Crystal Maiden's Freezing Field

If you had questions about Freezing Field just read the following answer:

The center of the explosion is determined by two true random variables: distance, which ranges from 140 to 710, and angle, which either ranges from (a) 0 to 90, (b) 90 to 180, (c) 180 to 270, or (d) 270 to 360. When you start casting the spell, a random distance and random angle from (a) is chosen for the first blast. This must therefore appear in the upper right quadrant of the circle. 0.1 seconds later, a new random distance and random angle from (b) is chosen (upper left quadrant), then from (c) (lower left quadrant), then (d) (lower right quadrant), and then it repeats. So every 0.4 seconds, Freezing Field completes one "circle" in this respect. Keep in mind the explosions have 255 triggered AoE (about 231 hardcoded AoE), so they can overlap even though each is forced to be in a different quadrant.

To be clear, these variables are chosen in a true random manner. So every distance between 140 and 710 is equally likely to be chosen. However, as you get closer to the center, the choice of angle matters less. For a simple example, if an enemy is standing right next to CM (at coordinates <x,y> = <48,0>), and the distance chosen is less than 207, then no matter what angle is chosen, that enemy will be hit. On the other hand, if the enemy is at <964,0>, and the distance chosen is 710, the angle chosen will need to be very close to 0 or the explosion will not hit the enemy. Therefore hitting nearby enemies is more likely than hitting distant enemies. This is a consequence of the area formula for polar coordinates having a factor of r.

However, as you get very close to the center, there is an opposite effect. Because the minimum distance is 140, you obviously have less chance of getting a distance in, say, the range 100-200 than in the range 200-300. This means fewer distances are available near the center that will hit. In spite of this, a very high volume (approx. 10^6) of statistical tests show that the chance of collision increases strictly and smoothly as distance decreases, with a maximum at r=0 (about 20% chance of collision) and minimum (0% chance) for all r > 965.

Probability of collision depends on only distance from CM, not on angle. If you are directly on the edge of two quadrants (let's say at theta = 90 degrees), you will have only half the usual probability of being hit by the first explosion, but then that same probability of being hit by the next one, meaning overall the probability is the same as if you had the usual probability for that distance on the first explosion and 0 on the next. Indeed, no matter where you are, the portion of the circle you "lose" due to the edge of the quadrant is equal to the one you "gain" on the next explosion. This only really matters in a couple of ways.

First of all, it tends to smooth out probabilities slightly. For instance, an enemy at a constant range won't have to roll the correct value between 0 and 360 (by "correct value," suppose at this predetermined range the angle must be between 30 and 60 degrees), but instead just roll the correct value between 0 and 90; however, you will only get the opportunity every fourth cycle. So after 4 seconds (40 comets), you will have 10 shots at a 33% roll instead of 40 shots at an 8.3% roll. It's the same expected damage, but you are less likely to get extreme results. For instance, it's impossible to be hit by all 40 comets, since you are only exposed to 10 at most, but it is also far less likely to get missed by all of them.

The other way it matters is at the end of the spell. If I stop casting just before the cycle reaches your quadrant, I miss one last chance to nuke you.

The distance and angle are chosen according to "true RNG," but the bounds for the angle do change periodically, and the method for calculation is like IceFrog's attempt to square the circle. Still, calling it "pseudo-random" is a bit of a stretch. The idea of a pseudo-random distribution is to reproduce the behavior of the uniform random distribution but with short tails. Freezing Field does not resemble the uniform random distribution at all.

At a reasonable distance, with 10% probability of collision, CM will deal 250 (* 310) magical dps with a level 3 Freezing Field and slow movement and attack speed by 30% (* 50%). Whether or not that is "a lot" is of course up for interpretation.



I know that not all are interested because they don't give a **** about mechanics of what spells do but hey I just wanted to clear some answers.

o_O

Nice DotaMaths, very nice. I tried to follow it all, but all i know is when you couple that with a 5 man Black Hole/double Ravage/double Reverse Polarity you have a dead team.

Though, it does seem less potent in my head now...

Not really. Crystal Maiden can kill a 5 man team with an Aghanim's Scepter level 3 Freezing Field, 1 Reverse Polarity and that's all.

I don't get it : the 250 (310 with Aghs) magical damage is the damage of each individual explosion. It's not the same thing as the average single target DPS of the skill. Is it *also* the average DPS? And if so, why?

For example on top of CM, your chance to be hit by any occurring explosion is ~20% (note that the graph is for DotA 1). So 1 in 5 explosions will hit you. There's 10 explosions per second, so you should be hit by 2 explosions per second.
At level 3 FF, that's 2x250 damage per second, or 500 dps.

If you are 530 distance away from CM, you will get hit by, on average, one explosion per second, so then the damage per explosion will be (numerically) equivalent to the dps. Stand further away and you take less damage, stand closer and you take more damage.

How much average dps you deal totally depends on where the enemies are. There's no sensible way to estimate that.

One note though: Since the first explosion always happens in the top right corner (the angle is, before, only pseudo-random), CM should try and position herself southwest of what she's trying to hit (unless it's a teamfight and you'll just have people on all sides). The difference is marginal, but you never know.

I managed to not hit a single shard against a solo hero during a full cast once. RNGeesuz hates me. q_q

It's still random after all. Very unlikely to happen, but not impossible.

So actually the YOLO CM playstyle to blink on top of people, BKB and ult is the right way to play her?

620 DPS boys. Free hugs. Better than a rapier Medusa xD

That's actually pretty stupid. I mean blink in, BKB, Crystal Nova and Freezing Field. With Aghs that's an 80% AOE slow (!!) and 620 DPS. Anything without a BKB dies in 3-4 seconds.

Clearly this means we've all been playing Crystal Maiden wrong. ICEfrog has been silently crying this whole time, wondering why we're using the strongest carry in the game as a support. :D

make Black King Bar.

Watch our favourite support's dreams crack and shatter into cold,cruel nothingness.