改进排版
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@@ -142,13 +142,14 @@ $$
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### 2.5叉积
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#### 2.5.1 定义
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对于两个三维矢量$\vec{a}, \vec{b}$，定义它们的叉积
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$$
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\vec{\boldsymbol a}\times\vec{\boldsymbol b}=\begin{vmatrix}
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$$\begin{aligned}
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\vec{\boldsymbol a}\times\vec{\boldsymbol b}&=\begin{vmatrix}
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\vec{\boldsymbol i}& \vec{\boldsymbol j} & \vec{\boldsymbol k}\cr
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a_x & a_y & a_z \cr
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b_x & b_y & b_z
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\end{vmatrix}=
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(a_yb_z-a_zb_y)\vec{\boldsymbol{i}}+(a_zb_x-a_xb_z)\vec{\boldsymbol{j}}+(a_xb_y-a_yb_x)\vec{\boldsymbol{k}}
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\end{vmatrix}\\
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&=(a_yb_z-a_zb_y)\vec{\boldsymbol{i}}+(a_zb_x-a_xb_z)\vec{\boldsymbol{j}}+(a_xb_y-a_yb_x)\vec{\boldsymbol{k}}
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\end{aligned}
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$$
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#### 2.5.2 叉积的几何意义
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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。
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@@ -7,19 +7,23 @@ $\boldsymbol A_{ij}$表示第$i$行第$j$列的元素，
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$\boldsymbol A_{i,*}$表示第$i$行所有元素，
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$\boldsymbol A_{*,j}$表示第$j$列所有元素，
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$$
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\boldsymbol A=\begin{bmatrix}
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\begin{aligned}
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\boldsymbol A&=\begin{bmatrix}
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A_{11}&A_{12}&A_{13}\\
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A_{21}&A_{22}&A_{23}\\
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A_{31}&A_{32}&A_{33}
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\end{bmatrix}=\begin{bmatrix}
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\end{bmatrix}\\
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&=\begin{bmatrix}
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\leftarrow & A_{1,*} & \rightarrow \\
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\leftarrow & A_{2,*} & \rightarrow \\
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\leftarrow & A_{3,*} & \rightarrow
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\end{bmatrix}=\begin{bmatrix}
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\end{bmatrix}\\
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&=\begin{bmatrix}
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\uparrow & \uparrow & \uparrow \\
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A_{*,1} & A_{*,2} & A_{*,3} \\
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\downarrow & \downarrow & \downarrow
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\end{bmatrix}
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\end{aligned}
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$$
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### 3.2 加法
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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
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@@ -72,11 +76,13 @@ $$
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#### 3.4.2 举例
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$$
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\boldsymbol A=\begin{bmatrix}
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\begin{aligned}
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\boldsymbol A&=\begin{bmatrix}
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2&-1&8\\3&6&-4\end{bmatrix}\\
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\boldsymbol A^T=\begin{bmatrix}
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\boldsymbol A^T&=\begin{bmatrix}
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2&3\\-1&6\\8&-4
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\end{bmatrix}
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\end{aligned}
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$$
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### 3.4.3 正交矩阵
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@@ -140,10 +146,13 @@ $$
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#### 3.8.2 举例
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$$
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\det{\begin{bmatrix}
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\begin{aligned}
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&\det{\begin{bmatrix}
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A_{11}&A_{12}\\
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A_{21}&A_{22}
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\end{bmatrix}}=A_{11}\det[A_{22}]-A_{12}\det[A_{21}]=A_{11}A_{22}-A_{12}A_{21}
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\end{bmatrix}}\\
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&=A_{11}\det[A_{22}]-A_{12}\det[A_{21}]=A_{11}A_{22}-A_{12}A_{21}
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\end{aligned}
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$$
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$$\begin{split}
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@@ -58,9 +58,11 @@ $$
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#### 3.2.1 **结果**
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围绕$x,y,z$轴旋转$\theta$的旋转矩阵
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$$
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\boldsymbol{R_x}=\begin{bmatrix}1&0&0\\0&\cos(\theta)&-\sin(\theta)\\0&\sin(\theta)&\cos(\theta)\end{bmatrix}
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\boldsymbol{R_y}=\begin{bmatrix}\cos(\theta)&0&\sin(\theta)\\ 0&1&0\\ -\sin(\theta)&0&\cos(\theta)\end{bmatrix}
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\boldsymbol{R_z}=\begin{bmatrix}\cos(\theta)&-\sin(\theta)&0 \\ \sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{bmatrix}
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\begin{aligned}
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\boldsymbol{R_x}&=\begin{bmatrix}1&0&0\\0&\cos(\theta)&-\sin(\theta)\\0&\sin(\theta)&\cos(\theta)\end{bmatrix}\\
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\boldsymbol{R_y}&=\begin{bmatrix}\cos(\theta)&0&\sin(\theta)\\ 0&1&0\\ -\sin(\theta)&0&\cos(\theta)\end{bmatrix}\\
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\boldsymbol{R_z}&=\begin{bmatrix}\cos(\theta)&-\sin(\theta)&0 \\ \sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{bmatrix}
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\end{aligned}
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$$
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#### 3.2.2 **求解过程**
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以$R_z$为例
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@@ -134,7 +136,7 @@ R_n(\vec{v})&=\vec{v}-\vec{v}+\vec{v}\cos(\theta)+(1-\cos(\theta))(\vec{v}\cdot\
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&=\vec{v}+(1-\cos(\theta))[(\vec{v}\cdot\vec{n})\vec{n}-\vec{v}]+(\vec{n}\times\vec{v})\sin(\theta)\\
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&=\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)
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\end{split}$$
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根据矢量[三重积公式](/graphics/math/math_01.html#_2-7-%E4%B8%89%E9%87%8D%E7%A7%AF)，可以得到
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根据矢量[三重积公式](/note/graphics/math/math_01.html#_2-7-%E4%B8%89%E9%87%8D%E7%A7%AF)，可以得到
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$$
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(\vec{v}\cdot\vec{n})\vec{n}-(\vec{n}\cdot\vec{n})\vec{v}=\vec{n}\times(\vec{n}\times\vec{v})
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$$
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@@ -142,7 +144,7 @@ $$
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$$
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R_n(\vec{v})=\vec{v}+(1-\cos(\theta))[\vec{n}\times(\vec{n}\times\vec{v})]+(\vec{n}\times\vec{v})\sin(\theta)
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$$
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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
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使用向量$\vec{n}$的[叉乘矩阵](/note/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
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$$
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\boldsymbol{M_n}=\begin{bmatrix}
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0&-n_z&n_y\\
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@@ -168,7 +170,7 @@ R_n(\vec{\boldsymbol{i}})&=c\vec{i}+(1-c)(\vec{i}\cdot\vec{n})\vec{n}
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R_n(\vec{\boldsymbol{j}})&=[(1-c)xy-sz,c+(1-c)y^2,(1-c)yz+sx]^T\\
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R_n(\vec{\boldsymbol{k}})&=[(1-c)xz+sy,(1-c)yz-sx,c+(1-c)z^2]^T
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\end{split}$$
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代入公式[1.3.1](/graphics/math/transform_01.html#_1-3-使用矩阵表示)得到
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代入公式[1.3.1](/note/graphics/math/transform_01.html#_1-3-使用矩阵表示)得到
|
||||
$$
|
||||
\boldsymbol{R_n}=\begin{bmatrix}
|
||||
c+(1-c)x^2& (1-c)xy-sz&(1-c)xz+sy\\
|
||||
|
||||
@@ -22,7 +22,14 @@ $$
|
||||
0&0&1
|
||||
\end{bmatrix}\begin{bmatrix}x\\y\\0\end{bmatrix}=\begin{bmatrix}x\\y\\0\end{bmatrix}
|
||||
$$
|
||||
两个矢量的和或者差$\vec{\boldsymbol{u}}\pm\vec{\boldsymbol{v}}=[u_x,u_y,u_z,0]\pm[v_x,v_y,v_z,0]=[u_x\pm v_x,u_y\pm v_y,u_z\pm v_z,0]$仍然是一个矢量，而两个点的差也是矢量，点和矢量的和或者差也是矢量
|
||||
两个矢量的和或者差
|
||||
$$
|
||||
\begin{aligned}
|
||||
\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]
|
||||
\end{aligned}
|
||||
$$
|
||||
仍然是一个矢量，而两个点的差也是矢量，点和矢量的和或者差也是矢量
|
||||
$$\begin{split}
|
||||
{vector}+{vector}&={vector} \\
|
||||
{point}-{point}&={vector} \\
|
||||
@@ -31,7 +38,10 @@ $$\begin{split}
|
||||
\end{split}$$
|
||||
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
|
||||
$$
|
||||
[x_1,y_1,1]+[x_2,y_2,1]=[x_1+x_2,y_1+y_2,2]=[\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, 1]
|
||||
\begin{aligned}
|
||||
[x_1,y_1,1]+[x_2,y_2,1]&=[x_1+x_2,y_1+y_2,2]\\
|
||||
&=[\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, 1]
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
### 4.2 齐次矩阵表达
|
||||
|
||||
@@ -11,7 +11,7 @@
|
||||
|
||||
### 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
|
||||
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](/note/graphics/math/math_02.html#_3-5-基本运算规则) ，可以得到
|
||||
$$\begin{split}
|
||||
0&=\mathrm{u}\mathrm{n}^T\\
|
||||
&=\mathrm{u}(\mathrm{AA^{-1}})\mathrm{n}^T \\
|
||||
|
||||
@@ -26,7 +26,7 @@ $$\begin{split}
|
||||
\end{split}$$
|
||||
|
||||
### 9.2 计算
|
||||
根据[坐标系变换公式](/graphics/math/transform_02.html#_6-2-3-%E7%BB%84%E5%90%88%E5%90%8E%E7%9A%84%E5%9D%90%E6%A0%87%E8%BD%AC%E6%8D%A2%E7%9F%A9%E9%98%B5)，设摄像机在世界坐标系中的位置为$e$，摄像机的三个轴的单位轴向量为$\vec{\boldsymbol{i}'}, \vec{\boldsymbol{j}'}, \vec{\boldsymbol{k}'}$视图变换矩阵为
|
||||
根据[坐标系变换公式](/note/graphics/math/transform_02.html#_6-2-3-%E7%BB%84%E5%90%88%E5%90%8E%E7%9A%84%E5%9D%90%E6%A0%87%E8%BD%AC%E6%8D%A2%E7%9F%A9%E9%98%B5)，设摄像机在世界坐标系中的位置为$e$，摄像机的三个轴的单位轴向量为$\vec{\boldsymbol{i}'}, \vec{\boldsymbol{j}'}, \vec{\boldsymbol{k}'}$视图变换矩阵为
|
||||
$$
|
||||
M_{view}=\begin{bmatrix}
|
||||
i_x'&i_y'&i_z'&-\vec{\boldsymbol{e}}\cdot\vec{\boldsymbol{i}}' \\
|
||||
@@ -161,12 +161,15 @@ $$\begin{split}
|
||||
#### 10.2.3 合并后的矩阵
|
||||
最终得到合并后的投影矩阵为
|
||||
$$
|
||||
\boldsymbol{M}_{\text{persp}\rightarrow\text{ccv}}=\boldsymbol{M}_{\text{ortho}\rightarrow\text{ccv}}\boldsymbol{M}_{\text{persp}\rightarrow\text{ortho}}=\begin{bmatrix}
|
||||
\begin{aligned}
|
||||
\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}
|
||||
\end{aligned}
|
||||
$$
|
||||
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 $aspect=w/h$
|
||||
|
||||
Reference in New Issue
Block a user