更新文章

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# 数学基础
* [矢量](./math_01.md)
* [矩阵](./math_02.md)
* [立体角](./math_03.md)
* [几何变换(一)](./transform_01.md)
* [几何变换(二)](./transform_02.md)
* [法线变换](./transform_03.md)
* [摄像机变换](./transform_04.md)
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# 数学基础(一)
----
## 1. 基础约定
### 1.1 坐标系
#### 1.1.1 左手坐标系
![](./axis_left.svg )
DirectX一般缺省使用左手坐标系
#### 1.1.2 右手坐标系
![](./axis_right.svg)
OpenGL缺省使用右手坐标系,图形学中缺省使用这种坐标系
### 1.2 左乘和右乘
#### 1.2.1 左乘,前乘(pre-multiply)
矢量为行向量,和矩阵的乘法为
$$
[x,y,z]\begin{bmatrix}
M_{11}&M_{12}&M_{13} \\
M_{21}&M_{22}&M_{23} \\
M_{31}&M_{32}&M_{33}
\end{bmatrix}
$$
DirectX缺省使用左乘
#### 1.2.2 右乘,后乘(post-multiply)
矢量为列向量,和矩阵的乘法为
$$
\begin{bmatrix}
M_{11}&M_{12}&M_{13} \\
M_{21}&M_{22}&M_{23} \\
M_{31}&M_{32}&M_{33}
\end{bmatrix}\begin{bmatrix}
x\\
y\\
z
\end{bmatrix}
$$
OpenGL缺省使用右乘,图形学中缺省使用这种方式。
### 1.3 行存储和列存储
#### 1.3.1 行存储
在计算机内存中,按照行优先存储一个矩阵,DirectX使用这种方式
比如对于矩阵
$$
\begin{bmatrix}
M_{11}&M_{12}&M_{13} \\
M_{21}&M_{22}&M_{23} \\
M_{31}&M_{32}&M_{33}
\end{bmatrix}
$$
在内存中为$M_{11},M_{12},M_{13},M_{21},M_{22}\ldots$
#### 1.3.2 列存储
按照列优先存储一个矩阵,OpenGL使用这种方式
比如对于矩阵
$$
\begin{bmatrix}
M_{11}&M_{12}&M_{13} \\
M_{21}&M_{22}&M_{23} \\
M_{31}&M_{32}&M_{33}
\end{bmatrix}
$$
在内存中为$M_{11},M_{21},M_{31},M_{12},M_{22}\ldots$
## 2. 矢量
### 2.1 基本操作
#### 2.1.1 定义
$$
\begin{split}
\vec{u}&=[u_x,u_y,u_z]^T, \vec{v}=[v_x,v_y,v_z]^T \\
\vec{u}\pm\vec{v}&=[u_x\pm v_x, u_y\pm v_y, u_z\pm v_z]^T \\
k\vec{u}&=[ku_x, ku_y, ku_z]^T \\
\end{split}
$$
#### 2.1.2 几何意义
![](./vector_op.svg)
### 2.2 标准矢量
![](./standard_basis.svg)
标准矢量(Standard Basis),定义$\vec{\boldsymbol i}=[1,0,0]^T, \vec{\boldsymbol j}=[0,1,0]^T, \vec{\boldsymbol k}=[0,0,1]^T$
### 2.3 长度
#### 2.3.1 定义
对于三维矢量$\vec{u}$,定义矢量的长度
$$
\|\vec{u}\|=\sqrt{u_x^2+u_y^2+u_z^2}
$$
有时候矢量的长度也可以用$|\vec{u}|$表示,但容易和绝对值混淆
#### 2.3.2 定义
对于三维矢量$\vec{u}$,定义矢量的单位矢量
$$
\hat{\vec{u}}=\frac{\vec{u}}{\|\vec{u}\|}=(\frac{u_x}{\|\vec{u}\|}, \frac{u_y}{\|\vec{u}\|},
\frac{u_z}{\|\vec{u}\|})
$$
### 2.4 点积
#### 2.4.1 定义
对于两个三维矢量$\vec{u}, \vec{v}$,定义它们的点积([Dot Product](https://en.wikipedia.org/wiki/Dot_product))为
$$
\vec{u}\cdot\vec{v}=u_xv_x+u_yv_y+u_zv_z
$$
#### 2.4.2 点积的几何意义
![](./dot_product.svg)
$$
\vec{a}\cdot\vec{b}=\|\vec{a}\|\|\vec{b}\|\cos{\theta}
$$
两个向量的点积可以看成两个向量相近的程度,向量和自身的点积是向量长度的平方
$$
\vec{u}\cdot\vec{u}=\|\vec{u}\|\|\vec{u}\|=u_x^2+u_y^2
$$
向量$\vec{a}$在向量$\vec{b}$上的投影的长度为
$$
a_b=\frac{\vec{a}\cdot\vec{b}}{\|b\|}
$$
#### 2.4.3 推论
判断两个矢量$\vec{\boldsymbol u},\vec{\boldsymbol v}$之间的夹角和它们之间点积的关系
$$
\cos(\theta)=\frac{\vec{\boldsymbol u}\cdot \vec{\boldsymbol v}}{\|\vec{\boldsymbol u}\|\|\vec{\boldsymbol v}\|}
$$
- 当$\vec{u}\cdot\vec{v}\gt 0$,夹角在[0,90)之间
- 当$\vec{u}\cdot\vec{v}=0$,两个矢量垂直
- 当$\vec{u}\cdot\vec{v}\lt 0$,两个矢量夹角在(90,180]之间
#### 2.4.3 基本运算规则
矢量的点积满足一般的交换律、分配律
- $\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$
- $\vec{a}\cdot(\vec{b}+\vec{c}) = \vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}$
- $(k\vec{a})\cdot\vec{b}=\vec{a}\cdot(k\vec{b})=k(\vec{a}\cdot\vec{b})$
### 2.5叉积
#### 2.5.1 定义
对于两个三维矢量$\vec{a}, \vec{b}$,定义它们的叉积
$$
\vec{\boldsymbol a}\times\vec{\boldsymbol b}=\begin{vmatrix}
\vec{\boldsymbol i}& \vec{\boldsymbol j} & \vec{\boldsymbol k}\cr
a_x & a_y & a_z \cr
b_x & b_y & b_z
\end{vmatrix}=
(a_yb_z-a_zb_y)\vec{\boldsymbol{i}}+(a_zb_x-a_xb_z)\vec{\boldsymbol{j}}+(a_xb_y-a_yb_x)\vec{\boldsymbol{k}}
$$
#### 2.5.2 叉积的几何意义
在三维几何中,向量a和向量b的叉乘结果是一个向量,更为熟知的叫法是法向量,该向量垂直于a和b向量构成的平面。
![](./cross_product.svg)
定义向量$\vec{\boldsymbol n}$是一个单位向量,方向同时垂直于向量$\vec{\boldsymbol u}$和$\vec{\boldsymbol v}$,且符合右手定则(右手坐标系),那么
$$
\vec{\boldsymbol u}\times\vec{\boldsymbol v}=\|\vec{\boldsymbol u}\|\|\vec{\boldsymbol v}\|\sin(\theta)\vec{\boldsymbol n}
$$
向量a和向量b的叉积还可以视作以$\vec{a}$和$\vec{b}$为边的平行四边形的面积
#### 2.5.3 基本运算规则
- $\vec{a}\times\vec{b}=-\vec{b}\times\vec{a}$
- $\vec{a}\times\vec{a}=\vec{0}$
- $\vec{a}\times(\vec{b}+\vec{c})=\vec{a}\times\vec{b}+\vec{a}\times\vec{c}$
- $\vec{a}\times(k\vec{b})=k(\vec{a}\times\vec{b})$
#### 2.5.4 叉积的矩阵形式
两个向量的叉积,可以表达成一个矩阵和向量的乘积的形式,例如向量$\vec{a}$和$\vec{b}$的叉积
$$
\vec{a}\times\vec{b}=\begin{bmatrix}
{(\vec{a}\times\vec{b}})_{x}\\
{(\vec{a}\times\vec{b}})_{y}\\
{(\vec{a}\times\vec{b}})_{z}
\end{bmatrix}=\begin{bmatrix}
a_yb_z-a_zb_y\\
a_zb_x-a_xb_z\\
a_xb_y-a_yb_x
\end{bmatrix}=\begin{bmatrix}
0&-a_z&a_y\\
a_z&0&-a_x\\
-a_y&a_x&0
\end{bmatrix}\begin{bmatrix}
b_x\\b_y\\b_z
\end{bmatrix}
$$
这里称这个矩阵为矢量$\vec{a}$的叉乘矩阵[cross-product matrix](https://en.wikipedia.org/wiki/Cross_product#Conversion_to_matrix_multiplication),写作$[\boldsymbol{a}]_{\times}$
$$
[\boldsymbol{a}]_{\times}\overset{def}{=}\begin{bmatrix}
0&-a_z&a_y\\
a_z&0&-a_x\\
-a_y&a_x&0
\end{bmatrix}
$$
### 2.6 混合积
对于三个矢量$\vec{a},\vec{b},\vec{c}$,定义混合积为
$$
(\vec{a}\times\vec{b})\cdot\vec{c}
$$
混合积满足
$$
(\vec{a}\times\vec{b})\cdot\vec{c}=(\vec{b}\times\vec{c})\cdot\vec{a}=(\vec{c}\times\vec{a})\cdot\vec{b}
$$
#### 2.6.1 混合积的几何意义
如图,以三个矢量为边的棱作平行六面体,$\|\vec{a}\times\vec{b}\|$可以视作以$\vec{a}$和$\vec{b}$为边的平行四边形的面积,$(\vec{a}\times\vec{b})\cdot\vec{c}$可以视作这个平行六面体的体积
![](./vector_trip.svg)
### 2.7 三重积
对于三个矢量$\vec{a},\vec{b},\vec{c}$,有如下公式
$$
\vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}\tag{2.6.1}
$$
证明:
设$\vec{d}=\vec{a}\times(\vec{b}\times\vec{c})$,可知$\vec{d}$垂直于$\vec{b}$和$\vec{c}$所组成的平面的法线,也就是$\vec{d}$平行于$\vec{b}$和$\vec{c}$所组成的平面。所以$\vec{d}$可以表达成$\vec{b}$和$\vec{c}$的线性组合
$$
\vec{d}=m\vec{b}+n\vec{c}
$$
由于$\vec{d}$垂直于$\vec{a}$,所以
$$\begin{split}
\vec{a}\cdot(m\vec{b}+n\vec{c})&=0\\
m(\vec{a}\cdot\vec{b})+n(\vec{a}\cdot\vec{c})&=0
\end{split}$$
所以一定存在一个非零的实数$p$,使得$m=p(\vec{a}\cdot\vec{c}), n=-p(\vec{a}\cdot\vec{b})$,所以
$$
\vec{d}=\vec{a}\times(\vec{b}\times\vec{c})=p(\vec{a}\cdot\vec{c})\vec{b}-p(\vec{a}\cdot\vec{b})\vec{c}
$$
由于这是一个恒等式,$p$的值和$\vec{a},\vec{b},\vec{c}$无关,当特殊情况$\vec{a}=\vec{c}$时,有
$$
\vec{a}\times(\vec{b}\times\vec{a})=p(\vec{a}\cdot\vec{a})\vec{b}-p(\vec{a}\cdot\vec{b})\vec{a}
$$
两边同时点积$\vec{b}$,可以得到
$$
\vec{a}\times(\vec{b}\times\vec{a})\cdot\vec{b}=p(\vec{a}\cdot\vec{a})(\vec{b}\cdot\vec{b})-p(\vec{a}\cdot\vec{b})(\vec{a}\cdot\vec{b})
$$
等号左侧应用混合积公式,可以得到
$$\begin{split}
\vec{a}\times(\vec{b}\times\vec{a})\cdot\vec{b}&=(\vec{b}\times\vec{a})\cdot(\vec{b}\times\vec{a})\\
&=\|a\|^2\|b\|^2sin^2(\theta)
\end{split}$$
等号右侧
$$\begin{split}
p(\vec{a}\cdot\vec{a})(\vec{b}\cdot\vec{b})-p(\vec{a}\cdot\vec{b})(\vec{a}\cdot\vec{b})&=p\|a\|^2\|b\|^2-p\|a\|^2\|b\|^2cos^2(\theta)\\
&=p\|a\|^2\|b\|^2(1-cos^2(\theta))\\
&=p\|a\|^2\|b\|^2sin^2(\theta)
\end{split}$$
由此可得$p=1$
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# 数学基础(二)
----
## 3. 矩阵
### 3.1 定义
一个$m\times n$的矩阵$\boldsymbol A$是一个有$m$行$n$列的数组,
$\boldsymbol A_{ij}$表示第$i$行第$j$列的元素,
$\boldsymbol A_{i,*}$表示第$i$行所有元素,
$\boldsymbol A_{*,j}$表示第$j$列所有元素,
$$
\boldsymbol A=\begin{bmatrix}
A_{11}&A_{12}&A_{13}\\
A_{21}&A_{22}&A_{23}\\
A_{31}&A_{32}&A_{33}
\end{bmatrix}=\begin{bmatrix}
\leftarrow & A_{1,*} & \rightarrow \\
\leftarrow & A_{2,*} & \rightarrow \\
\leftarrow & A_{3,*} & \rightarrow
\end{bmatrix}=\begin{bmatrix}
\uparrow & \uparrow & \uparrow \\
A_{*,1} & A_{*,2} & A_{*,3} \\
\downarrow & \downarrow & \downarrow
\end{bmatrix}
$$
### 3.2 加法
如果矩阵$\boldsymbol A$是一个$m\times n$的矩阵,矩阵$\boldsymbol B$是一个$m\times n$的矩阵,那么$\boldsymbol A +\boldsymbol B$是一个$m\times n$的矩阵$\boldsymbol C$,并且
$$
C_{ij}=A_{ij}+ B_{ij}
$$
### 3.3 乘法
#### 3.3.1 定义
矩阵之间的乘法,如果矩阵$\boldsymbol A$是一个$m\times n$的矩阵,矩阵$\boldsymbol B$是一个$n\times p$的矩阵,那么$\boldsymbol{AB}$是一个$m\times p$的矩阵$\boldsymbol C$,并且
$$
C_{ij}=A_{i,*}\cdot B_{*,j}
$$
一般来说$\boldsymbol{AB}\neq \boldsymbol{BA}$
#### 3.3.2 举例
$$\begin{split}
\boldsymbol{AB}&=\begin{bmatrix}
-1&5&4\\3&2&1
\end{bmatrix}\begin{bmatrix}
2&1&0\\0&-2&1\\-1&2&3
\end{bmatrix}\\ &= \begin{bmatrix}
(-1,5,-4)\cdot(2,0,-1) & (-1,5,-4)\cdot(1,-2,2) & (-1,5,-4)\cdot(0,1,3)\\
(3,2,1)\cdot(2,0,-1) & (3,2,1)\cdot(1,-2,2) & (3,2,1)\cdot(1,-2,2)
\end{bmatrix}\\ &=\begin{bmatrix}
2&-19&-7\\
5&1&5
\end{bmatrix}
\end{split}
$$
#### 3.3.3 矢量和矩阵相乘
以三维为例
$$\begin{split}
\vec{\boldsymbol u}\boldsymbol A &=
[x,y,z]\begin{bmatrix}
A_{11}&A_{12}&A_{13}\\
A_{21}&A_{22}&A_{23}\\
A_{31}&A_{32}&A_{33}
\end{bmatrix}\\&=[x,y,z]\begin{bmatrix}
\uparrow & \uparrow & \uparrow \\
A_{*,1} & A_{*,2} & A_{*,3} \\
\downarrow & \downarrow & \downarrow
\end{bmatrix}\\&=x\boldsymbol{A}_{*,1}+y\boldsymbol{A}_{*,2}+z\boldsymbol{A}_{*,3}
\end{split}
$$
### 3.4 转置矩阵
#### 3.4.1 定义
将一个$m\times n$的矩阵$\boldsymbol A$的行和列颠倒,变成一个$n\times m$的矩阵,成为$\boldsymbol A$的转置矩阵,记为$\boldsymbol A^T$
#### 3.4.2 举例
$$
\boldsymbol A=\begin{bmatrix}
2&-1&8\\3&6&-4\end{bmatrix}\\
\boldsymbol A^T=\begin{bmatrix}
2&3\\-1&6\\8&-4
\end{bmatrix}
$$
### 3.4.3 正交矩阵
如果一个矩阵满足$\boldsymbol{A}\boldsymbol{A}^T=\boldsymbol{I}$,则称$\boldsymbol{A}$为正交矩阵([Orthogonal matrix](https://en.wikipedia.org/wiki/Orthogonal_matrix))。
正交矩阵满足如下特性:
* 必须是一个方阵,即n行n列;
* 矩阵中的每一列若视作向量,则这些向量均两两相互垂直;
* 矩阵中的每一列若视作向量,则这些向量的长度均为1;
### 3.5 基本运算规则
- $\boldsymbol A+\boldsymbol B=\boldsymbol B+\boldsymbol A$
- $(\boldsymbol A+\boldsymbol B)+\boldsymbol C=\boldsymbol A+(\boldsymbol B+\boldsymbol C)$
- $r(\boldsymbol A+\boldsymbol B)=r\boldsymbol A+r\boldsymbol B$
- $(r+s)\boldsymbol A=r\boldsymbol A+s\boldsymbol A$
- $\boldsymbol{A}(\boldsymbol{B}+\boldsymbol{C})=\boldsymbol{AB}+\boldsymbol{AC}$
- $(\boldsymbol A+\boldsymbol B)\boldsymbol{C}=\boldsymbol{AC}+\boldsymbol{BC}$
- $(\boldsymbol{AB})\boldsymbol{C}=\boldsymbol{A}(\boldsymbol{BC})$
- $(\boldsymbol A+\boldsymbol B)^T=\boldsymbol{A}^T+\boldsymbol{B}^T$
- $(c\boldsymbol A)^T=c\boldsymbol A^T$
- $(\boldsymbol{AB})^T=\boldsymbol{B}^T\boldsymbol{A}^T$
- $(\boldsymbol{A}^T)^T=\boldsymbol{A}$
- $(\boldsymbol{A}^{-1})^T=(\boldsymbol{A}^T)^{-1}$
对于矩阵,交换律不生效,大部分情况下
$\boldsymbol{A}\boldsymbol{B}\neq\boldsymbol{B}\boldsymbol{A}$
### 3.6 单位矩阵
#### 3.6.1 定义
一个$n\times n$的矩阵,从左上角到右下角都是1,其他元素都是0,称为单位矩阵,用$\boldsymbol I$表示,如果$\boldsymbol A$是一个$m\times n$矩阵,$\boldsymbol B$是一个$n\times p$矩阵,那么
$$
\boldsymbol{AI}=\boldsymbol{A}, \boldsymbol{B}=\boldsymbol{B}
$$
### 3.7 子式矩阵(Minor Matrix)
#### 3.7.1 定义
对于一个$n\times n$的矩阵,把第i行和第j列去掉之后的子矩阵,称为Minor Matrix(中文翻译比较乱),记为$\overline{\boldsymbol A}_{ij}$,比如对于一个3x3的矩阵
$$
\boldsymbol A=\begin{bmatrix}
A_{11}&A_{12}&A_{13}\\
A_{21}&A_{22}&A_{23}\\
A_{31}&A_{32}&A_{33}
\end{bmatrix}
$$
$$
\overline{\boldsymbol A}_{11}=\begin{bmatrix}
A_{22}&A_{23}\\
A_{32}&A_{33}
\end{bmatrix},
\overline{\boldsymbol A}_{13}=\begin{bmatrix}
A_{21}&A_{22}\\
A_{31}&A_{32}
\end{bmatrix}
$$
### 3.8 行列式(Determinant of a Matrix)
#### 3.8.1 定义
一个矩阵$\boldsymbol A$的行列式是一个数值,记为$\det{\boldsymbol A}$
$$
\det{\boldsymbol A}=\sum_{j=1}^n(\boldsymbol{A}_{1j}(-1)^{1+j}\det{\overline{\boldsymbol A}}_{1j})
$$
当矩阵是一个1x1的矩阵时,$\det{[A_{11}]}=A_{11}$
#### 3.8.2 举例
$$
\det{\begin{bmatrix}
A_{11}&A_{12}\\
A_{21}&A_{22}
\end{bmatrix}}=A_{11}\det[A_{22}]-A_{12}\det[A_{21}]=A_{11}A_{22}-A_{12}A_{21}
$$
$$\begin{split}
&\det{\begin{bmatrix}
A_{11}&A_{12}&A_{13}\\
A_{21}&A_{22}&A_{23}\\
A_{31}&A_{32}&A_{33}
\end{bmatrix}}\\&=A_{11}\det\begin{bmatrix}
A_{22}&A_{23}\\
A_{32}&A_{33}
\end{bmatrix}-A_{12}\det\begin{bmatrix}
A_{21}&A_{23}\\
A_{31}&A_{33}
\end{bmatrix}+A_{13}\det\begin{bmatrix}
A_{21}&A_{22}\\
A_{31}&A_{32}
\end{bmatrix}\\
&=A_{11}A_{22}A_{33}-A_{11}A_{23}A_{32}-A_{12}A_{21}A_{33}+A_{12}A_{23}A_{31}+A_{13}A_{21}
A_{32}-A_{13}A_{22}A_{31}
\end{split}$$
### 3.9 伴随矩阵(Adjoint of a Matrix)
#### 3.9.1 定义
对于$n\times n$的矩阵$\boldsymbol A$,定义另一个$n\times n$矩阵$\boldsymbol C$,使得
$$
C_{ij}=(-1)^{i+j}\det\overline{\boldsymbol A}_{ij}
$$
那么$\boldsymbol C$的转置矩阵称为$\boldsymbol A$的伴随矩阵,记为$\boldsymbol A^*$
$$
\boldsymbol A^*=\boldsymbol C_A^T
$$
#### 3.9.2 举例
$$\begin{split}
\boldsymbol A&=\begin{bmatrix}a&b\\c&d\end{bmatrix} \\
\boldsymbol A^*&=\begin{bmatrix}d&-b\\-c&a\end{bmatrix}
\end{split}$$
#### 3.9.3 基本运算
- $(\boldsymbol A^*)^T=(\boldsymbol A^T)^*$
- $(k\boldsymbol A)^*=k^{n-1}\boldsymbol A^*$
- $\boldsymbol{AA}^*=\boldsymbol{A}^*\boldsymbol{A}=\det(\boldsymbol A)\boldsymbol I$
### 3.10 逆矩阵(Inverse of a Matrix)
#### 3.10.1 **定义**
只有正方形矩阵有逆矩阵,对于一个$n\times n$的矩阵$\boldsymbol M$,定义它的逆矩阵为$\boldsymbol M^{-1}$,满足$\boldsymbol{MM}^{-1}=\boldsymbol{M}^{-1}\boldsymbol{M}=\boldsymbol{I}$
#### 3.10.2 计算方法
$$
\boldsymbol M^{-1}=\frac{\boldsymbol M^{*}}{\det(\boldsymbol A)}
$$
#### 3.10.3 正交矩阵
对于正交矩阵,根据定义有$\boldsymbol M^{-1}=\boldsymbol M^{T}$
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# 数学基础(三)
----
## 4. 立体角
### 4.1 定义
三维空间的弧度,以r为半径的球,在球表面面积为$A=r^2$的圆对应的立体角为**单位立体角**,
![](./Steradian.svg)
### 4.2 和平面角的对应关系
下图中方形面积对应的立体角为
$$
d\omega=\dfrac{dA}{R^2}=\dfrac{R\sin\theta d\phi\times R d\theta}{R^2}=\sin\theta d\phi d\theta
$$
![](./solid_angle.svg)
### 4.3 积分
针对立体角的半球体积分可以转换为球面角
$$
\int_{\Omega}f(\vec{\omega})d\omega=\int_{0}^{2\pi}\int_{0}^{\pi/2}f(\theta,\phi)\sin\theta d\theta d\phi
$$
比如求半球体的面积
$$
\int_{\Omega}d\omega=\int_{0}^{2\pi}\int_{0}^{\pi/2}\sin\theta d\theta d\phi=2\pi
$$
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# 几何变换(一)
----
## 1. 线性变换
### 1.1 定义
假设有某种数学函数$\tau$,输入矢量$\vec{\boldsymbol u}$,输出的仍然是一个矢量,且满足以下条件
$$\begin{split}
\tau(\boldsymbol u + \boldsymbol v)&=\tau(\boldsymbol u) +\tau(\boldsymbol v) \\
\tau(k\boldsymbol u)&=k\tau(\boldsymbol u)
\end{split}$$
那么定义$\tau$为线性变换
### 1.2 举例
比如$\tau(u_x,u_y,u_z)=(u_x^2,u_y^2,u_z^2)$就不是线性变换,因为不满足第二条
### 1.3 使用矩阵表示
一般使用矩阵表示一个线性变换,设一个三维矢量$\vec{\boldsymbol{v}}=[x,y,z]^T=x\vec{\boldsymbol{i}}+y\vec{\boldsymbol{j}}+z\vec{\boldsymbol{k}}$,那么对其做线性变换
$$\begin{split}
\tau(\vec{\boldsymbol{v}})&=\tau(x\vec{\boldsymbol{i}}+y\vec{\boldsymbol{j}}+z\vec{\boldsymbol{k}})\\
&=\tau(\vec{\boldsymbol{i}})x+\tau(\vec{\boldsymbol{j}})y+\tau(\vec{\boldsymbol{k}})z\\
&=\begin{bmatrix}
\uparrow & \uparrow & \uparrow \\
\tau(\vec{\boldsymbol{i}}) & \tau(\vec{\boldsymbol{j}}) & \tau(\vec{\boldsymbol{k}})\\
\downarrow & \downarrow & \downarrow
\end{bmatrix}\begin{bmatrix}x\\y\\z \end{bmatrix}\\
&=\begin{bmatrix}
T_{11}&T_{12}&T_{13}\\
T_{21}&T_{22}&T_{23}\\
T_{31}&T_{32}&T_{33}
\end{bmatrix}\begin{bmatrix}x\\y\\z \end{bmatrix}
\end{split}\tag{1.3.1}$$
## 2. 缩放变换
### 2.1 定义
定义放缩变换为
$$
S(x,y,z)=(S_xx,S_yy, S_zz)
$$
用矩阵表示
$$
\boldsymbol S=\begin{bmatrix}
S_x&0&0 \\
0&S_y&0 \\
0&0&S_z
\end{bmatrix}
$$
## 3. 旋转变换
**以右手坐标系为例**
### 3.1 正方向约定
描述一个旋转角度时,一般约定沿着坐标轴负方向向原点看,逆时针为正方向
![](./rotate_dir.svg)
### 3.2 围绕标准轴旋转
#### 3.2.1 **结果**
围绕$x,y,z$轴旋转$\theta$的旋转矩阵
$$
\boldsymbol{R_x}=\begin{bmatrix}1&0&0\\0&\cos(\theta)&-\sin(\theta)\\0&\sin(\theta)&\cos(\theta)\end{bmatrix}
\boldsymbol{R_y}=\begin{bmatrix}\cos(\theta)&0&\sin(\theta)\\ 0&1&0\\ -\sin(\theta)&0&\cos(\theta)\end{bmatrix}
\boldsymbol{R_z}=\begin{bmatrix}\cos(\theta)&-\sin(\theta)&0 \\ \sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{bmatrix}
$$
#### 3.2.2 **求解过程**
以$R_z$为例
![](./rotate_standard.svg)
标准矢量$\vec{\boldsymbol{i}}=[1,0,0]^T,\vec{\boldsymbol{j}}=[0,1,0]^T,\vec{\boldsymbol{k}}=[0,0,1]^T,$经过旋转$\theta$后,结果为
$$\begin{split}
R_z(\vec{\boldsymbol{i}})&=[\cos(\theta),\sin(\theta),0]^T \\
R_z(\vec{\boldsymbol{j}})&=[-\sin(\theta),\cos(\theta),0]^T \\
R_z(\vec{\boldsymbol{k}})&=[0,0,1]^T
\end{split}
$$
所以根据公式1.3.1
$$
\boldsymbol{R_z}=\begin{bmatrix}\cos(\theta)&-\sin(\theta)&0 \\ \sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{bmatrix}
$$
### 3.3 围绕任意轴
#### 3.3.1 **结果**
有穿过原点单位轴$\vec{\boldsymbol{n}}$
![](./rotate_axis.svg)
求围绕该轴旋转$\theta$的旋转矩阵
#### 3.3.2 **求解过程**
![](./rotate_axis_1.svg)![](./rotate_axis_2.svg)
将矢量$\vec{\boldsymbol{v}}$分解成两个矢量的和,一个是平行于$\vec{\boldsymbol{n}}$的分量$\vec{\boldsymbol{v}}_1$和垂直于$\vec{\boldsymbol{n}}$的分量$\vec{\boldsymbol{v}}_2$
$$\begin{split}
\vec{\boldsymbol{v}} &= \vec{\boldsymbol{v}}_1+\vec{\boldsymbol{v}}_2\\
\|\vec{\boldsymbol{v}}_1\|&=\|\vec{\boldsymbol{v}}\|\cos(\alpha) \\
\|\vec{\boldsymbol{v}}_2\|&=\|\vec{\boldsymbol{v}}\|\sin(\alpha)
\end{split}$$
由于$\vec{\boldsymbol{n}}$是单位矢量,$\|\vec{\boldsymbol{n}}\|=1$,所以
$$
\vec{\boldsymbol{v}}\cdot\vec{\boldsymbol{n}}=\|\vec{\boldsymbol{v}}\|\|\vec{\boldsymbol{n}}\|\cos(\alpha)=\|\vec{\boldsymbol{v}}\|\cos(\alpha)=\|\vec{\boldsymbol{v}}_1\| \tag{3.3.2.1}
$$
$$
\|\vec{\boldsymbol{v}}\times\vec{\boldsymbol{n}}\| =\|\vec{\boldsymbol{v}}\|\|\vec{\boldsymbol{n}}\| \sin(\alpha)=\|\vec{\boldsymbol{v}}\| \sin(\alpha)=\|\vec{\boldsymbol{v}}_2\| \tag{3.2.2.2}
$$
所以根据公式3.3.2.1,$\vec{\boldsymbol{v}}_1$和$\vec{\boldsymbol{n}}$同方向,且长度和$\vec{\boldsymbol{v}}\cdot\vec{\boldsymbol{n}}$相等,所以
$$
\vec{\boldsymbol{v}}_1=\|\vec{\boldsymbol{v}}_1\|\vec{\boldsymbol{n}}=(\vec{\boldsymbol{v}}\cdot\vec{\boldsymbol{n}})\vec{\boldsymbol{n}}\tag{3.3.2.3}
$$
由于$\vec{\boldsymbol{v}}_1$不受旋转影响,所以
$$
R_n(\vec{\boldsymbol{v}})=\vec{\boldsymbol{v}}_1+R_n(\vec{\boldsymbol{v}}_2)\tag{3.3.2.4}
$$
$R_n(\vec{\boldsymbol{v}}_2)$可以分解成两部分,一部分是平行于$\vec{\boldsymbol{v}}_2$的分量$\vec{\boldsymbol{v}}_2\cos(\theta)$另一部分是垂直于$\vec{\boldsymbol{v}}_2$的分量$\vec{\boldsymbol{v}}_2\sin(\theta)$,联合公式3.2.2.2
$$
R_n(\vec{\boldsymbol{v}}_2)=\vec{\boldsymbol{v}}_2\cos(\theta)+(\vec{\boldsymbol{n}}\times\vec{\boldsymbol{v}})\sin(\theta)\tag{3.3.2.5}
$$
代入3.3.2.4可以得到
$$\begin{split}
R_n(\vec{\boldsymbol{v}})&=\vec{\boldsymbol{v}}_1+R_n(\vec{\boldsymbol{v}}_2)\\
&=\vec{\boldsymbol{v}}_1+\vec{\boldsymbol{v}}_2\cos(\theta)+(\vec{\boldsymbol{n}}\times\vec{\boldsymbol{v}})\sin(\theta)\\
&=\vec{\boldsymbol{v}}_1+(\vec{\boldsymbol{v}}-\vec{\boldsymbol{v}}_1)\cos(\theta)+(\vec{\boldsymbol{n}}\times\vec{\boldsymbol{v}})\sin(\theta)\\
&=\vec{\boldsymbol{v}}\cos(\theta)+(1-\cos(\theta))\vec{\boldsymbol{v}}_1+(\vec{\boldsymbol{n}}\times\vec{\boldsymbol{v}})\sin(\theta)\\
&=\vec{\boldsymbol{v}}\cos(\theta)+(1-\cos(\theta))(\vec{\boldsymbol{v}}\cdot\vec{\boldsymbol{n}})\vec{\boldsymbol{n}}+(\vec{\boldsymbol{n}}\times\vec{\boldsymbol{v}})\sin(\theta)
\end{split}\tag{3.3.2.6}$$
这个公式称为罗德里格旋转公式[Rodrigues' rotation formula](https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula),使用以下方法得到其矩阵形式
#### 3.3.3 一般形式
$$\begin{split}
R_n(\vec{v})&=\vec{v}-\vec{v}+\vec{v}\cos(\theta)+(1-\cos(\theta))(\vec{v}\cdot\vec{n})\vec{n}+(\vec{n}\times\vec{v})\sin(\theta)\\
&=\vec{v}-(1-\cos(\theta))\vec{v}+(1-\cos(\theta))(\vec{v}\cdot\vec{n})\vec{n}+(\vec{n}\times\vec{v})\sin(\theta)\\
&=\vec{v}+(1-\cos(\theta))[(\vec{v}\cdot\vec{n})\vec{n}-\vec{v}]+(\vec{n}\times\vec{v})\sin(\theta)\\
&=\vec{v}+(1-\cos(\theta))[(\vec{v}\cdot\vec{n})\vec{n}-(\vec{n}\cdot\vec{n})\vec{v}]+(\vec{n}\times\vec{v})\sin(\theta)
\end{split}$$
根据矢量[三重积公式](/graphics/math/math_01.html#_2-7-%E4%B8%89%E9%87%8D%E7%A7%AF),可以得到
$$
(\vec{v}\cdot\vec{n})\vec{n}-(\vec{n}\cdot\vec{n})\vec{v}=\vec{n}\times(\vec{n}\times\vec{v})
$$
所以
$$
R_n(\vec{v})=\vec{v}+(1-\cos(\theta))[\vec{n}\times(\vec{n}\times\vec{v})]+(\vec{n}\times\vec{v})\sin(\theta)
$$
使用向量$\vec{n}$的[叉乘矩阵](./graphics/math/math_01.html#_2-5-4-%E5%8F%89%E7%A7%AF%E7%9A%84%E7%9F%A9%E9%98%B5%E5%BD%A2%E5%BC%8F)$[\boldsymbol{n}]_{\times}$来代替叉乘,这里记矩阵$\boldsymbol{M_n}$为向量$\vec{n}$的叉积矩阵
$$
\boldsymbol{M_n}=\begin{bmatrix}
0&-n_z&n_y\\
n_z&0&-n_x\\
-n_y&n_x&0
\end{bmatrix}
$$
使用矩阵形式表达罗德里格旋转公式
$$\begin{split}
R_n(\vec{v})&=\boldsymbol{I}\vec{v}+(1-\cos(\theta))[\boldsymbol{M_n}(\boldsymbol{M_n}\vec{v})]+\boldsymbol{M_n}\vec{v}\sin(\theta)\\
&=[\boldsymbol{I}+(1-\cos(\theta)){\boldsymbol{M_n^2}}+\boldsymbol{M_n}\sin(\theta)]\vec{v}
\end{split}$$
所以矩阵形式的罗德里格旋转公式的公式为
$$
R_n=\boldsymbol{I}+\boldsymbol{M_n^2}(1-\cos(\theta))+\boldsymbol{M_n}\sin(\theta)
$$
#### 3.3.4 三维行列式
设$\vec{\boldsymbol{n}}=[x,y,z]^T, c=\cos(\theta), s=\sin(\theta)$分别将$\vec{\boldsymbol{i}}, \vec{\boldsymbol{j}}, \vec{\boldsymbol{k}}$带入3.3.2.6,得到
$$\begin{split}
R_n(\vec{\boldsymbol{i}})&=c\vec{i}+(1-c)(\vec{i}\cdot\vec{n})\vec{n}
+(\vec{n}\times\vec{i})s\\
&=[c+(1-c)x^2, (1-c)xy+sz, (1-c)xz-sy]^T\\
R_n(\vec{\boldsymbol{j}})&=[(1-c)xy-sz,c+(1-c)y^2,(1-c)yz+sx]^T\\
R_n(\vec{\boldsymbol{k}})&=[(1-c)xz+sy,(1-c)yz-sx,c+(1-c)z^2]^T
\end{split}$$
代入公式[1.3.1](/graphics/math/transform_01.html#_1-3-使用矩阵表示)得到
$$
\boldsymbol{R_n}=\begin{bmatrix}
c+(1-c)x^2& (1-c)xy-sz&(1-c)xz+sy\\
(1-c)xy+sz&c+(1-c)y^2&(1-c)yz-sx\\
(1-c)xz-sy&(1-c)yz+sx&c+(1-c)z^2
\end{bmatrix}
$$
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# 几何变换(二)
----
## 4. 仿射变换
### 4.1 齐次坐标(Homogeneous Coordinates)
#### 4.1.1 定义
齐次坐标就是将一个原本是n维的向量或者一个n维的点用一个n+1维坐标来表示,例如二维点$[x,y]$,用$[x,y,w]$表示,它等同于笛卡尔坐标系中的$[x/w,y/w]$
$$
[x,y,w]\Leftrightarrow[\frac{x}{w},\frac{y}{w}]
$$
#### 4.1.2 齐次的含义
笛卡尔坐标$[x,y]$,等同于齐次坐标$[kx,ky,k]$,所以称作该坐标是"齐次",Homogeneous一词有“同质”的含义,译作同质坐标可能更准确一些
#### 4.1.3 点和矢量
在图形学中,一般用$[x,y,0]$表示一个二维矢量,用$[x,y,z,1]$表示一个二维点。 这样可以保证矢量的“平移不变性”,也就是一个矢量经过平移变换,仍然保持和原先一致。
$$
\begin{bmatrix}
1&0&t_x\\
0&1&t_y\\
0&0&1
\end{bmatrix}\begin{bmatrix}x\\y\\0\end{bmatrix}=\begin{bmatrix}x\\y\\0\end{bmatrix}
$$
两个矢量的和或者差$\vec{\boldsymbol{u}}\pm\vec{\boldsymbol{v}}=[u_x,u_y,u_z,0]\pm[v_x,v_y,v_z,0]=[u_x\pm v_x,u_y\pm v_y,u_z\pm v_z,0]$仍然是一个矢量,而两个点的差也是矢量,点和矢量的和或者差也是矢量
$$\begin{split}
{vector}+{vector}&={vector} \\
{point}-{point}&={vector} \\
{point}+{point}&={?} \\
{point}+{vector}&={point} \\
\end{split}$$
其次坐标表示的两个点的和也是有意义的,表示两个点的中点
$$
[x_1,y_1,1]+[x_2,y_2,1]=[x_1+x_2,y_1+y_2,2]=[\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, 1]
$$
### 4.2 齐次矩阵表达
#### 4.2.1 线性变换无法表示图形学中所有的几何变换,因此需要定义一种变换为
$$
\alpha(\vec{\boldsymbol{v}})=\tau(\vec{\boldsymbol{v}})+\vec{\boldsymbol{b}}
$$
用矩阵表达为
$$\begin{split}
\alpha(\vec{\boldsymbol{v}})&=\boldsymbol{A}\vec{\boldsymbol{v}}+\vec{\boldsymbol{b}} \\
&=\begin{bmatrix}
A_{11}&A_{12}&A_{13}\\
A_{21}&A_{22}&A_{23}\\
A_{31}&A_{32}&A_{33}
\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}+\begin{bmatrix}b_x\\b_y\\b_z\end{bmatrix}
&=\begin{bmatrix}x'\\y'\\z'\end{bmatrix}
\end{split}$$
常用4X4齐次矩阵来做这种运算
$$
\begin{bmatrix}
A_{11}&A_{12}&A_{13}&b_x\\
A_{21}&A_{22}&A_{23}&b_y\\
A_{31}&A_{32}&A_{33}&b_z\\
0&0&0&1
\end{bmatrix}\begin{bmatrix}x\\y\\z\\1\end{bmatrix}=\begin{bmatrix}x'\\y'\\z'\\1\end{bmatrix}
$$
## 5. 平移变换
### 5.1 定义
空间中的某一点$\boldsymbol u$,移动到$\boldsymbol{u}+\vec{\boldsymbol{b}}$,那么用齐次矩阵表示这个变换为
$$
\boldsymbol T=\begin{bmatrix}
1&0&0&b_x \\
0&1&0&b_y \\
0&0&1&b_z \\
0&0&0&1
\end{bmatrix}
$$
## 6. 组合变换
### 6.1 定义
图形学中将所有变换统一使用4X4的齐次坐标组合
$$
R=R_n\cdots R_3R_2R_1
$$
这样,一系列变换可以组合到一个矩阵中完成
$$
\cdots(R_3(R_2(R_1\vec{\boldsymbol{u}}))=(\cdots R_3R_2R_1)\vec{\boldsymbol{u}}
$$
### 6.2 坐标系变换
一个物体在不同坐标系中的变换,比如常用的局部坐标系到世界坐标系的变换
![](./transform_axis.svg)
设新坐标系x'y'z'的原点在原坐标系的$\vec{\boldsymbol{O}}(x_o,y_o,z_o)$处,新坐标系的单位坐标矢量在原坐标系中的矢量为$\vec{\boldsymbol{i}'},\vec{\boldsymbol{j}'},\vec{\boldsymbol{k}'}$,那么将物体的坐标从原先坐标系xyz转换到新坐标系x'y'z'的转换可以分为两部分
#### 6.2.1 平移矩阵
构建平移矩阵,目的是将物体在原坐标系中的坐标做一个偏移,这个偏移可以视作将物体和新坐标系一起移动,将新坐标系的原点$\boldsymbol{O}$挪到原坐标系的原点处,可以得到
![](./transform_axis_move.svg)
$$
\boldsymbol{T}=\begin{bmatrix}
1&0&0&-x_0 \\
0&1&0&-y_0 \\
0&0&1&-z_0 \\
0&0&0&1
\end{bmatrix}
$$
#### 6.2.2 构建旋转矩阵
接下来要构建一个旋转矩阵$\boldsymbol{R}$,目的是将片以后的物体坐标和新坐标系旋转到和原坐标系一致,也就是经过这个矩阵,可以将新坐标系中的$x'$轴旋转到原坐标系的$x$轴,也就是:
$$
\boldsymbol{R}\begin{bmatrix}i_x'\\i_y'\\i_z'\\0\end{bmatrix}=\begin{bmatrix}1\\0\\0\\0\end{bmatrix}
$$
同理,也需要将$y'$轴旋转到$y$轴,$z'$轴旋转到$z$轴
![](./transform_axis_roate.svg)
这个矩阵并不好直接写出,但是它的逆矩阵就比较容易,也就是能够将写出
$$
\boldsymbol{R}^{-1}=\begin{bmatrix}
i_x'&j_x'&k_x'&0 \\
i_y'&j_y'&k_y'&0 \\
i_z'&j_z'&k_z'&0 \\
0&0&0&1
\end{bmatrix}
$$
这个矩阵满足
$$\begin{split}
\boldsymbol{R}^{-1}[1,0,0,0]^T&=[i_x',i_y',i_z',0]^T\\
\boldsymbol{R}^{-1}[0,1,0,0]^T&=[j_x',j_y',j_z',0]^T\\
\boldsymbol{R}^{-1}[0,0,1,0]^T&=[k_x',k_y',k_z',0]^T
\end{split}$$
由于这是一个正交矩阵,所以
$$
\boldsymbol{R}=(\boldsymbol{R}^{-1})^T=\begin{bmatrix}
i_x'&i_y'&i_z'&0 \\
j_x'&j_y'&j_z'&0 \\
k_x'&k_y'&k_z'&0 \\
0&0&0&1
\end{bmatrix}
$$
#### 6.2.3 组合后的坐标转换矩阵
$$\begin{split}
\boldsymbol{RT}&=\begin{bmatrix}
i_x'&i_y'&i_z'&-(x_0i_x'+y_0i_y'+z_0i_z') \\
j_x'&j_y'&j_z'&-(x_0j_x'+y_0j_y'+z_0j_z') \\
k_x'&k_y'&k_z'&-(x_0k_x'+y_0k_y'+z_0k_z') \\
0&0&0&1
\end{bmatrix}\\&=\begin{bmatrix}
i_x'&i_y'&i_z'&-\vec{O}\cdot\vec{i'} \\
j_x'&j_y'&j_z'&-\vec{O}\cdot\vec{j'} \\
k_x'&k_y'&k_z'&-\vec{O}\cdot\vec{k'} \\
0&0&0&1
\end{bmatrix}\end{split}
$$
验证
$$\begin{split}
\boldsymbol{RT}[i_x',i_y',i_z',0]^T&=[1,0,0,0]^T\\
\boldsymbol{RT}[j_x',j_y',j_z',0]^T&=[0,1,0,0]^T\\
\boldsymbol{RT}[k_x',k_y',k_z',0]^T&=[0,0,1,0]^T\\
\boldsymbol{RT}[x_0,y_0,z_0,1]^T&=[0,0,0,1]^T
\end{split}$$
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# 几何变换(三)
----
## 7. 法线变换
做几何变换时,法线比较特殊,需要保证法线经过变换之后仍然垂直于原先的切平面,如果使用和顶点变换相同的变换,无法保证这一点
### 7.1 举例
![](./transform_normal.svg)
如图,线段ab经过放缩变换之后成为a'b',但法线经过同样的放缩矩阵变换之后发生错误,实际法线需要另外的矩阵计算
### 7.2 法线变换矩阵
以上图为例,设$\vec{\boldsymbol{u}}=\vec{ab}$,那么$\vec{\boldsymbol{u}}\cdot\vec{\boldsymbol{n}}=0$,用矩阵的方式表达,$\mathrm{u}=[u_x, u_y, u_z]$, $\mathrm{n}=[n_x, n_y, n_z]$,就是$\mathrm{u}\mathrm{n}^T=0$
设从$\vec{ab}$到$\vec{a'b'}$的变换矩阵为$A$,根据转置矩阵的[基本运算规则](/graphics/math/math_02.html#_3-5-基本运算规则) ,可以得到
$$\begin{split}
0&=\mathrm{u}\mathrm{n}^T\\
&=\mathrm{u}(\mathrm{AA^{-1}})\mathrm{n}^T \\
&=(\mathrm{uA})(\mathrm{A^{-1}n^T}) \\
&=(\mathrm{uA})((\mathrm{A^{-1}n^T})^T)^T \\
&=(\mathrm{uA})(\mathrm{n(A^{-1})^T})^T \\
&=(\vec{\boldsymbol{u}}\mathrm{A})\cdot(\vec{\boldsymbol{n}}(\mathrm{A}^{-1})^T) \\
&= (\vec{\boldsymbol{u}}\mathrm{A})\cdot(\vec{\boldsymbol{n}}\mathrm B)
\end{split}$$
所以法线转换矩阵为
$$
\mathrm{B}=(\mathrm{A}^{-1})^T
$$
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# 几何变换(四)
----
## 8. 视图变换管线概览
一个物体从独立的物体最终转换为2D图像,期间经过如下坐标变换
![](./transformation_pipeline.png)
* Modeling transformation: 由局部坐标系(Local Space)转换为世界坐标系(World Space)
* Viewing transformation: 由世界坐标系(World Space)转换为摄像机坐标系(View Space)
* Projection transformation: 由摄像机坐标系(View Space)转换为标准设备坐标系NDC(Normalized Device Coordinate)
* Viewport transformation: 由NDC坐标系转换为视口坐标系(Screen coordinate System)
## 9. 摄像机视图变换(Viewing/Camera Transformation)
### 9.1 定义
视图变换是将物体的世界坐标系转换到摄像机的坐标系
![](./view_transform.svg)
摄像机坐标系以观察点(eye)为原点,向前方向为-z轴,x轴指向右侧(右手坐标系)
所以
$$\begin{split}
\vec{\boldsymbol{z}}'&=lookat-eye\\
\vec{\boldsymbol{x}}'&=\vec{up}\times\vec{\boldsymbol{z}}' \\
\vec{\boldsymbol{y}}'&=\vec{\boldsymbol{z}}'\times\vec{\boldsymbol{x}}'
\end{split}$$
### 9.2 计算
根据[坐标系变换公式](/graphics/math/transform_02.html#_6-2-3-%E7%BB%84%E5%90%88%E5%90%8E%E7%9A%84%E5%9D%90%E6%A0%87%E8%BD%AC%E6%8D%A2%E7%9F%A9%E9%98%B5),设摄像机在世界坐标系中的位置为$e$,摄像机的三个轴的单位轴向量为$\vec{\boldsymbol{i}'}, \vec{\boldsymbol{j}'}, \vec{\boldsymbol{k}'}$视图变换矩阵为
$$
M_{view}=\begin{bmatrix}
i_x'&i_y'&i_z'&-\vec{\boldsymbol{e}}\cdot\vec{\boldsymbol{i}}' \\
j_x'&j_y'&j_z'&-\vec{\boldsymbol{e}}\cdot\vec{\boldsymbol{j}}' \\
k_x'&k_y'&k_z'&-\vec{\boldsymbol{e}}\cdot\vec{\boldsymbol{k}}' \\
0&0&0&1
\end{bmatrix}
$$
## 10. 透视投影变换(Perspective Projective Transformations)
### 10.1 定义
投影变换是将物体从摄像机坐标系转换到标准设备坐标系(NDC),也就是坐标值在一个$[-1,1]^3$的盒子里(Canonical view volume)
可以将这个过程分为两步,首先构造一个坐标转换,将摄像机的摄影范围构成的视锥体(Frustum)变形成一个立方体(Cuboid),然后再将这个立方体转化为成标准设备坐标系下(NDC)的标准观察体CCV(Canonical View Volume)
![](./proj_transform.svg)
NDC坐标系在不同的系统上有不同的约定,在DirectX中,近平面到远平面映射到$[0,1]$,在OpenGL则是把近平面到远平面映射到$[-1,1]$,但一般都把NDC坐标系设为左手坐标系。
### 10.2 求解过程
#### 10.2.1 从视锥体到立方体
![](./proj_transform_1.svg)
设摄像机坐标系中的点$p(x,y,z)$,转换后的坐标为$(x',y',z')$,由定义可知$p$投影到摄像机近裁剪面上的点为$p'(x',y')$,由相似三角形可知
$$\begin{split}
y'&=\frac{ny}{z}\\
x'&=\frac{nx}{z}
\end{split}$$
设这个转换矩阵为$\boldsymbol{M}_{\text{persp}\rightarrow\text{ortho}}$, 可知
$$
\boldsymbol{M}_{\text{persp}\rightarrow\text{ortho}}\begin{bmatrix}
x\\y\\z\\1\end{bmatrix}=\begin{bmatrix}
\frac{nx}{z}\\\frac{ny}{z}\\z'\\1\end{bmatrix}
$$
对于表示一个点的其次坐标,所有数值乘以一个同样的数,所表达的意义相同,所以$[nx/z,ny/z,z',1]=[nx,ny,z'z,z]$,这里先暂时忽略$z$轴,可以得到
$$
\boldsymbol{M}_{\text{persp}\rightarrow\text{ortho}}\begin{bmatrix}
x\\y\\z\\1\end{bmatrix}=\begin{bmatrix}
nx\\ny\\?\\z\end{bmatrix}
$$
由此可以推断出,这个矩阵的第1,2和4行
$$
\begin{bmatrix}
n&0&0&0\\
0&n&0&0\\
?&?&?&?\\
0&0&1&0
\end{bmatrix}\begin{bmatrix}
x\\y\\z\\1\end{bmatrix}=\begin{bmatrix}
nx\\ny\\?\\z\end{bmatrix}
$$
考虑特殊情况,近平面上的任意一点$[x,y,n]$经过这个矩阵转换后
$$
\begin{bmatrix}
n&0&0&0\\
0&n&0&0\\
?&?&?&?\\
0&0&1&0
\end{bmatrix}\begin{bmatrix}
x\\y\\n\\1\end{bmatrix}=\begin{bmatrix}
nx\\ny\\?\\n\end{bmatrix}
$$
由于近平面上的点经过转换后坐标不变,所以$[nx,ny,?,n]=[x,y,n,1]$,而$[x,y,n,1]$作为其次坐标,所有数值都乘以$n$,可以得到$[x,y,n,1]==[nx,ny,n^2,n]$,所以
$$
\begin{bmatrix}
n&0&0&0\\
0&n&0&0\\
?&?&?&?\\
0&0&1&0
\end{bmatrix}\begin{bmatrix}
x\\y\\n\\1\end{bmatrix}=\begin{bmatrix}
nx\\ny\\n^2\\n\end{bmatrix}
$$
同理,远平面上的任意一个点$[x,y,f,1]$转换后为$[x',y',f,1]$,可以得到
$$
\begin{bmatrix}
n&0&0&0\\
0&n&0&0\\
?&?&?&?\\
0&0&1&0
\end{bmatrix}\begin{bmatrix}
x\\y\\f\\1\end{bmatrix}=\begin{bmatrix}
x'\\y'\\f^2\\n\end{bmatrix}
$$
由此可知,第三行的数值和$x,y$无关,一定是形如$[0,0,A,B]$这样的形式,且
$$\begin{split}
An+B=n^2\\
Af+B=f^2
\end{split}$$
可以得到
$$\begin{split}
A&=f+n\\
B&=-nf
\end{split}$$
得到这个转换矩阵为
$$
\boldsymbol{M}_{\text{persp}\rightarrow\text{ortho}}=\begin{bmatrix}
n&0&0&0\\
0&n&0&0\\
0&0&f+n&-nf\\
0&0&1&0
\end{bmatrix}
$$
#### 10.2.2 从立方体到标准观察体
这一步相对简单很多,只要将观察体的中心的$z$挪到原点,做一定的放缩,并且将$z$轴反转即可
$$\begin{split}
\boldsymbol{M}_{\text{ortho}\rightarrow\text{ccv}}&=\begin{bmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&-1&0\\
0&0&0&1
\end{bmatrix}\begin{bmatrix}
2/w&0&0&0\\
0&2/h&0&0\\
0&0&2/(n-f)&0\\
0&0&0&1
\end{bmatrix}\begin{bmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&1&-(n+f)/2\\
0&0&0&1
\end{bmatrix}\\
&=\begin{bmatrix}
2/w&0&0&0\\
0&2/h&0&0\\
0&0&-2/(n-f)&(n+f)/(n-f)\\
0&0&0&1
\end{bmatrix}
\end{split}$$
#### 10.2.3 合并后的矩阵
最终得到合并后的投影矩阵为
$$
\boldsymbol{M}_{\text{persp}\rightarrow\text{ccv}}=\boldsymbol{M}_{\text{ortho}\rightarrow\text{ccv}}\boldsymbol{M}_{\text{persp}\rightarrow\text{ortho}}=\begin{bmatrix}
\frac{2n}{w}&0&0&0\\
0&\frac{2n}{h}&0&0\\
0&0&-\frac{n+f}{n-f}&\frac{2nf}{n-f}\\
0&0&1&0
\end{bmatrix}
$$
在大部分工程项目中,摄像机的输入参数一般是如下几个
* 宽高比 $aspect=w/h$
* 视场角fov,$\tan(fov/2)=-h/2n$
* 近裁剪距离 $near=\left|n\right|=-n$
* 远裁剪距离 $far=\left|f\right|=-f$
将这些值带入上面的公式,可以得到
$$
\boldsymbol{M}_{\text{persp}\rightarrow\text{ccv}}=\begin{bmatrix}
\frac{-1}{\text{aspect}.\tan(fov/2)}&0&0&0\\
0&\frac{-1}{\tan(fov/2)}&0&0\\
0&0&-\frac{near+far}{near-far}&\frac{-2near.far}{near-far}\\
0&0&1&0
\end{bmatrix}
$$
一般来说,当针对一个空间点做矩阵转换时,如果$\boldsymbol{M}[u_x, u_y, u_z, 1]^T=[v_x,v_y,v_z,1]^T$,那么
$$
(k\boldsymbol{M})[u_x, u_y, u_z, 1]^T=k[v_x,v_y,v_z,1]^T=[kv_x,kv_y,kv_z,k]^T=[v_x,v_y,v_z,1]^T
$$
可知$k\boldsymbol{M}$在针对空间点转换矩阵中等同于$\boldsymbol{M}$,所以为了表达方便,经常将上面得到的这个矩阵乘以-1,得到
$$
\boldsymbol{M}_{\text{persp}\rightarrow\text{ccv}}=\begin{bmatrix}
\frac{1}{\text{aspect}.\tan(fov/2)}&0&0&0\\
0&\frac{1}{\tan(fov/2)}&0&0\\
0&0&\frac{near+far}{near-far}&\frac{2near.far}{near-far}\\
0&0&-1&0
\end{bmatrix}
$$
#### 10.2.4 深度插值
![](./proj_transform_2.svg)
设摄像机坐标系中有一个线段$(x_1,z_1),(x_2,z_2)$,在摄像机近平面上的投影为$(x_1',n),(x_2',n)$,线段上有一点为$(x,z)$,投影为$(x',n)$
可知
$$\begin{split}
\frac{x}{x'}=\frac{z}{n}\\
x=\frac{zx'}{n}
\end{split}$$
设线段的方程为$ax+bz=c$,带入可得
$$\begin{split}
a\frac{zx'}{n}+bz=c \\
\frac{1}{z}=\frac{ax'}{cn}+\frac{b}{c}
\end{split}$$
当投影点$x'$在投影线段上线性移动时,$x'=(1-t)x_1'+tx_2'$,可得
$$\begin{split}
\frac{1}{z}&=\frac{a}{cn}\left((1-t)x_1'+tx_2'\right)+\frac{b}{c} \\
&=(\frac{ax_1'}{cn}+\frac{b}{c})(1-t)+(\frac{ax_2'}{cn}+\frac{b}{c})t \\
&=\frac{1}{z_1}(1-t)+\frac{1}{z_2}t
\end{split}$$
也就是说,当投影点$x'$在投影线段上线性移动时,原线段上的点的深度值的倒数是线性变换的
## 11. 正交投影变换(Orthogonal Projection Transformations)
### 11.1 定义
![](./proj_transform_3.svg)
正交投影没有“远大近小”的效果
### 11.2 求解过程
相比透视投影,正交投影只需要将一个立方体形状的摄影范围最终映射到标准设备坐标系(NDC)下的一个标准立方体(Cuboid)即可
$$\begin{split}
\boldsymbol{M}_{\text{ortho}\rightarrow\text{ccv}}&=\begin{bmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&-1&0\\
0&0&0&1
\end{bmatrix}\begin{bmatrix}
2/(r-l)&0&0&0\\
0&2/(t-b)&0&0\\
0&0&2/(n-f)&0\\
0&0&0&1
\end{bmatrix}\begin{bmatrix}
1&0&0&-(l+r)/2\\
0&1&0&-(t+b)/2\\
0&0&1&-(n+f)/2\\
0&0&0&1
\end{bmatrix}\\
&=\begin{bmatrix}
2/(r-l)&0&0&(l+r)/(l-r)\\
0&2/(t-b)&0&(b+t)/(b-t)\\
0&0&-2/(n-f)&(n+f)/(n-f)\\
0&0&0&1
\end{bmatrix}
\end{split}$$
同理,在工程中一般标注一个正交投影摄像机的主要参数如下
* 近裁剪距离 $near=\left|n\right|=-n$
* 远裁剪距离 $far=\left|f\right|=-f$
* 左侧裁剪面坐标 $left=l$
* 右侧裁剪面坐标 $left=l$
* 上侧裁剪面坐标 $top=t$
* 下侧裁剪面坐标 $bottom=t$
带入上面的公式可以得到
$$
\boldsymbol{M}_{\text{ortho}\rightarrow\text{ccv}}=\begin{bmatrix}
\frac{2}{right-left}&0&0&\frac{left+right}{left-right}\\
0&\frac{2}{top-bottom}&0&\frac{bottom+top}{bottom-top}\\
0&0&\frac{2}{near-far}&\frac{near+far}{near-far}\\
0&0&0&1
\end{bmatrix}
$$
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