The following is an attempt to describe the maximum speed at which a player within the osu! community can execute a micro pattern. This description assumes a linear micro pattern which involves back and forth movements or a V shape. This description ignores tapping and human reaction. This is the first of the several upcoming descriptions in an effort to describe the difficulty of maps in the standard gameplay mode.
Let's assume the micro pattern is as long as possible. The game's distance is measured in osu!pixels, which is 1 pixel in 600x480 resolution. Therefore, the max distance of any 2 hit circles can be is
sqrt(600^2 + 480^2) = 768.37 osu!px
Since the real world is not measured in osu!px, and is based on the unknown distance the stylus or mouse goes from one end to the other, we need a basis that describes how fast a player's hand or fingers can move.
Since the real world is not measured in osu!px, and is based on the unknown distance the stylus or mouse goes from one end to the other, we need a basis that describes how fast a player's hand or fingers can move.
Staiain has an impressive finger speed, capable of spamming up to 32 keys/second with 4 keys with remarkable accuracy. To figure out how fast one of his fingers moves on average, we divide the rate by 4. So Staiain can tap 8 times/second. Let's keep it in milliseconds since we will be measuring the time between the notes in milliseconds in the upcoming descriptions. So Staiain can tap 0.008 times/ms, or a tap rate of 0.008 kHz.
Putting it in terms of frequency gives us the ability to calculate the speed at which a player's finger is moving the fastest. I know that mouse players don't use fingers to move the mouse and use their wrists instead, but at this point I am assuming the human muscle within fingers and wrist are capable of achieving close to the same speed. This is where the max distance in osu!px comes in as that will be the amplitude. It doesn't matter how far to the side the stylus or mouse is moving in reality since a player can change the mouse sensitivity or tablet area, but it does matter onscreen.
We start by determining how long one period of the oscillation lasts.
period = 1/frequency = 1/0.008 kHz = 125 ms
Therefore, it will take the player a minimum of 125 ms to go from point A to point B and then back to point A at max distance. To find out how long it would to go just from point A to point B, we divide by 2 and get 62.5 ms. We will need this number later in future descriptions, but for now we need the time at which the cursor is moving the fastest.
Therefore, it will take the player a minimum of 125 ms to go from point A to point B and then back to point A at max distance. To find out how long it would to go just from point A to point B, we divide by 2 and get 62.5 ms. We will need this number later in future descriptions, but for now we need the time at which the cursor is moving the fastest.
At 62.5 ms the the cursor is slowing down to 0 osu!px/ms and is about to turn back to point A. The point at which the cusor is moving the fastest is right in the middle of point A and point B. So we divide by 2, giving us 31.25 ms. Therefore, it takes 31.25 ms to accelerate to the fastest speed. To find how many osu!px that is, we need to half the distance from point A to point B. So 768.37 osu!px/2 is 384.19 osu!px. To find the average speed, we divide the distance by the time:
384.19 osu!px / 31.25 ms = 12.29 osu!px/ms
So we found out that the fastest speed a player can do a micro pattern without screwing up due to inertia and strain is 9.6 osu!px/ms. But let's take a step back and take a look at the minimum time the player can preform such micro pattern: 62.5 ms. Reacting to that is impossible due to the human reaction time being 180 ms minimum (220 ms average). Therefore, this speed is the max speed for which a player could do if movement is memorized.
Now for your feedback! Did you think I did something wrong? Are my calculations correct? Is my assumption to use Staiain's tapping speed reasonable? Is my assumption the max wrist speed is similar to finger speed correct? Can you, the players, even achieve 12.29 osu!px/ms? Try setting up a map with a V shaped micro pattern and try doing it with the relax mod.
but its so generic
