Computer Graphics Linear Algebra

资料来源：https://sites.cs.ucsb.edu/~lingqi/teaching/games101.html

前段时间学习完成了Games101的计算机图形学入门课程作业，一直没抽时间回顾下知识点，所以以个人理解，写的精简点，用来回顾搜索。

本篇以线性代数(Linear Algebra)基础知识点开始.

Vectors(带方向的长度，指向)

\[\vec a = \overrightarrow{AB} = B - A\]

Magnitude (length(长度) - 向量也等于与自身点积的平方根)

\[\begin{Vmatrix} \vec a \end{Vmatrix} = \sqrt{A^2 + B^2}\] \[\begin{Vmatrix} \vec b \end{Vmatrix} = \sqrt{B^2 + B^2}\]

Eigen
----
VectorXf v(2);
MatrixXf m(2,2), n(2,2);
  
  v << -1,
       2;
  
  m << 1,-2,
       -3,4;
       
Output:

v.norm() = 2.23607
m.norm() = 5.47723

norm() = sqrt(dot(A, B))

In Eigen，由于向量是矩阵的一个特例，它们也被隐含地处理在那里，所以矩阵-向量积实际上只是矩阵-矩阵积的一个特例，向量-向量外积也是如此

- Scalar * Vector use *
- Scalar * Martix use *
- Martix * Martix use *
- Martix * vector use *
- vector * vector use dot 
- vector * vector use cross

Unit vector(单位向量)

\[\hat a = \vec a / \vec a\]

a = a.normalized() = a.norm() / a.norm() = sqrt(dot(a, a)) / sqrt(dot(a, a))
a = |a| / |a|
#向量自身的长度相除

Dot(scalar) product(点乘，标量乘)

\[\vec a · \vec b = \begin{Vmatrix} \vec a \end{Vmatrix} \begin{Vmatrix} \vec b \end{Vmatrix} cos \Theta\] \[cos \Theta = \vec a · \vec b \div \begin{Vmatrix} \vec a \end{Vmatrix} \begin{Vmatrix} \vec b \end{Vmatrix}\]

For unit vectors

\[\vec a = 1\] \[\vec b = 1\] \[cos \Theta = \vec a · \vec b \div \begin{Vmatrix} 1 \end{Vmatrix} \begin{Vmatrix} 1 \end{Vmatrix}\] \[cos \Theta = \vec a · \vec b\]

dot(a,b)
#用于求两向量夹角角度

Cross (vector) Product(叉乘，向量乘)

\[\vec x \times \vec y = + \vec z\] \[\vec y \times \vec x = - \vec z\] \[a \times b = \begin{bmatrix} & YaZb - YbZa & \\ & ZaXb - XaZb & \\ & XaYb - YaXb & \end{bmatrix}\]

Determine left / right

Determine inside / outside

crossProduct

Matrices

Array of numbers (m × n = m rows, n columns)

\[\begin{matrix} 1 & 2 \\ 4 & 5 \\ 7 & 8 \end{matrix}\]

# (number of) columns in A must = # rows in B 

一个矩阵列的数量必须等于另一个矩阵行的数量

或

一个矩阵行的数量必须等于另一个矩阵列的数量

(M x N) (N x P) = (M x P)

\[\left \{ \begin{matrix} 1 & 2 \\ 4 & 5 \\ 7 & 8 \end{matrix} \right \}\] \[\left \{ \begin{matrix} 1 & 2 & 3 & 4 \\ 4 & 5 & 6 & 8 \\ \end{matrix} \right \}\] \[\left \{ \begin{matrix} ? & ? & ? & ? \\ ? & ? & ? & ? \\ ? & ? & ? & ? \\ \end{matrix} \right \}\]

(3 x 2)(2 x 4) = (3 x 4)

Matrix-Matrix Multiplication

矩阵相乘顺序不能调换

(AB and BA are different in general)

Associative and distributive

A(B+C) = AB + AC

Matrix-Vector Multiplication

\[\begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} -x\\ y \end{bmatrix}\]

Transpose of a Matrix

Switch rows and columns

\[\begin{matrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{matrix} ^ \intercal = \begin{matrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{matrix}\]

Property

\[(AB)^\intercal = B^\intercal A^\intercal\]

Identity Matrix and Inverses

\[I_{3\times 3} = \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\] \[AA^{-1} = A^{-1}A = I\] \[(AB)^{-1} = B^{-1}A^{-1}\]

Vector multiplication in Matrix form

Dot product?

\[a·b = a^\intercal b = \left \{ \begin{matrix} Xa & Ya & Za \\ \end{matrix} \right\} \left \{ \begin{matrix} Xb \\ Yb \\ Zb \end{matrix} \right\} = \left \{ \begin{matrix} XaXb + YaYb + ZaZb \\ \end{matrix} \right\}\]

Cross product?

\[\vec a \times \vec b = A *b = \left \{ \begin{matrix} 0 & -Za & Ya & Xb \\ Za & 0 & -Xa & Yb \\ -Ya & Xa & 0 & Zb \end{matrix} \right\}\]

向量a的对偶矩阵