Previous description: Maximum Speed a Player Can Handle
See the formated Google Doc version here.
This description talks about player reaction time to a 2 point jump from a random point to a hitcircle. This description is an attempt to factor any possible variable that may hinder a player's ability to position from one point to another in the least amount of time. This description will also attempt to estimate the rough minimum and maximum times a player might take to react from the time the note starts fading in to the time the player is over the note.
Time taken to jump from a random point to a hitcircle (C_t)
Suppose the coordinates of the hitcircles A = (x_1, y_1) and B = (x_2, y_2). Let C_t be the time it takes to move the cursor to hit B when going from A. The cursor needs to be on the edge of the hitcircle to be able to hit it. This would make C_t be equal to the time it takes to get from point A to the point on the edge of circle B closest to point A. Therefore, the distance Ctdepends on would be:
Where d_AB is the distance between point A and circle B and R_CS is the hitcircle radius (dependent on circle size). The formula for distance between 2 points is
And the formula for converting CS to osu!px (diameter) is*
Therefor the entire formula for the min distance the player need to move the cursor from circle A to circle B is:
The previous description derived the theoretical max speed at which a player can move the cursor. However, it is an unknown value for any given player. For now let it be V_p. Therefore,
Reaction to fading in: (f_t)
As the note is fading in, it's hard to notice it in the first several ms. Therefore, there is a time at which the hitcircle becomes noticeable. The non HD modded hitcircle is 100% faded in at:
Then the time interval at which the circle becomes noticeable is
Where f_p is the fade percent (0.0 <= f_p <= 1.0) at which the player starts to notice the hitcircle. That value is unknown due to uncertainties in screen contrast, skin, and the player's ability to differentiate between shades and colors.
Latency: (L_h, L_s)
Hardware and visual latency is another thing to consider. That implies that information in and out of the game has a delay. Hardware latency is usually small, but does impact the amount of time a player has to react. This latency can be anywhere from 8ms to under 1 ms. The screen latency is affected by the monitor's refresh rate and the game's FPS. This value can be anywhere from 5.5 ms to 42 ms (calculated from 180FPS to 24 FPS).
Natural human reaction time: (P_R)
The final thing to consider is the player's natural responce time. This value can be anywhere from 180 ms to 380 ms*. This value is completely unknown, yet affects the full reaction time considerably.
Putting it all together:
The final formula for the reaction time would be all of the time periods added up:
= [sqrt((x_2-x_1)^2 + (y_2-y_1)^2) - 104 + 8*cs]/V_p + f_p*min(400, AR_t) + L_h + L_s + P_R
Setting the limit:
Let t_0 be the point in time at which the note starts fading in and t_H the point in time at which the note needs to be hit. Then t_H-t_0 is the total time the player has to react and move towards the note. There is also some room for the player to hit a bit early or late due to OD. By taking account the left bound OD, the formula becomes the minimal time possible for a perfect hit. Therefore, if the player is to have a chance of hitting the note,
Where the formula for OD_t is*
Finding the min and max:
Note that some of these values are guessed and/or estimated. These are to provide a rough idea to where the limit falls within high and low difficulties and considers the player will not miss unless reaction time is more the note duration time (from fading in to hit):
Map independent times:
Hardware Latency: 1ms < L_h < 8ms
Screen Latency: 5.5ms < L_s < 42ms
Player Reaction: 180ms < P_R < 380ms
HARD: CS = 10, AR=11, Distance: 384.19 osu!px, OD = 10
The list for reaction times is as follows:
Cursor Time: 30.28ms < C_t < 93.05ms, assuming 4 osu!px/ms to 12.29 osu!px/ms speed
Fade Time: 40ms < f_t < 100ms, assuming 10% to 25% fade
Total time: 256.78ms < T < 623.05ms
(OD=19.5ms) factoring: 276.28ms < T < 642.55ms
So for the most brutal conditions, for a perfect player, the notes could be 276.28ms apart to barely hit it without accuracy reduction. For a slower player, that number would be 642.55ms.
EASY: CS = 5, AR=8, Distance: 100 osu!px, OD = 6
The list for reaction times is as follows:
Cursor Time: 5.53ms < C_t < 17ms, assuming 4 osu!px/ms to 12.29 osu!px/ms speed
Fade Time: 40ms < f_t < 100ms, assuming 10% to 25% fade
Total time: 232.03ms < T < 547ms
(OD=43.5ms) factoring: 275.53ms < T < 590.5ms
So for the most easier conditions, for a perfect player, the notes could be 275.53ms apart to barely hit it without accuracy reduction. For a slower player, that number would be 590.5ms.
Summary:
An interesting thing to note is that the easy or hard maps for a skilled player barely vary the reaction time by 1 ms despite the C_t having about a 25ms difference. From the estimated results, a skillful player would be able to perform 2.5 to 2.7 faster than a disadvantaged player. As seen, there are a lot of unknown variables which makes finding the difficulty of map problematic. We can only base these value on extremes known to estimate the range in which a player may be able to due the 2 point jump. Therefore the difficulty of the map is very subjective; One difficulty value for a pro player may mean entirely different to another less skilled player. If there is to be a measurement of difficulty, the best way to do it would be through odds of getting a certain performance because otherwise a precise value is not accurate.
CS was not considered within the limits, but would greatly affect the player's performance since hitting 10 CS from the other end of the screen within that time frame is difficult, and would be considered in future descriptions. Future descriptions will also examine the unstable rate due to strain and uncertainty, odds of missing, more complex patterns, hitcircle tracking, etc.
-------------------------------------------------------------------------------------------------------------------------------------
I admit this description doesn't give a lot of interesting things. I was thinking what its actual purpose would be if I posted it. This description just shows you how many uncertainties there are when trying to figure out a map's difficulty. If this is just the reaction time, then there are a lot of other things that will be uncertain and I'll have fun with... yay, joy (=_=)/
Go ahead and comment what you think on the bottom (just note the message on top). I'm sure several errors slipped by, so go ahead and point those out if you catch any.
.
.
.
.
.
.
