weapon AND a critical headshot being a thing. 

Above 1.67x critical damage and below 50% critical chance we have an area where it can be better to shoot the crystals. For example, with 2x critical damage it's better to go for the crystals if below 11.1% critical chance. For 3x CD it's 26.7% CC or lower; For 4x it's 33.3%; For 5x it's 37.0%; for 5x it's 39.4%; and for 10.5x, which is the Kuva Nukor with Primed Target Cracker, it's 44.2%. See attached graph (green line)

Notably, this means that if you are using Arcane Avenger, it is ALWAYS better for this category of weapon to aim for a headshot instead while the Arcane is procced. Red crits may look nice, but they're just a type of crit, not an end in itself.

We can adapt the inqeuality to address less favourable headshot circumstances.

When using a weapon without a headshot (3x) bonus, the equation becomes simply:

CC÷3 < (CD - 1) ÷ (CD * 2 - 1) [blue line on attached graph]

This moves our always headshot breakpoint to 150%+ critical chance, or below 1x critical damage. Nothing should have a 1x critical damage multiplier so the critical chance is always the relevant term, and 150% is no small amount. Weapons this applies to are things like the Fulmin semi-auto, Arca Plasmor, and Catchmoon. The game won't tell you which weapons this affects though, you just have to know/check the wiki.

Likewise when attacking an enemy without a critical headshot bonus, but still using a headshot capable weapon, we instead get the even simpler:

CC < (CD - 5/3) / (CD - 1) [black line on attached graph]

Just as the original inequality reveals, a critical damage of 1.67 or lower means we should always go for the headshot. However the lower scaling divisor means that only with 100%+ critical chance can we be certain a headshot will be a better choice. Otherwise there's a sliding factor. Aim for the crystals with crit chance below these thresholds: 2x CD below 33.3%; 3x CD below 66.6%; 4x below 77.7%; 5x below 83.3%; and 10.5x below 93.0%. So for units like the Corpus with their helmets blocking critical damage bonuses, going for the crystals is often the smarter idea.

Finally then the equation for if you have no critical headshot bonus AND no headshot bonus is simply:

CC < 300% [red line on attached graph]

Because we've taken out everything that otherwise makes the headshot important. 

MOST weapons that care about the crystals will MOSTLY want to shoot for the head instead, but the myriad of hidden bonuses and debuffs means there's not a simple general case to give. This is all because Warframe is complicated and doesn't tell you half of what is going on.

I'm attaching a graph of all 4 of these cases to visually demonstrate the boundaries.