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[[File:0t sprintjump graph.png|thumb|600x600px|Movement formulas applied to a 3b jump.]]
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The player's movement can be accurately calculated with [https://en.wikipedia.org/wiki/Sequence sequences].


The following formulas come from analyzing the game's [[SourceCode|source code]].




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Note that these formulas are '''not exact''', due to how [https://en.wikipedia.org/wiki/Floating-point_arithmetic floats] are computed. When used for calculations, only the first 4-6 decimals should be considered accurate. For a completely accurate simulation, you would need to replicate the source code.


<translate><!--T:4--> [[File:Collision box (player and block).png|left|thumb|image 1]]</translate>




In this article, we only consider standard movement, and ignore mechanics specific to certain blocks.






You will find further documentation of movement physics in these articles:


* [[Soulsand]]
* [[Ladders and Vines|Ladders and Vines]]
* [[Slime Block|Slime Blocks]]
* [[Cobweb|Cobwebs]]




'''Note:''' Minecraft's coordinate system is oriented differently: 0° points towards "positive Z", and 90° points towards "negative X". We choose to work in the standard coordinate system to make calculations more intuitive. If need be, we can simply invert the X axis to match Minecraft's coordinate system.


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<br />

== Vertical Movement ==

===Jump Formula===

*<math>V\displaystyle _{Y,1} = 0.42</math>
* <math>V\displaystyle _{Y,t} = \left (V_{Y,t-1} - \underset{gravity}{0.08} \right ) \times \underset{drag}{0.98}</math>


:If <math display="inline"> \left | V\displaystyle _{Y,t} \right | < 0.005 </math>, <math display="inline">V\displaystyle _{Y,t}</math> is set to 0 instead (the player's height doesn't change for that tick)

:In 1.9+, it's compared to 0.003 instead.



'''Notes'''

*<math display="inline"> V\displaystyle _{Y,0}</math> isn't assigned a value because it has no importance. By convention, the 0<sup>th</sup> tick corresponds to the player's initial velocity before jumping.
*<math display="inline"> V\displaystyle _{Y,1}</math> corresponds to the initial jump motion. It is increased by 0.1 per level of [[Status Effects|Jump Boost]]
*Terminal velocity is -3.92 m/t
*When the player collides vertically with a block, vertical momentum is cancelled and only the acceleration is left.



===Vertical Position===

: To get the position on a given tick, you simply need to sum <math display="inline">V\displaystyle _{Y}</math>

:<math display="inline">Y(n) = \sum_{t=1}^{n} V_{Y,t}</math>


===Jump duration===

:The '''duration''' of a jump is the number of ticks between jumping and landing.

:It also corresponds to the period of that jump's cycle when performed repeatedly.

:This notion is linked to the notion of [[Tiers]].

:{| class="wikitable"
!Description
!Duration
|-
|Flat Jump
|12 t
|-
|3bc Jump
|11 t
|-
|<nowiki>+0.5 Jump</nowiki>
|10 t
|-
| +1 Jump
|9 t
|-
|2.5bc Jump
|6 t
|-
|2bc Jump
|3 t
|-
|1.8125bc Jump
|2 t
|}



=== Source code ===
'''from [[SourceCode:EntityLivingBase|EntityLivingBase]]'''

: <syntaxhighlight lang="java">
/* Code unrelated to vertical movement is cut out */

protected float getJumpUpwardsMotion(){
return 0.42F;
}


protected void jump()
{
this.motionY = this.getJumpUpwardsMotion();
if (this.isPotionActive(Potion.jump))
{
this.motionY += (this.getActivePotionEffect(Potion.jump).getAmplifier() + 1) * 0.1F;
}
this.isAirBorne = true;
}


public void moveEntityWithHeading(float strafe, float forward)
{
... /* also moves the player horizontally */

this.motionY -= 0.08;
this.motionY *= 0.98;
}


public void onLivingUpdate()
{
if (this.jumpTicks > 0)
--this.jumpTicks;

if (Math.abs(this.motionY) < 0.005D)
this.motionY = 0.0D;


if (this.isJumping)
{
... /* different if in water or lava */

if (this.onGround && this.jumpTicks == 0)
{
this.jump();
this.jumpTicks = 10; //activate autojump cooldown (0.5s)
}
}
else
{
this.jumpTicks = 0; //reset autojump cooldown
}

...
this.moveEntityWithHeading(this.moveStrafing, this.moveForward);
}

</syntaxhighlight>

== Horizontal Movement ==

Horizontal Movement is a bit more complex than Vertical Movement, as it relies on many more factors: player actions, direction, and ground slipperiness.

On every tick, the game does these three steps:
# Acceleration is added to the player's velocity.
# The player is moved (new position = position + velocity).
# The player's velocity is reduced to simulate drag.

We'll start by introducing Multipliers in an effort to make formulas more readable.


=== Multipliers ===

: '''Movement Multiplier''' (See [[45 Strafe|45° Strafe]])


:: <math>M_{t} = \begin{Bmatrix}1.3 & \textrm{Sprinting} \\ 1.0 & \textrm{Walking}\\ 0.3 & \textrm{Sneaking}\\ 0.0 & \textrm{Stopping} \end{Bmatrix} \times \begin{Bmatrix}0.98 & \textrm{Default}\\ 1.0 & \textrm{45° Strafe} \\ 0.98 \sqrt{2} & \textrm{45° Sneak} \end{Bmatrix}</math>


: '''Effects Multiplier''' (See [[Status Effects]])


:: <math>E_{t} = (\underset{Decreases \; by \; 15\% \; per \; level \; of \; Slowness}{\underset{Increases \; by \; 20\% \; per \; level \; of \; Speed}{\underbrace{\left ( 1 + 0.2\times Speed \right ) \: \times\: \left ( 1 - 0.15\times Slowness \right )}}} \geq 0</math>


: '''Slipperiness Multiplier''' (See [[Slipperiness]])


:: <math>S_{t} = \begin{Bmatrix}0.6 & \textrm{Default}\\ 0.8 & \textrm{Slime}\\ 0.98& \textrm{Ice} \\ 1.0 & \textrm{Airborne} \end{Bmatrix}</math>
:

<br />
=== Linear Formulas ===

: These simplified formulas only apply to linear movement (no change in direction).
:While this condition might seem very restrictive, these formulas are very useful to analyze conventional jumps and momentum
:We'll later expand on these formulas by including angles.
:<br />'''Definition''':
:* <math>V_{H,0}</math> is the player's initial speed (default = 0).
:* <math display="inline">V_{H,t}</math> is the player's speed on tick <math display="inline">t</math>.



: '''Ground Speed''':

::<math>V_{H,t} = \underset{Momentum}{\underbrace{\underset{ }{V_{H,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}} \right )^{3}}} </math>


: '''Jump Speed''':

::<math>V_{H,t} = \underset{Momentum}{\underbrace{\underset{ }{V_{H,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}} \right )^{3}}} + \underset{Sprintjump \; Boost}{\underbrace{\begin{Bmatrix}0.2 & \textrm{Sprinting}\\ 0.0 & \textrm{Else}\end{Bmatrix} }}</math>


: '''Air Speed''':

::<math>V_{H,t} = \underset{Momentum}{\underbrace{\underset{ }{V_{H,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{\underset{ }{0.02 \times M_{t}}}} </math>


=== Complete Formulas ===

: Let's introduce two more variables:
:*<math display="inline"> D_{t} </math> , The player's Direction in degrees (defined by their inputs and rotation)
:* <math display="inline"> F_{t} </math> , The player's Facing in degrees (defined by their rotation only)


In reality, angles aren't as simple as that, as there are a limited number of significant angles (see [[Facing and Angles]]).

For the purpose of simplicity, we'll ignore this fact.




: '''Definition''':
:* <math display="inline">V_{X,0}</math> and <math display="inline">V_{Z,0}</math> correspond to the player's initial velocity.
:* <math display="inline">V_{X,t}</math> and <math display="inline">V_{Z,t}</math> correspond to the player's velocity on tick <math display="inline">t</math> <br /> '''Ground Velocity''':

::<math>V\displaystyle _{X,t} = \underset{ }{V\displaystyle _{X,t-1} \times S_{t-1} \times 0.91 } \: + \: 0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}} \right )^{3} \times \sin (D_{t}) </math>
::<math>V\displaystyle _{Z,t} = \underset{Momentum}{\underbrace{\underset{ }{V\displaystyle _{Z,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}} \right )^{3}}} \times \cos (D_{t}) </math>

: '''Jump Velocity''':

::<math>V\displaystyle _{X,t} = \underset{ }{V\displaystyle _{X,t-1} \times S_{t-1} \times 0.91 } \: + \: 0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}} \right )^{3} \times \sin (D_{t}) + \begin{Bmatrix} 0.2 & \textrm{Sprinting}\\ 0.0 & \textrm{Else}\end{Bmatrix} \times \sin (F_{t}) </math>
::<math>V\displaystyle _{Z,t} = \underset{Momentum}{\underbrace{\underset{ }{V\displaystyle _{Z,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}} \right )^{3}}} \times \cos (D_{t}) + \underset{Sprintjump \; Boost }{\underbrace{\begin{Bmatrix} 0.2 & \textrm{Sprinting}\\ 0.0 & \textrm{Else}\end{Bmatrix} \times \cos (F_{t})}} </math>


: '''Air Velocity''':

::<math>V\displaystyle _{X,t} = \underset{ }{V\displaystyle _{X,t-1} \times S_{t-1} \times 0.91 } \: + \: 0.02 \times M_{t} \times \sin (D_{t}) </math>
::<math>V\displaystyle _{Z,t} = \underset{Momentum}{\underbrace{\underset{ }{V\displaystyle _{Z,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{\underset{}{0.02 \times M_{t}}}} \times \cos (D_{t}) </math>

=== Stopping Conditions ===

: Horizontal speed is set to 0 if the player hits a wall, or if the speed is considered to be negligible.


: '''Wall Collision''':

:: If the player hits a X-facing wall, then <math display="inline">V\displaystyle _{X,t}</math> is set to 0 and the player is placed against the wall.
:: If the player hits a Z-facing wall, then <math display="inline">V\displaystyle _{Z,t}</math> is set to 0 and the player is placed against the wall.


: '''Negligible Speed Threshold''':

:: If <math display="inline"> \left | V\displaystyle _{X,t} \times S_{t} \times 0.91 \right | < 0.005 </math> , momentum is cancelled and only the acceleration is left.
:: If <math display="inline"> \left | V\displaystyle _{Z,t} \times S_{t} \times 0.91 \right | < 0.005 </math> , momentum is cancelled and only the acceleration is left.

:: In 1.9+, they are compared to 0.003 instead.


=== Source Code ===
[...]


<br />

== Non-Recursive Formulas ==
Since [https://en.wikipedia.org/wiki/Arithmetico%E2%80%93geometric_sequence arithmetico-geometric sequences] have explicit formulas, we can build non-recursive formulas to calculate simple but useful results, such as the height of the player on any given tick, or the distance of a jump in terms of the initial speed and duration.

Numeric approximations are given with 6 digits of precision.



'''Definitions:'''

* <math display="inline">v_0</math> is the player's initial speed (speed on <math>t_0</math>, before jumping)
* <math display="inline">t</math> is the number of ticks considered (ex: t=12 on flat ground, see '''Jump Duration''')
*<math display="inline">J</math> is the "jump bonus" (0.3274 for sprintjump, 0.291924 for strafed sprintjump, 0.1 for 45° no-sprint jump...)
*<math display="inline">M</math> is the movement multiplier after jumping (1.3 for 45° sprint, 1.274 for normal sprint, 1.0 for no-sprint 45°...)


=== Vertical Movement (jump) [1.8] ===
Vertical speed after jumping (<math>t \geq 6</math>)
{| class="wikitable"
!<math display="inline">\textrm{V}_Y(t) = 4 \times 0.98^{t-5} - 3.92</math>
|}
Relative height after jumping (<math>t \geq 6</math>)
{| class="wikitable"
!<math display="inline">\textrm{Y}_{rel}(t) = \underset{\textrm{jump peak}}{\underbrace{197.4 - 217 \times 0.98^5}} + 200 (0.98-0.98^{t-4}) - 3.92 (t-5)</math>
|-
|num. approx: <math>\textrm{Y}_{rel}(t) \approx 216.849187 - 216.833157 \times 0.98^t - 3.92t</math>
|}

For <math display="inline">t<6</math>, see below.


=== Vertical Movement (jump) [1.9+] ===
Vertical speed after jumping (<math>t \geq 1</math>)
{| class="wikitable"
!<math display="inline">\textrm{V}_Y(t) = 0.42 \times 0.98^{t-1} + 4 \times 0.98^t - 3.92</math>
|-
|num. approx: <math>\textrm{V}_Y(t) \approx 4.428571 \times 0.98^t - 3.92</math>
|}

Relative height after jumping (<math>t \geq 0</math>)
{| class="wikitable"
!<math display="inline">\textrm{Y}_{rel}(t) = 217 \times (1 - 0.98^t) - 3.92 t</math>
|}



=== Horizontal Movement (instant jump) ===
Assuming the player was airborne before jumping.

Horizontal speed after sprintjumping (<math>t \geq 2</math>)
{| class="wikitable"
!<math display="inline">\textrm{V}_H(v_0,t) = \frac{0.02 M}{0.09} + 0.6 \times 0.91^t \times \left ( v_0 + \frac{J}{0.91} - \frac{0.02 M}{0.6 \times 0.91 \times 0.09} \right )</math>
|-
|45° sprint: <math>\textrm{V}_H(v_0,t) \approx 0.288889 + 0.6 \times 0.91^t \times \left ( v_0 - 0.169320 \right )</math>
|-
|reg. sprint: <math display="inline">\textrm{V}_H(v_0,t) \approx 0.283111 + 0.6 \times 0.91^t \times \left ( v_0 - 0.158738 \right )</math>
|}
Sprintjump distance (<math>t \geq 2</math>)
{| class="wikitable"
!<math display="inline">\textrm{Dist}(v_0,t) = 1.91 v_0 + J + \frac{0.02 M}{0.09} (t-2) + \frac{0.6 \times 0.91^2}{0.09} \times (1 - 0.91^{t-2}) \times \left ( v_0 + \frac{J}{0.91} - \frac{0.02 M}{0.6 \times 0.91 \times 0.09} \right )</math>
|-
|45° sprint: <math display="inline">\textrm{Dist}(v_0,t) \approx 7.430667 v_0 + 0.288889 t -1.185138 - 6.666667 \times 0.91^t \left ( v_0 - 0.169320 \right )</math>
|-
|reg. sprint: <math display="inline">\textrm{Dist}(v_0,t) \approx 7.430667 v_0 + 0.283111 t -1.115163 - 6.666667 \times 0.91^t \left ( v_0 - 0.158738 \right )</math>
|}

'''Note:''' These formulas are accurate for most values of <math>v_0</math>, but some negative values can wind up activating the speed threshold and reset the player's speed at some point, thus rendering these formulas inaccurate.


=== Horizontal Movement (delayed jump) ===
Assuming the player is on ground before jumping (at least 1 tick since landing).

Horizontal speed after sprintjumping (<math>t \geq 2</math>)
{| class="wikitable"
!<math display="inline">\textrm{V}^*_H(v_0,t) = \frac{0.02 M}{0.09} + 0.6 \times 0.91^t \times \left ( 0.6 v_0 + \frac{J}{0.91} - \frac{0.02 M}{0.6 \times 0.91 \times 0.09} \right )</math>
|-
|45° sprint: <math>\textrm{V}^*_H(v_0,t) \approx 0.288889 + 0.36 \times 0.91^t \times \left ( v_0 -0.282201 \right )</math>
|-
|reg. sprint: <math display="inline">\textrm{V}^*_H(v_0,t) \approx 0.283111 + 0.36 \times 0.91^t \times \left ( v_0 - 0.264563 \right )</math>
|}
Sprintjump distance (<math>t \geq 2</math>)
{| class="wikitable"
!<math display="inline">\textrm{Dist}^*(v_0,t) = 1.546 v_0 + J + \frac{0.02 M}{0.09} (t-2) + \frac{0.6 \times 0.91^2}{0.09} \times (1 - 0.91^{t-2}) \times \left ( 0.6v_0 + \frac{J}{0.91} - \frac{0.02 M}{0.6 \times 0.91 \times 0.09} \right )</math>
|-
|45° sprint: <math display="inline">\textrm{Dist}^*(v_0,t) \approx 4.8584 v_0 + 0.288889t -1.185138 + 4 \times 0.91^t (v_0 - 0.169320)</math>
|-
|reg. sprint: <math display="inline">\textrm{Dist}^*(v_0,t) \approx 4.8584 v_0 + 0.283111t -1.115163 + 4 \times 0.91^t (v_0 - 0.158738)</math>
|}



=== Advanced Formulas ===
Horizontal speed after <math>n</math> consecutive sprintjumps on a momentum of period <math display="inline">T</math> (<math>n \geq 0</math>, <math>T \geq 2</math>).

If the first sprintjump is delayed, multiply <math display="inline">v_0</math> by 0.6
{| class="wikitable"
!<math display="inline">\textrm{V}^{\,n}_H(v_0,T,n) = \left ( 0.6 \times 0.91^T \right )^n v_0 + \left ( 0.6 \times 0.91^{T-1} J + 0.02M \frac{1 - 0.91^{T-1}}{0.09} \right ) \frac{1- (0.6 \times 0.91^T)^n}{1 - 0.6 \times 0.91^T} </math>
|}


=== Examples ===

*<math>Y_{rel}(60) </math> gives the relative height of the player 3 seconds (60 ticks) after jumping.
*<math>\textrm{Dist}(0,12)</math> gives the jump distance on flat ground with no initial speed.
*<math>\textrm{Dist}( V_H(0,12) ,9)</math> gives the distance of a +1 jump with one sprintjump of flat momentum as initial speed.
*<math>\textrm{Dist}(V^{10}_H(0,2), 12) </math> gives the jump distance with 10 sprintjumps of momentum under a trapdoor-headhitter.

Latest revision as of 04:37, 19 August 2021

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