games101笔记-02线性代数回顾

02 Review of Linear Algebra

概述

线性代数回顾
A Swift and Brutal Introduction to Linear Algebra!

向量

图形学中均为列向量
表示方向和长度
没有绝对的起点（任意两个方向和长度相等的向量可认为是同一个向量）
A B → = B − A \overrightarrow{AB} = B - A AB =B−A

单位向量

大小为1的向量
被用来表示方向

点乘（Dot）

结果为一个数
a ⃗ ⋅ b ⃗ = ∥ a ⃗ ∥ ∥ b ⃗ ∥ cos ⁡ θ \vec{a}\cdot\vec{b}=\lVert\vec{a}\rVert\lVert\vec{b}\rVert \cos\theta a ⋅b =∥a ∥∥b ∥cosθ
=>
cos ⁡ θ = a ⃗ ⋅ b ⃗ ∥ a ⃗ ∥ ∥ b ⃗ ∥ \cos \theta = \frac{\vec{a} \cdot \vec{b}}{\lVert \vec{a} \rVert \lVert \vec{b} \rVert} cosθ=∥a ∥∥b ∥a ⋅b
cos ⁡ θ = a ^ ⋅ b ^ , a ^ is unit vector, a ^ = a ⃗ ∥ a ⃗ ∥ \cos \theta = \hat{a} \cdot \hat{b} ,\text { $\\hat{a}$ is unit vector, $\\hat{a}=\\frac{\\vec{a}}{\\lVert \\vec{a} \\rVert}$ } cosθ=a^⋅b^, a^ is unit vector, a^=∥a ∥a

属性

a ⃗ ⋅ b ⃗ = b ⃗ ⋅ a ⃗ \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} a ⋅b =b ⋅a
a ⃗ ( b ⃗ + c ⃗ ) = a ⃗ ⋅ b ⃗ + a ⃗ ⋅ c ⃗ \vec{a}(\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} a (b +c )=a ⋅b +a ⋅c
( k a ⃗ ) ⋅ b ⃗ = a ⃗ ⋅ ( k b ⃗ ) = k a ⃗ ⋅ b ⃗ (k\vec{a})\cdot \vec{b}= \vec{a} \cdot (k\vec{b}) = k{\vec{a} \cdot \vec{b}} (ka )⋅b =a ⋅(kb )=ka ⋅b

点乘-笛卡尔坐标系

二维
a ⃗ ⋅ b ⃗ = ( x a y a ) ⋅ ( x b y b ) = x a x b + y a y b \vec{a} \cdot \vec{b}=\begin{pmatrix} x_a \\ y_a \\ \end{pmatrix} \cdot \begin{pmatrix} x_b \\ y_b \\\end{pmatrix} = x_ax_b + y_ay_b a ⋅b =(xaya)⋅(xbyb)=xaxb+yayb

三维
a ⃗ ⋅ b ⃗ = ( x a y a z a ) ⋅ ( x b y b z b ) = x a x b + y a y b + z a z b \vec{a} \cdot \vec{b}=\begin{pmatrix} x_a \\ y_a \\ z_a \end{pmatrix} \cdot \begin{pmatrix} x_b \\ y_b \\ z_b \end{pmatrix} = x_ax_b + y_ay_b + z_az_b a ⋅b = xayaza ⋅ xbybzb =xaxb+yayb+zazb

点乘在图形学中应用

向量投影
分解向量（计算投影后可根据平行四边形法则得到垂直向量）
计算两个向量是同向还是反向

叉乘（Cross）

结果为一个向量
叉积向量与初始向量正交
方向符合右手螺旋法则
可以用来构建坐标系

a ⃗ × b ⃗ = − b ⃗ × a ⃗ \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} a ×b =−b ×a
∥ a ⃗ × b ⃗ ∥ = ∥ a ⃗ ∥ ∥ b ⃗ ∥ sin ⁡ θ \lVert \vec{a} \times \vec{b} \rVert = \lVert \vec{a} \rVert \lVert \vec{b} \rVert \sin \theta ∥a ×b ∥=∥a ∥∥b ∥sinθ

属性

x ⃗ × y ⃗ = + z ⃗ \vec{x} \times \vec{y} = +\vec{z} x ×y =+z
y ⃗ × x ⃗ = − z ⃗ \vec{y} \times \vec{x} = -\vec{z} y ×x =−z

a ⃗ × b ⃗ = − b ⃗ × a ⃗ \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} a ×b =−b ×a
a ⃗ × ( b ⃗ + c ⃗ ) = a ⃗ × b ⃗ + a ⃗ × c ⃗ \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} a ×(b +c )=a ×b +a ×c
a ⃗ × ( k b ⃗ ) = k ( a ⃗ × b ⃗ ) \vec{a} \times (k\vec{b}) = k(\vec{a} \times \vec{b}) a ×(kb )=k(a ×b )

叉乘-笛卡尔坐标系

a ⃗ × b ⃗ = ( y a z b − y b z a z a x b − x a z b x a y b − y a x b ) \vec{a} \times \vec{b} = \begin{pmatrix} y_az_b-y_bz_a \\ z_ax_b-x_az_b \\ x_ay_b-y_ax_b \\ \end{pmatrix} a ×b = yazb−ybzazaxb−xazbxayb−yaxb
a ⃗ × b ⃗ = A ∗ b ( 0 − z a y a z a 0 − x a − y a x a 0 ) ( x b y b z b ) \vec{a} \times \vec{b} = A^*b \begin{pmatrix} 0 & -z_a & y_a \\ z_a & 0 & -x_a \\ -y_a & x_a & 0 \\ \end{pmatrix} \begin{pmatrix} x_b \\ y_b \\ z_b \\ \end{pmatrix} a ×b =A∗b 0za−ya−za0xaya−xa0 xbybzb

叉乘在图形学中应用

判断向量在左还是右（相对而言）
判断向量在内还是外，判断p点在三角形内

矩阵

属性

矩阵往往没有交换律
具有分配律

矩阵转置

( A B ) T = B T A T (AB)^T=B^TA^T (AB)T=BTAT