From eafd08058e034de739ee32171c7c6e59603b3167 Mon Sep 17 00:00:00 2001 From: InkSoul Date: Tue, 12 Sep 2023 22:22:05 +0800 Subject: [PATCH] =?UTF-8?q?=E5=AE=8C=E6=88=90=E4=BA=8C=E6=AC=A1=E5=9E=8B?= =?UTF-8?q?=E5=85=AC=E5=BC=8F?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...崩坏:星穹铁道卡渲还原》基于blender材质.md | 29 ++++ content/mathematics/公式大全.md | 137 +++++++++++++++++- static/images/计算机图形学/小预告AO.png | Bin 0 -> 1419883 bytes static/images/计算机图形学/小预告AO.webp | Bin 0 -> 156724 bytes 4 files changed, 165 insertions(+), 1 deletion(-) create mode 100644 content/computergraphic/《崩坏:星穹铁道卡渲还原》基于blender材质.md create mode 100644 static/images/计算机图形学/小预告AO.png create mode 100644 static/images/计算机图形学/小预告AO.webp diff --git a/content/computergraphic/《崩坏:星穹铁道卡渲还原》基于blender材质.md b/content/computergraphic/《崩坏:星穹铁道卡渲还原》基于blender材质.md new file mode 100644 index 0000000..be84654 --- /dev/null +++ b/content/computergraphic/《崩坏:星穹铁道卡渲还原》基于blender材质.md @@ -0,0 +1,29 @@ +--- +title: "《崩坏:星穹铁道卡渲还原》基于blender材质(锐意制作中)" +date: 2023-09-12T21:08:00+08:00 + +--- + +本文章内容基于以下教程视频 + +[给你柠檬椰果养乐多你会跟我玩吗](https://space.bilibili.com/32704665/video) + +制作的 + +[【Unity/虚幻5/Blender】3种引擎 崩坏: 星穹铁道风格 卡通渲染 从球谐光照到眉毛透过刘海 完整流程](https://www.bilibili.com/video/BV1CN411C7qx/?share_source=copy_web&vd_source=07c9e60b3641af29e1c4b6ce08afeeb3) + +和 + +[Rorinais](https://space.bilibili.com/351668489/video) + +制作的 + +[【已更新完成】blender卡通渲染流程教程小白向](https://www.bilibili.com/video/BV1Rg4y1c7bT/?share_source=copy_web&vd_source=07c9e60b3641af29e1c4b6ce08afeeb3) + + +--- + +小预告(已完成至AO) + + +![webp support test](../../images/%E8%AE%A1%E7%AE%97%E6%9C%BA%E5%9B%BE%E5%BD%A2%E5%AD%A6/%E5%B0%8F%E9%A2%84%E5%91%8AAO.webp) diff --git a/content/mathematics/公式大全.md b/content/mathematics/公式大全.md index 18fbbfb..e5ce711 100644 --- a/content/mathematics/公式大全.md +++ b/content/mathematics/公式大全.md @@ -128,12 +128,16 @@ date: 2022-12-24T13:05:40+08:00 - [二次型定义](#二次型定义) - [二次型的矩阵表达式](#二次型的矩阵表达式) - [二次型的标准形与规范形](#二次型的标准形与规范形) + - [实二次型的标准形](#实二次型的标准形) + - [实二次型的规范形](#实二次型的规范形) - [惯性定理](#惯性定理) - [配方法化二次型为标准型](#配方法化二次型为标准型) - [正交变换法化实二次型为标准形](#正交变换法化实二次型为标准形) - [正定二次型及其判定](#正定二次型及其判定) - [正定二次型](#正定二次型) - [二次型正定的判定](#二次型正定的判定) + - [判断正定性的充分必要条件](#判断正定性的充分必要条件) + - [实对称矩阵A正定的必要条件](#实对称矩阵a正定的必要条件) @@ -1500,18 +1504,149 @@ $$ #### 二次型定义 +n个变量$x_1,x_2,\cdots,x_n$的一个二次齐次多项式 + +$$ +\begin{aligned} + f(x_1,x_2,\cdots,x_n) & = a_{11}x_1^2+2a_{12}x_1x_2+3a_{13}x_1x_3+\cdots+2a_{1n}x_1x_n \\\\ + & + a_{22}x_2^2 + 2a_{23}x_2x_3+\cdots+2a_{2n}x_2x_n \\\\ + & + a_{33}x_3^2 + \cdots + 2a_{3n}x_3x_n \\\\ + & + \cdots + a_{nn}x_n^2 +\end{aligned} +$$ + +称为一个关于$x_1,x_2,\cdots,x_n$的二次型 + #### 二次型的矩阵表达式 +设$X=(x_1,x_2,\cdots,x_n),A=\begin{bmatrix} + a_{11} & a_{12} & \cdots & a_{1n} \\\\ + a_{21} & a_{22} & \cdots & a_{2n} \\\\ + \vdots & \vdots & \ddots & \vdots \\\\ + a_{n1} & a_{n2} & \cdots & a_{nn} +\end{bmatrix}$ + +则 + +$f(x_1,x_2,\cdots,x_n)=X^TAX$ + +称A为二次型对应的矩阵 + #### 二次型的标准形与规范形 +##### 实二次型的标准形 + +若 +$$ +\begin{aligned} +f(X)&=X^TAX\overset{X=CY}{=}Y^T(C^TAC)Y \\\\ +& =d_1y_1^2+d_2y_2^2+\cdots+d_ny_n^2 \\\\ +& =Y^T \cdot diag(d_1,d_2,\cdots,d_n) \cdot Y +\end{aligned} +$$ + +则称平方和 +$$ +d_1y_1^2+d_2y_2^2+\cdots+d_ny_n^2 +$$ + +为二次型$f(X)=X^TAX$的一个标准形 + +任一个实二次型$f(X)=X^TAX$经过适当的可逆线性替换X=CY总可化成标准形(即平方和)即实对称矩阵总可与一个对角矩阵合同; + +$$ +C^TAC=B=diag(d_1,d_2,\cdots,d_n) +$$ + +##### 实二次型的规范形 + +形如 + + +$f(X)=X^TAX \overset{X=DZ}{=} z_1^2+z_2^2+\cdots+z_p^2-z_{p+1}^2-\cdots-z_{p+q}^2+0 \cdot z_{p+q+1}^2+\cdots+0 \cdot z_n^2$ + +的标准型称为$f(X)$的规范形,其中p称为正惯性指数,q称为负惯性指数 + + #### 惯性定理 +任意一个实系数的二次型$f(X)=X^TAX$总可经过一个适当的可逆线性替换化成规范形,其规范形是唯一的,与所选的坐标变换无关,即正平方向个数P,负平方项个数q由原二次型$f(X)=X^TAX$唯一确定 + +用矩阵的语言来讲:实对称矩阵总可与对角阵 + +$diag(1,\cdots,1,-1,\cdots,-1,0,\cdots,0)$ + +合同,且p+q=r(A),其中p,q为不变的量 + #### 配方法化二次型为标准型 +任一个实二次型总可用配方的方法通过一个适当的可逆线性替换化为标准形 + #### 正交变换法化实二次型为标准形 +设$f(X)=X^TAX$,因为A为实对称矩阵,由实对称矩阵必可通过正交变换化为对角阵A,其主对角线元素必为A的全部特征值,而正交变换$(Q^TAQ=Q^{-1}AQ=\Lambda)$中包含了合同变换,因此当作变换X=QY后,二次型$f(X)$即变为 + +$$ +\begin{aligned} +g(Y)&=Y^T(Q^TAQ)Y=Y^T\Lambda Y \\\\ +& =d_1y_1^2+d_2y_2^2+\cdots+d_ny_n^2 +\end{aligned} +$$ + +任一个实二次型$f(X)=X^TAX$总可通过变量间的正交替换X=QY化为标准形(平方和),其平方项前的系数必是A的全部特征值 + ### 正定二次型及其判定 #### 正定二次型 -#### 二次型正定的判定 \ No newline at end of file +设$f(X)=X^TAX$是一个实二次型,若对于任意$X_0\not ={0},f(X_0)=X_0^TAX_0>0$恒成立,则称$f(X)=X^TAX$为正定二次型,称对应矩阵A为正定矩阵 + +#### 二次型正定的判定 + +##### 判断正定性的充分必要条件 + +1. 实二次型$f(X)=X^TAX$正定$\iff$正惯性指数p=n +2. 实二次型$f(X)=X^TAX$正定$\iff$A与单位矩阵合同,即存在可逆矩阵P,使$P^TAP=E$成立 +3. 实二次型$f(X)=X^TAX$正定$\iff$存在可逆矩阵C,使$A=C^TC$ +4. 实二次型$f(X)=X^TAX$正定$\iff$A的特征值全大于0 +5. 实二次型$f(X)=X^TAX$正定$\iff$A的和各阶顺序主子式全部大于0,即 + +$$ +\begin{vmatrix} + a_{11} +\end{vmatrix}> 0 +$$ + +$$ +\begin{vmatrix} + a_{11} & a_{12} \\\\ + a_{21} & a_{22} +\end{vmatrix}> 0 +$$ + +$$ +\begin{vmatrix} + a_{11} & a_{12} & a_{13} \\\\ + a_{21} & a_{22} & a_{23} \\\\ + a_{31} & a_{32} & a_{33} +\end{vmatrix}> 0 +$$ + +$$ +\cdots +$$ + +$$ +\begin{vmatrix} + a_{11} & a_{12} & \cdots & a_{1n} \\\\ + a_{21} & a_{22} & \cdots & 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