水平运动公式

在每 tick，游戏会进行如下三个步骤：

1. 将加速度加到玩家的速度中。
2. 移动玩家（新的位置 = 原位置 + 速度）。
3. 降低玩家的速度以模拟阻力。

我们将从介绍各种乘数开始，尽力使公式可读性更强。

乘数

：移动乘数（参见 45°斜跑）

Failed to parse (unknown function "\begin{Bmatrix}"): {\displaystyle M_{t} = \begin{Bmatrix}1.3 & \textrm{疾跑} \\ 1.0 & \textrm{行走}\\ 0.3 & \textrm{潜行}\\ 0.0 & \textrm{停止} \end{Bmatrix} \times \begin{Bmatrix}0.98 & \textrm{正常}\\ 1.0 & \textrm{45°斜跑} \\ 0.98 \sqrt{2} & \textrm{45°潜行} \end{Bmatrix}}

：效果乘数（参见状态效果）

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}

Slipperiness Multiplier (See Slipperiness)

$S_{t}={\begin{Bmatrix}0.6&{\textrm {Default}}\\0.8&{\textrm {Slime}}\\0.98&{\textrm {Ice}}\\1.0&{\textrm {Airborne}}\end{Bmatrix}}$

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.

Definition:

• $V_{H,0}$ is the player's initial speed (default = 0).
• ${\textstyle V_{H,t}}$ is the player's speed on tick ${\textstyle t}$.

Ground Speed:

Failed to parse (syntax error): {\displaystyle 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}}} }

Jump Speed:

Failed to parse (syntax error): {\displaystyle 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} }}}

Air Speed:

Failed to parse (syntax error): {\displaystyle 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}}}} }

Complete Formulas

Let's introduce two more variables:

• ${\textstyle D_{t}}$ , The player's Direction in degrees (defined by their inputs and rotation)
• ${\textstyle F_{t}}$ , 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:

• ${\textstyle V_{X,0}}$ and ${\textstyle V_{Z,0}}$ correspond to the player's initial velocity.
• ${\textstyle V_{X,t}}$ and ${\textstyle V_{Z,t}}$ correspond to the player's velocity on tick ${\textstyle t}$
  
  

Ground Velocity:

Failed to parse (syntax error): {\displaystyle 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}) }

Failed to parse (syntax error): {\displaystyle 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}) }

Jump Velocity:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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})}} }

Air Velocity:

Failed to parse (syntax error): {\displaystyle 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}) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}) }

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 momentum is cancelled and ${\textstyle V\displaystyle _{X,t}}$ only includes acceleration.

If the player hits a Z-facing wall, then momentum is cancelled and ${\textstyle V\displaystyle _{Z,t}}$ only includes acceleration.

In either of these case, the player stops sprinting.

Negligible Speed Threshold:

If ${\textstyle \left|V\displaystyle _{X,t}\times S_{t}\times 0.91\right|<0.005}$ , momentum is cancelled and only the acceleration is left.

If ${\textstyle \left|V\displaystyle _{Z,t}\times S_{t}\times 0.91\right|<0.005}$ , momentum is cancelled and only the acceleration is left.

In 1.9+, they are compared to 0.003 instead.

Source Code

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