From 791c17ebdfbaa8dd5944ac4009491fcf7ffcca8e Mon Sep 17 00:00:00 2001
From: thejinchao <thejinchao@gmail.com>
Date: Sun, 23 Feb 2025 20:51:53 +0800
Subject: [PATCH] =?UTF-8?q?=E6=94=B9=E8=BF=9B=E6=8E=92=E7=89=88?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 .../luminosity/luminosity.md                  |   5 +-
 .../microfacets/microfacet_3.svg              | 248 +++---
 .../microfacets/microfacet_4.svg              | 711 +++++++++---------
 .../microfacets/microfacet_7.svg              | 191 ++---
 .../microfacets/microfacet_beckmann.svg       | 498 ++++++------
 .../microfacets/microfacet_ggx.svg            | 402 +++++-----
 .../microfacets/microfacet_gp.svg             | 466 ++++++------
 .../microfacets/microfacet_phong.svg          | 570 +++++++-------
 .../microfacets/microfacets_3.md              |  29 +-
 .../graphics/image_based_lighting/ibl_01.md   |  10 +-
 .../graphics/image_based_lighting/ibl_02.md   |  20 +-
 .../graphics/image_based_lighting/ibl_02.svg  | 243 +++---
 .../graphics/image_based_lighting/ibl_03.svg  | 208 ++---
 docs/note/graphics/math/math_01.md            |   9 +-
 docs/note/graphics/math/math_02.md            |  23 +-
 docs/note/graphics/math/proj_transform.svg    |  73 +-
 docs/note/graphics/math/proj_transform_3.svg  |  30 +-
 docs/note/graphics/math/transform_01.md       |  14 +-
 docs/note/graphics/math/transform_02.md       |  14 +-
 docs/note/graphics/math/transform_03.md       |   2 +-
 docs/note/graphics/math/transform_04.md       |   7 +-
 21 files changed, 1946 insertions(+), 1827 deletions(-)

diff --git a/docs/note/graphics/illumination_model/luminosity/luminosity.md b/docs/note/graphics/illumination_model/luminosity/luminosity.md
index ed8fc93..90d3669 100644
--- a/docs/note/graphics/illumination_model/luminosity/luminosity.md
+++ b/docs/note/graphics/illumination_model/luminosity/luminosity.md
@@ -134,7 +134,10 @@ d^2\Phi=L_r dA_r d\omega_r \cos\theta_r
 $$
 代入前面的立体角公式可得
 $$
-\displaystyle{ d^2\Phi=L_sdA_s\cos\theta_s\frac{dA_r\cos\theta_r}{r^2}=L_rdA_r\cos\theta_r\frac{dA_s\cos\theta_s}{r^2}}
+\begin{aligned}
+d^2\Phi&=\displaystyle{L_sdA_s\cos\theta_s\frac{dA_r\cos\theta_r}{r^2}}\\
+&=\displaystyle{L_rdA_r\cos\theta_r\frac{dA_s\cos\theta_s}{r^2}}
+\end{aligned}
 $$
 整理可得
 $$
diff --git a/docs/note/graphics/illumination_model/microfacets/microfacet_3.svg b/docs/note/graphics/illumination_model/microfacets/microfacet_3.svg
index 886a630..f96d9b5 100644
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+++ b/docs/note/graphics/illumination_model/microfacets/microfacet_3.svg
@@ -2,21 +2,21 @@
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+         points="694,201 968,201 968,25 694,25 " />
     </clipPath>
     <polyline
        style="fill:none;stroke:#000000;stroke-width:1.29231px;stroke-opacity:1"
@@ -2342,12 +2344,12 @@
   <g
      id="g1058"
      clip-path="url(#cp148)"
-     transform="translate(-72,-11.073587)">
+     transform="translate(-136.119,-11.073587)">
     <clipPath
        id="cp148">
       <polygon
          id="polygon1053"
-         points="968,25 694,25 694,201 968,201 " />
+         points="694,201 968,201 968,25 694,25 " />
     </clipPath>
     <polyline
        style="fill:none;stroke:#000000;stroke-width:1.29231px;stroke-opacity:1"
@@ -2357,12 +2359,12 @@
   <g
      id="g1065"
      clip-path="url(#cp149)"
-     transform="translate(-72,-11.073587)">
+     transform="translate(-136.119,-11.073587)">
     <clipPath
        id="cp149">
       <polygon
          id="polygon1060"
-         points="968,25 694,25 694,201 968,201 " />
+         points="694,201 968,201 968,25 694,25 " />
     </clipPath>
     <polyline
        style="fill:none;stroke:#000000;stroke-width:1.29231px;stroke-opacity:1"
@@ -2372,12 +2374,12 @@
   <g
      id="g1072"
      clip-path="url(#cp150)"
-     transform="translate(-72,-11.073587)">
+     transform="translate(-136.119,-11.073587)">
     <clipPath
        id="cp150">
       <polygon
          id="polygon1067"
-         points="968,25 694,25 694,201 968,201 " />
+         points="694,201 968,201 968,25 694,25 " />
     </clipPath>
     <polyline
        style="fill:none;stroke:#000000;stroke-width:1.29231px;stroke-opacity:1"
@@ -2387,12 +2389,12 @@
   <g
      id="g1079"
      clip-path="url(#cp151)"
-     transform="translate(621.83175,25.650101)">
+     transform="translate(557.71275,25.650101)">
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        id="cp151">
       <polygon
          id="polygon1074"
-         points="16.1683,-0.723688 0.168252,-0.723688 0.168252,13.2763 16.1683,13.2763 " />
+         points="0.168252,13.2763 16.1683,13.2763 16.1683,-0.723688 0.168252,-0.723688 " />
     </clipPath>
     <text
        style="font-size:10px;font-family:'Times New Roman';fill:#000000"
@@ -2404,12 +2406,12 @@
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      id="g1086"
      clip-path="url(#cp152)"
-     transform="translate(-72,-11.073587)">
+     transform="translate(-136.119,-11.073587)">
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        id="cp152">
       <polygon
          id="polygon1081"
-         points="968,25 694,25 694,201 968,201 " />
+         points="694,201 968,201 968,25 694,25 " />
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     <polyline
        style="fill:none;stroke:#000000;stroke-width:1.29231px;stroke-opacity:1"
@@ -2419,12 +2421,12 @@
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      id="g1093"
      clip-path="url(#cp153)"
-     transform="translate(-72,-11.073587)">
+     transform="translate(-136.119,-11.073587)">
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        id="cp153">
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          id="polygon1088"
-         points="968,25 694,25 694,201 968,201 " />
+         points="694,201 968,201 968,25 694,25 " />
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     <polyline
        style="fill:none;stroke:#000000;stroke-width:1.29231px;stroke-opacity:1"
@@ -2434,12 +2436,12 @@
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      id="g1100"
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-     transform="translate(-72,-11.073587)">
+     transform="translate(-136.119,-11.073587)">
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        id="cp154">
       <polygon
          id="polygon1095"
-         points="968,25 694,25 694,201 968,201 " />
+         points="694,201 968,201 968,25 694,25 " />
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        style="fill:none;stroke:#000000;stroke-width:1.29231px;stroke-opacity:1"
@@ -2449,12 +2451,12 @@
   <g
      id="g1107"
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-     transform="translate(-72,-11.073587)">
+     transform="translate(-136.119,-11.073587)">
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        id="cp155">
       <polygon
          id="polygon1102"
-         points="968,25 694,25 694,201 968,201 " />
+         points="694,201 968,201 968,25 694,25 " />
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     <polyline
        style="fill:none;stroke:#000000;stroke-width:1.29231px;stroke-opacity:1"
diff --git a/docs/note/graphics/illumination_model/microfacets/microfacets_3.md b/docs/note/graphics/illumination_model/microfacets/microfacets_3.md
index e5ac08e..7125d43 100644
--- a/docs/note/graphics/illumination_model/microfacets/microfacets_3.md
+++ b/docs/note/graphics/illumination_model/microfacets/microfacets_3.md
@@ -25,9 +25,11 @@ G_1=\dfrac{x}{y}=\dfrac{2\sin\gamma\sin\alpha}{\sin(\beta+\pi/2)}
 $$
 其中
 $$
-\sin\alpha=\vec{n}\cdot\vec{h} \\
-\sin(\beta+\pi/2)=\cos\beta=\vec{v}\cdot\vec{h} \\
-\sin\gamma=\vec{n}\cdot\vec{l}
+\begin{aligned}
+\sin\alpha&=\vec{n}\cdot\vec{h} \\
+\sin(\beta+\pi/2)&=\cos\beta=\vec{v}\cdot\vec{h} \\
+\sin\gamma&=\vec{n}\cdot\vec{l}
+\end{aligned}
 $$
 所以可以求得
 $$
@@ -39,16 +41,21 @@ G_2=\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{v})}{\vec{v}\cdot\vec{h}}
 $$
 所以光线受几何衰减因子的影响共有三种可能性，完全不受影响，照射光被阻挡，反射光被阻挡，最终的表达式为
 $$
-G(\vec{l},\vec{v},\vec{h})=\min\{1,G_1,G_2\}=\min\left\{1,\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{l})}{\vec{v}\cdot\vec{h}}
+\begin{aligned}
+G(\vec{l},\vec{v},\vec{h})&=\min\{1,G_1,G_2\}\\
+&=\min\left\{1,\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{l})}{\vec{v}\cdot\vec{h}}
 ,\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{v})}{\vec{v}\cdot\vec{h}}
 \right\}
+\end{aligned}
 $$
 
 ### 7.3 Schlick-Beckmann
 在Unreal Engine中，使用了这种模型，在这种模型中
 $$
-G(\vec{l},\vec{v},\vec{h})=G_1(\vec{l})G_1(\vec{v}) \\
-G_1(\vec{v})=\dfrac{\vec{n}\cdot\vec{v}}{(\vec{n}\cdot\vec{v})(1-k)+k}
+\begin{aligned}
+G(\vec{l},\vec{v},\vec{h})&=G_1(\vec{l})G_1(\vec{v}) \\
+G_1(\vec{v})&=\dfrac{\vec{n}\cdot\vec{v}}{(\vec{n}\cdot\vec{v})(1-k)+k}
+\end{aligned}
 $$
 在原始的Schlick算法中，定义
 $$
@@ -60,11 +67,17 @@ k=\frac{\alpha}{2}
 $$
 根据所计算的光源的不同，$\alpha$的定义也不同，当使用间接光源(IBL)时
 $$
-\alpha=roughness^2, k=\dfrac{roughness^2}{2}
+\begin{aligned}
+\alpha&=roughness^2\\
+k&=\dfrac{roughness^2}{2}
+\end{aligned}
 $$
 使用直接光源时，
 $$
-\alpha=\left(\frac{roughness+1}{2}\right)^2, k=\dfrac{(roughness+1)^2}{8}
+\begin{aligned}
+\alpha&=\left(\frac{roughness+1}{2}\right)^2\\
+k&=\dfrac{(roughness+1)^2}{8}
+\end{aligned}
 $$
 
 ## 8. 菲尼尔反射方程
diff --git a/docs/note/graphics/image_based_lighting/ibl_01.md b/docs/note/graphics/image_based_lighting/ibl_01.md
index f4df597..70b0e65 100644
--- a/docs/note/graphics/image_based_lighting/ibl_01.md
+++ b/docs/note/graphics/image_based_lighting/ibl_01.md
@@ -83,7 +83,10 @@ $$\begin{split}
 \end{split}$$
 其中
 $$
-\phi_i=2\pi\xi, \theta_i=\dfrac{\pi}{2}\xi
+\begin{aligned}
+\phi_i&=2\pi\xi\\
+\theta_i&=\dfrac{\pi}{2}\xi
+\end{aligned}
 $$
 $\xi$表示均匀分布在[0,1]之间的随机变量
 
@@ -101,7 +104,10 @@ $$\begin{split}
 \end{split}$$
 其中
 $$
-\phi_i=2\pi\xi, \theta_i=\arccos(1-\xi)
+\begin{aligned}
+\phi_i&=2\pi\xi\\
+\theta_i&=\arccos(1-\xi)
+\end{aligned}
 $$
 ````cpp :no-line-numbers
 vec3 irradiance = vec3(0.0);  
diff --git a/docs/note/graphics/image_based_lighting/ibl_02.md b/docs/note/graphics/image_based_lighting/ibl_02.md
index 7119e54..8ea3427 100644
--- a/docs/note/graphics/image_based_lighting/ibl_02.md
+++ b/docs/note/graphics/image_based_lighting/ibl_02.md
@@ -8,11 +8,17 @@ L_{os}(\vec{\omega_o})=\int_{\Omega}f_s(\vec{\omega_i}, \vec{\omega_o})L_i(\vec{
 $$
 同样使用蒙特卡洛进行近似积分
 $$
-\int_{\Omega}f_s(\vec{\omega_i}, \vec{\omega_o})L_i(\vec{\omega_i})(\vec{\omega}_i\vec{n})d\omega_i \approx \dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})L_i(\vec{\omega_k})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k})}
+\begin{aligned}
+&\int_{\Omega}f_s(\vec{\omega_i}, \vec{\omega_o})L_i(\vec{\omega_i})(\vec{\omega}_i\vec{n})d\omega_i \\
+\approx &\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})L_i(\vec{\omega_k})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k})}
+\end{aligned}
 $$
 EPIC在此基础上提出了一种近似[计算方法](http://blog.selfshadow.com/publications/s2013-shading-course/karis/s2013_pbs_epic_notes_v2.pdf)
 $$
-\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})L_i(\vec{\omega_k})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k}, \vec{\omega_o})}\approx\left(\dfrac{1}{N}\sum_{k=1}^N L_i(\vec{\omega_k})\right)\left(\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k})}\right)
+\begin{aligned}
+&\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})L_i(\vec{\omega_k})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k}, \vec{\omega_o})}\\
+\approx&\left(\dfrac{1}{N}\sum_{k=1}^N L_i(\vec{\omega_k})\right)\left(\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k})}\right)
+\end{aligned}
 $$
 这种方法的好处是可以通过分开预计算，尽量减少实际渲染时的计算量，这个公式可以理解为“Env. lighting x BRDF”，左边的第一部分是小平面受到的所有环境光的采样值，第二部分只跟材质的BRDF属性相关。
 
@@ -71,7 +77,10 @@ $$\begin{split}
 \end{split}$$
 使用Mathmatica计算可得
 $$
-\text{cdf}(\theta)=\dfrac{\sin^2\theta}{\cos^2\theta(\alpha^2-1)+1}=\dfrac{1-\cos^2\theta}{\cos^2\theta(\alpha^2-1)+1} 
+\begin{aligned}
+\text{cdf}(\theta)&=\dfrac{\sin^2\theta}{\cos^2\theta(\alpha^2-1)+1}\\
+&=\dfrac{1-\cos^2\theta}{\cos^2\theta(\alpha^2-1)+1} 
+\end{aligned}
 $$
 根据前面的概率知识，可以得到微表面的法线的球坐标$\theta$和$\phi$的采样函数为
 $$\begin{split}
@@ -85,7 +94,7 @@ $$
 \dfrac{1}{N}\sum_{k=1}^N L_i(\vec{\omega}_k)\approx\dfrac{1}{\sum_{k=1}^{N}(\vec{n}\cdot\vec{\omega}_k)}\sum_{k=1}^{N}{L_i(\vec{\omega}_k)(\vec{n}\cdot\vec{\omega}_k)}
 $$
 
-```cpp
+```cpp :no-line-numbers
 // make the simplyfying assumption that V equals R equals the normal 
 vec3 R = N;
 vec3 V = R;
@@ -96,7 +105,8 @@ float totalWeight = 0.0;
 
 for(uint i = 0u; i < SAMPLE_COUNT; ++i)
 {
-    // generates a sample vector that's biased towards the preferred alignment direction (importance sampling).
+    // generates a sample vector that's biased towards the preferred 
+    //alignment direction (importance sampling).
     vec2 Xi = Hammersley(i, SAMPLE_COUNT);
     vec3 H = ImportanceSampleGGX(Xi, N, roughness);
     vec3 L  = normalize(2.0 * dot(V, H) * H - V);
diff --git a/docs/note/graphics/image_based_lighting/ibl_02.svg b/docs/note/graphics/image_based_lighting/ibl_02.svg
index c357b80..438c30a 100644
--- a/docs/note/graphics/image_based_lighting/ibl_02.svg
+++ b/docs/note/graphics/image_based_lighting/ibl_02.svg
@@ -2,21 +2,21 @@
 <!-- Created with Inkscape (http://www.inkscape.org/) -->
 
 <svg
-   xmlns:dc="http://purl.org/dc/elements/1.1/"
-   xmlns:cc="http://creativecommons.org/ns#"
-   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
-   xmlns:svg="http://www.w3.org/2000/svg"
-   xmlns="http://www.w3.org/2000/svg"
-   xmlns:xlink="http://www.w3.org/1999/xlink"
-   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
-   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
-   width="600"
+   width="510"
    height="300"
-   viewBox="0 0 600 300"
+   viewBox="0 0 510 300"
    version="1.1"
    id="svg8"
-   inkscape:version="0.92.3 (2405546, 2018-03-11)"
-   sodipodi:docname="ibl_02.svg">
+   inkscape:version="1.4 (86a8ad7, 2024-10-11)"
+   sodipodi:docname="ibl_02.svg"
+   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+   xmlns:xlink="http://www.w3.org/1999/xlink"
+   xmlns="http://www.w3.org/2000/svg"
+   xmlns:svg="http://www.w3.org/2000/svg"
+   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+   xmlns:cc="http://creativecommons.org/ns#"
+   xmlns:dc="http://purl.org/dc/elements/1.1/">
   <sodipodi:namedview
      id="base"
      pagecolor="#ffffff"
@@ -25,25 +25,31 @@
      inkscape:pageopacity="0.0"
      inkscape:pageshadow="2"
      inkscape:zoom="1.4778532"
-     inkscape:cx="303.36494"
-     inkscape:cy="115.86812"
+     inkscape:cx="303.48075"
+     inkscape:cy="116.04671"
      inkscape:document-units="px"
      inkscape:current-layer="layer2"
      showgrid="true"
      units="px"
-     inkscape:window-width="1920"
-     inkscape:window-height="1017"
-     inkscape:window-x="-8"
-     inkscape:window-y="-8"
+     inkscape:window-width="2560"
+     inkscape:window-height="1494"
+     inkscape:window-x="-11"
+     inkscape:window-y="-11"
      inkscape:window-maximized="1"
      inkscape:snap-grids="false"
      inkscape:snap-nodes="false"
-     inkscape:snap-intersection-paths="false">
+     inkscape:snap-intersection-paths="false"
+     inkscape:showpageshadow="2"
+     inkscape:pagecheckerboard="0"
+     inkscape:deskcolor="#d1d1d1">
     <inkscape:grid
        spacingy="10"
        spacingx="10"
        id="grid10"
-       type="xygrid" />
+       type="xygrid"
+       originx="0"
+       originy="0"
+       units="px" />
   </sodipodi:namedview>
   <defs
      id="defs2">
@@ -58,7 +64,7 @@
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          inkscape:connector-curvature="0"
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-         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
          d="M 0,0 5,-5 -12.5,0 5,5 Z"
          id="path5360" />
     </marker>
@@ -73,7 +79,7 @@
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          inkscape:connector-curvature="0"
          transform="matrix(0.4,0,0,0.4,4,0)"
-         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
          d="M 0,0 5,-5 -12.5,0 5,5 Z"
          id="path5254" />
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@@ -90,7 +96,7 @@
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-         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
          transform="matrix(0.4,0,0,0.4,4,0)" />
     </marker>
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@@ -106,7 +112,7 @@
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+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
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+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
          transform="matrix(-0.4,0,0,-0.4,-4,0)" />
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@@ -136,7 +142,7 @@
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+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
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+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
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          id="path4834" />
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+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
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+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
          transform="matrix(-0.4,0,0,-0.4,-4,0)" />
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@@ -199,7 +205,7 @@
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+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
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+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
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+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1"
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@@ -244,7 +250,6 @@
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         <dc:type
            rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
-        <dc:title />
       </cc:Work>
     </rdf:RDF>
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diff --git a/docs/note/graphics/math/math_01.md b/docs/note/graphics/math/math_01.md
index ce9d24e..9441656 100644
--- a/docs/note/graphics/math/math_01.md
+++ b/docs/note/graphics/math/math_01.md
@@ -142,13 +142,14 @@ $$
 ### 2.5叉积
 #### 2.5.1 定义
 对于两个三维矢量$\vec{a}, \vec{b}$，定义它们的叉积
-$$
-\vec{\boldsymbol a}\times\vec{\boldsymbol b}=\begin{vmatrix}
+$$\begin{aligned}
+\vec{\boldsymbol a}\times\vec{\boldsymbol b}&=\begin{vmatrix}
 \vec{\boldsymbol i}& \vec{\boldsymbol j} & \vec{\boldsymbol k}\cr
 a_x   & a_y & a_z \cr
 b_x & b_y & b_z
-\end{vmatrix}=
-(a_yb_z-a_zb_y)\vec{\boldsymbol{i}}+(a_zb_x-a_xb_z)\vec{\boldsymbol{j}}+(a_xb_y-a_yb_x)\vec{\boldsymbol{k}}
+\end{vmatrix}\\
+&=(a_yb_z-a_zb_y)\vec{\boldsymbol{i}}+(a_zb_x-a_xb_z)\vec{\boldsymbol{j}}+(a_xb_y-a_yb_x)\vec{\boldsymbol{k}}
+\end{aligned}
 $$
 #### 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。
diff --git a/docs/note/graphics/math/math_02.md b/docs/note/graphics/math/math_02.md
index ac4c0b3..6c8ffef 100644
--- a/docs/note/graphics/math/math_02.md
+++ b/docs/note/graphics/math/math_02.md
@@ -7,19 +7,23 @@ $\boldsymbol A_{ij}$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，
 $$
-\boldsymbol A=\begin{bmatrix}
+\begin{aligned}
+\boldsymbol A&=\begin{bmatrix}
 A_{11}&A_{12}&A_{13}\\
 A_{21}&A_{22}&A_{23}\\
 A_{31}&A_{32}&A_{33}
-\end{bmatrix}=\begin{bmatrix}
+\end{bmatrix}\\
+&=\begin{bmatrix}
 \leftarrow & A_{1,*} & \rightarrow \\
 \leftarrow & A_{2,*} & \rightarrow \\
 \leftarrow & A_{3,*} & \rightarrow 
-\end{bmatrix}=\begin{bmatrix}
+\end{bmatrix}\\
+&=\begin{bmatrix}
 \uparrow & \uparrow & \uparrow \\
 A_{*,1} & A_{*,2} & A_{*,3} \\
 \downarrow & \downarrow & \downarrow 
 \end{bmatrix}
+\end{aligned}
 $$
 ### 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
@@ -72,11 +76,13 @@ $$
 
 #### 3.4.2 举例
 $$
-\boldsymbol A=\begin{bmatrix}
+\begin{aligned}
+\boldsymbol A&=\begin{bmatrix}
 2&-1&8\\3&6&-4\end{bmatrix}\\
-\boldsymbol A^T=\begin{bmatrix}
+\boldsymbol A^T&=\begin{bmatrix}
 2&3\\-1&6\\8&-4
 \end{bmatrix}
+\end{aligned}
 $$
 
 ### 3.4.3 正交矩阵
@@ -140,10 +146,13 @@ $$
 
 #### 3.8.2 举例
 $$
-\det{\begin{bmatrix}
+\begin{aligned}
+&\det{\begin{bmatrix}
 A_{11}&A_{12}\\
 A_{21}&A_{22}
-\end{bmatrix}}=A_{11}\det[A_{22}]-A_{12}\det[A_{21}]=A_{11}A_{22}-A_{12}A_{21}
+\end{bmatrix}}\\
+&=A_{11}\det[A_{22}]-A_{12}\det[A_{21}]=A_{11}A_{22}-A_{12}A_{21}
+\end{aligned}
 $$
 
 $$\begin{split}
diff --git a/docs/note/graphics/math/proj_transform.svg b/docs/note/graphics/math/proj_transform.svg
index b5ab53c..eeb78f7 100644
--- a/docs/note/graphics/math/proj_transform.svg
+++ b/docs/note/graphics/math/proj_transform.svg
@@ -2,9 +2,9 @@
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-   width="800"
-   height="600"
-   viewBox="0 0 211.66666 158.75001"
+   width="450"
+   height="850"
+   viewBox="0 0 119.0625 224.89585"
    version="1.1"
    id="svg8"
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+     inkscape:cx="386.94455"
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+     inkscape:current-layer="svg8"
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       <g
@@ -1083,9 +1061,32 @@
         </g>
       </g>
     </g>
+    <g
+       inkscape:label=""
+       transform="rotate(90,-132.74809,193.6589)"
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+      <g
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+         fill-opacity="1"
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+           transform="translate(1.993,9.573)"
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+          <path
+             d="m 15,-6.75 c 0.859375,0.734375 1.921875,1.28125 2.59375,1.578125 -0.734375,0.34375 -1.75,0.875 -2.59375,1.59375 H 1.875 c -0.34375,0 -0.734375,0 -0.734375,0.421875 0,0.40625 0.375,0.40625 0.71875,0.40625 H 14.0625 C 13.078125,-1.796875 12,0.015625 12,0.296875 c 0,0.21875 0.265625,0.21875 0.390625,0.21875 0.171875,0 0.3125,0 0.390625,-0.171875 0.4375,-0.78125 1.015625,-1.875 2.359375,-3.078125 1.421875,-1.25 2.8125,-1.8125 3.890625,-2.125 0.34375,-0.125 0.359375,-0.140625 0.40625,-0.1875 0.04687,-0.015625 0.04687,-0.078125 0.04687,-0.125 0,-0.03125 0,-0.078125 -0.03125,-0.109375 l -0.0625,-0.046875 C 19.359375,-5.34375 19.328125,-5.375 18.9375,-5.5 c -2.78125,-0.828125 -4.84375,-2.703125 -6,-4.90625 -0.234375,-0.421875 -0.25,-0.4375 -0.546875,-0.4375 -0.125,0 -0.390625,0 -0.390625,0.21875 0,0.28125 1.0625,2.078125 2.0625,3.046875 H 1.859375 c -0.34375,0 -0.71875,0 -0.71875,0.40625 C 1.140625,-6.75 1.53125,-6.75 1.875,-6.75 Z m 0,0"
+             id="path2-1-2-8"
+             style="stroke:none;stroke-width:2.835;stroke-dasharray:none;stroke-opacity:1" />
+        </g>
+      </g>
+    </g>
   </g>
   <g
-     id="g26">
+     id="g26"
+     transform="translate(-87.931676,63.402272)">
     <path
        style="fill:none;stroke:#0000ff;stroke-width:0.264583px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
        d="m 145.48648,83.40865 31.32432,7.80112 -3.58316,31.04593"
@@ -1527,7 +1528,8 @@
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   </g>
   <g
-     id="g25">
+     id="g25"
+     transform="translate(-87.30805,69.014933)">
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        sodipodi:nodetypes="ccccc"
        inkscape:connector-curvature="0"
@@ -2128,7 +2130,8 @@
      y="0"
      x="0" />
   <g
-     id="g23">
+     id="g23"
+     transform="translate(10.809568,2.078763)">
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        d="m 25.324104,43.60594 a 8.5879527,13.208604 0 0 1 0.978868,3.63203"
diff --git a/docs/note/graphics/math/proj_transform_3.svg b/docs/note/graphics/math/proj_transform_3.svg
index 0469c83..9fd2403 100644
--- a/docs/note/graphics/math/proj_transform_3.svg
+++ b/docs/note/graphics/math/proj_transform_3.svg
@@ -2,9 +2,9 @@
 <!-- Created with Inkscape (http://www.inkscape.org/) -->
 
 <svg
-   width="800"
-   height="350"
-   viewBox="0 0 211.66666 92.604173"
+   width="400"
+   height="520"
+   viewBox="0 0 105.83333 137.58334"
    version="1.1"
    id="svg8"
    inkscape:version="1.4 (86a8ad7, 2024-10-11)"
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-     inkscape:cx="390.4312"
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+     inkscape:cx="359.6433"
+     inkscape:cy="321.7505"
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@@ -344,17 +344,17 @@
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+     transform="translate(0,-191.167)"
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+       inkscape:transform-center-x="6.6241936"
+       inkscape:transform-center-y="-2.6854843">
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          fill="#000000"
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@@ -373,7 +373,8 @@
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          inkscape:connector-curvature="0"
@@ -778,7 +779,8 @@
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-       id="g14">
+       id="g14"
+       transform="translate(-99.183849,51.203215)">
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diff --git a/docs/note/graphics/math/transform_01.md b/docs/note/graphics/math/transform_01.md
index a5c60d5..039db5e 100644
--- a/docs/note/graphics/math/transform_01.md
+++ b/docs/note/graphics/math/transform_01.md
@@ -58,9 +58,11 @@ $$
 #### 3.2.1 **结果**
 围绕$x,y,z$轴旋转$\theta$的旋转矩阵
 $$
-\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}
+\begin{aligned}
+\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}
+\end{aligned}
 $$
 #### 3.2.2 **求解过程**
 以$R_z$为例
@@ -134,7 +136,7 @@ R_n(\vec{v})&=\vec{v}-\vec{v}+\vec{v}\cos(\theta)+(1-\cos(\theta))(\vec{v}\cdot\
 &=\vec{v}+(1-\cos(\theta))[(\vec{v}\cdot\vec{n})\vec{n}-\vec{v}]+(\vec{n}\times\vec{v})\sin(\theta)\\
 &=\vec{v}+(1-\cos(\theta))[(\vec{v}\cdot\vec{n})\vec{n}-(\vec{n}\cdot\vec{n})\vec{v}]+(\vec{n}\times\vec{v})\sin(\theta)
 \end{split}$$
-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
+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
 $$
 (\vec{v}\cdot\vec{n})\vec{n}-(\vec{n}\cdot\vec{n})\vec{v}=\vec{n}\times(\vec{n}\times\vec{v})
 $$
@@ -142,7 +144,7 @@ $$
 $$
 R_n(\vec{v})=\vec{v}+(1-\cos(\theta))[\vec{n}\times(\vec{n}\times\vec{v})]+(\vec{n}\times\vec{v})\sin(\theta)
 $$
-使用向量$\vec{n}$的[叉乘矩阵](./graphics/math/math_01.html#_2-5-4-%E5%8F%89%E7%A7%AF%E7%9A%84%E7%9F%A9%E9%98%B5%E5%BD%A2%E5%BC%8F)$[\boldsymbol{n}]_{\times}$来代替叉乘，这里记矩阵$\boldsymbol{M_n}$为向量$\vec{n}$的叉积矩阵
+使用向量$\vec{n}$的[叉乘矩阵](/note/graphics/math/math_01.html#_2-5-4-%E5%8F%89%E7%A7%AF%E7%9A%84%E7%9F%A9%E9%98%B5%E5%BD%A2%E5%BC%8F)$[\boldsymbol{n}]_{\times}$来代替叉乘，这里记矩阵$\boldsymbol{M_n}$为向量$\vec{n}$的叉积矩阵
 $$
 \boldsymbol{M_n}=\begin{bmatrix}
 0&-n_z&n_y\\
@@ -168,7 +170,7 @@ R_n(\vec{\boldsymbol{i}})&=c\vec{i}+(1-c)(\vec{i}\cdot\vec{n})\vec{n}
 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}$$
-代入公式[1.3.1](/graphics/math/transform_01.html#_1-3-使用矩阵表示)得到
+代入公式[1.3.1](/note/graphics/math/transform_01.html#_1-3-使用矩阵表示)得到
 $$
 \boldsymbol{R_n}=\begin{bmatrix}
 c+(1-c)x^2& (1-c)xy-sz&(1-c)xz+sy\\
diff --git a/docs/note/graphics/math/transform_02.md b/docs/note/graphics/math/transform_02.md
index c37c4c7..51998c6 100644
--- a/docs/note/graphics/math/transform_02.md
+++ b/docs/note/graphics/math/transform_02.md
@@ -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
diff --git a/docs/note/graphics/math/transform_03.md b/docs/note/graphics/math/transform_03.md
index f9e21a6..66905ee 100644
--- a/docs/note/graphics/math/transform_03.md
+++ b/docs/note/graphics/math/transform_03.md
@@ -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
 $$\begin{split}
 0&=\mathrm{u}\mathrm{n}^T\\
 &=\mathrm{u}(\mathrm{AA^{-1}})\mathrm{n}^T \\
diff --git a/docs/note/graphics/math/transform_04.md b/docs/note/graphics/math/transform_04.md
index d69cfcb..adf4d12 100644
--- a/docs/note/graphics/math/transform_04.md
+++ b/docs/note/graphics/math/transform_04.md
@@ -26,7 +26,7 @@ $$\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 $aspect=w/h$