Dot (scalar) Product

$$ \vec{a} \cdot \vec{b} = \lVert \vec{a} \rVert \lVert \vec{b} \rVert \cos \theta

\\

\cos \theta = \frac{\vec{a} \cdot \vec{b}} {\lVert \vec{a} \rVert \lVert \vec{b} \rVert},

\quad

\cos \theta = \hat{a} \cdot \hat{b} $$

Application in Graphics

• Find angle between two vectors (e.g. cosine of angle between light source and surface)
• Finding projection of one vector on another (投影)
• Measure how close two directions are (判断夹角)
• Decompose a vector (分解向量)
• Determine forward / backword (通过点乘结果正负判断方向)

Cross Product

$$ \vec{a} \times \vec{b}

\begin{pmatrix} y_a z_b - y_b z_a \\ z_a x_b - x_a z_b \\ x_a y_b - y_a x_b \end{pmatrix} $$

$$ \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} $$

• 叉乘的向量坐标可以用行列式理解记忆 (虽说基本不需要自己手算吧…)

• $\vec{a} \times \vec{b}$ 的方向是右手螺旋法，即右手四指从 $\vec{a}$ 的方向握向 $\vec{b}$ 的方向时，大拇指的朝向。

• $\vec{a} \times \vec{b}$ 的模长就是 $\lVert \vec{a} \rVert \lVert \vec{b} \rVert \sin \theta$ 。

  

叉乘的应用：判断左和右，判断内和外 (视频 37:02)

• 判断左和右：$\vec{a} \times \vec{b}$ 的结果为正，说明 $\vec{a}$ 在 $\vec{b}$ 的左侧，反之同理。)

• 判断内和外

  P 在三角形内部等价于在三条边的同一侧， $\vec{PA}$ 分别和 $\vec{a}, \vec{b}, \vec{c}$ 做三次叉乘即可

  procedure cross(a, b):
      return a.x * b.y - b.x * a.y
  
  procedure is_point_in_triangle(P, A, B, C):
      AB = B - A
      BC = C - B
      CA = A - C
  
      AP = P - A
      BP = P - B
      CP = P - C
  
      c1 = cross(AB, AP)
      c2 = cross(BC, BP)
      c3 = cross(CA, CP)
  		
  		// We must check each sign, think about "- - +"
      all_positive = c1 > 0 or c2 > 0 or c3 > 0
      all_negative = c1 < 0 or c2 < 0 or c3 < 0
  
      if all_positive or all_negative:
          return true
      else:
          return false
  

Matrix

m x n → m rows, n colsy