Horizontal Movement Formulas/zh

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 (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 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 (syntax error): {\displaystyle E_{t} = (\underset{每级缓慢减少 \; 15\%}{\underset{每级速度增加 \; 20\%}{\underbrace{\left ( 1 + 0.2\times 速度 \right ) \: \times\: \left ( 1 - 0.15\times 缓慢 \right )}}} \geq 0}

滑度乘数（参见滑度）

Failed to parse (unknown function "\begin{Bmatrix}"): {\displaystyle S_{t} = \begin{Bmatrix}0.6 & \textrm{正常}\\ 0.8 & \textrm{史莱姆}\\ 0.98& \textrm{冰} \\ 1.0 & \textrm{空中} \end{Bmatrix}}

线性运动公式

这些简化的公式只适用于线性运动（方向不变）。 尽管这个条件似乎很有限制性，但这些公式对分析常规的跳跃和助跑非常有用。 之后我们将扩展这些公式以将角度囊括其中。

定义：

• $V_{H,0}$ 是玩家的初始速度（正常 = 0）。
• ${\textstyle V_{H,t}}$ 是玩家在第 ${\textstyle t}$ tick 的速度。

地面速度：

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_{H,t} = \underset{助跑}{\underbrace{\underset{ }{V_{H,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{加速度}{\underbrace{0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}} \right )^{3}}} }

跳跃速度：

Failed to parse (syntax error): {\displaystyle V_{H,t} = \underset{助跑}{\underbrace{\underset{ }{V_{H,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{加速度}{\underbrace{0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}} \right )^{3}}} + \underset{疾跑跳跃加成}{\underbrace{\begin{Bmatrix}0.2 & \textrm{Sprinting}\\ 0.0 & \textrm{Else}\end{Bmatrix} }}}

空中速度：

Failed to parse (syntax error): {\displaystyle V_{H,t} = \underset{助跑}{\underbrace{\underset{ }{V_{H,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{加速度}{\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 (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}) }

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 (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}) + \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 (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{\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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