From 86e4530fff8a185562156c5123434ac3460a7f9b Mon Sep 17 00:00:00 2001 From: GZ-Metal-Cell <1904817346@qq.com> Date: Sat, 28 Oct 2023 10:30:27 +0800 Subject: [PATCH] Site updated: 2023-10-28 10:30:23 --- .../1_1.png" | Bin 36066 -> 34676 bytes .../2_1.png" | Bin 10521 -> 9574 bytes .../3_1.png" | Bin 0 -> 118330 bytes .../3_2.png" | Bin 0 -> 51149 bytes .../4_1.png" | Bin 0 -> 8271 bytes .../4_2.png" | Bin 0 -> 84744 bytes .../4_3.png" | Bin 0 -> 17138 bytes .../4_4.png" | Bin 0 -> 38253 bytes .../4_5.png" | Bin 0 -> 24481 bytes .../4_6.png" | Bin 0 -> 21855 bytes .../4_7.png" | Bin 0 -> 64159 bytes .../4_8.png" | Bin 0 -> 19331 bytes .../HW0_1.png" | Bin 0 -> 25454 bytes .../HW0_2.png" | Bin 0 -> 179529 bytes .../index.html" | 340 +++++++++++++++++- search.xml | 2 +- 16 files changed, 338 insertions(+), 4 deletions(-) create mode 100644 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"b/2023/10/25/CG-GAMES101-\347\216\260\344\273\243\350\256\241\347\256\227\346\234\272\345\233\276\345\275\242\345\255\246\345\205\245\351\227\250-\351\227\253\344\273\244\347\220\252\357\274\2101\357\274\211/index.html" @@ -336,7 +336,10 @@

CG-GAMES101
  • 资源
  • Lecture 01 Overview of Computer Graphics
  • Lecture 02 Review of Linear Algebra
  • +
  • Lecture 03 Transformation
  • +
  • Lecture 04 Transformation Cont.
  • HW0
  • +
  • HW1
  • 资源

      @@ -345,6 +348,7 @@

      资源

      Lecture 01 Overview of Computer Graphics

      What is Computer Graphics? 图形学的应用:

        @@ -524,7 +528,7 @@

        Lecture 02 Review of Linear A

        构建右手坐标系:

        Matrix 矩阵

        Matrix-Matrix Multiplication 矩阵乘法

        -

        一般不满足交换律(一般不满足 $AB\= BA$)

        +

        一般不满足交换律(一般不满足 $AB\ne BA$)

        满足结合律和分配律:

        • $(AB)C=A(BC)$
        • @@ -539,8 +543,338 @@

          Lecture 02 Review of Linear A
          • Cross product 叉乘
          -

          HW0

          image-20231026155230885

          - +

          Lecture 03 Transformation

          2D Transform 二维变换

          +

          Scale Transform 缩放

          +

          等比缩放:

          +

          用矩阵乘法表示:

          +

          Scale (Non-Uniform) 任意缩放

          +

          Reflection Matrix 镜像

          +

          Horizontal reflection 水平镜像:

          +

          用矩阵乘法表示:

          +

          Shear Matrix 斜切:

          +

          png

          +

          Hints:

          +
            +
          • Horizontal shift is $0$ at $y=0$
          • +
          • Horizontal shift is $a$ at $y=1$
          • +
          • Vertical shift is always $0$
          • +
          +

          Rotation Matrix 旋转

          +

          旋转相反的角等于矩阵的逆,等于矩阵的转置:

          +

          旋转矩阵是正交矩阵

          +

          Linear Transforms = Matrices 线性变换 = 矩阵

          +

          简写为:

          +

          Translation 平移

          +

          Why Homogeneous Coordinates 为什么引入齐次坐标

          +
            +
          • Translation cannot be represented in matrix form 平移变换不能用矩阵来表示
          • +
          +

          因此,平移变换不是线性变换!

          +
            +
          • But we don’t want translation to be a special case 但我们不希望平移变换成为特例

            +
          • +
          • Is there a unified way to represent all transformations? (and what’s the cost?) 用统一的方式描述所有变换?(代价是什么?)

            +
          • +
          +

          Solution: Homogenous Coordinates 解决办法:齐次坐标

          +

          Add a third coordinate (w-coordinate) 增加一个维度

          +
            +
          • 2D point = $(x, y, {\color{Red}1})^T$
          • +
          • 2D vector = $(x, y, {\color{Red}0})^T$
          • +
          +

          Matrix representation of translations 用矩阵描述所有变换

          +

          Valid operation if w-coordinate of result is 1 or 0 所增加维度的值不是 1 就是 0,这是因为:

          +
            +
          • vector + vector = vector
          • +
          • point – point = vector
          • +
          • point + vector = point
          • +
          • point + point = 它们的中点
          • +
          +

          因此定义,在齐次坐标中:

          +

          Affine Transformation 仿射变换

          +

          Affine map = linear map + translation 仿射变换 = 线性变换 + 平移

          +

          Using homogenous coordinates: 使用齐次坐标

          +

          2D Transformations 2D 变换

          +

          Scale 缩放

          +

          Rotation 旋转

          +

          Translation 平移

          +

          Inverse Transform 逆变换

          +

          Composing Transforms 复合变换

          +

          Sequence of affine transforms $A_1, A_2, A_3, …$ 对于仿射变换序列 $A_1, A_2, A_3, …$

          +
            +
          • Compose by matrix multiplication 通过矩阵乘法进行合成
              +
            • Very important for performance! 对性能非常重要!
            • +
            +
          • +
          +

          预乘 $n$ 个矩阵以获得表示组合变换的单个矩阵。

          +

          Decomposing Complex Transforms 分解复合变换

          +

          png

          +

          How to rotate around a given point $c$? 如何绕空间任意一点 $c$ 旋转?

          +
            +
          1. Translate center to origin 平移回原点
          2. +
          3. Rotate 旋转
          4. +
          5. Translate back 平移回去
          6. +
          +

          3D Transforms

          +

          Use homogeneous coordinates again 再次使用齐次坐标:

          +
            +
          • 3D point = $(x, y, z, {\color{Red}1})^T$
          • +
          • 3D vector = $(x, y, z, {\color{Red}0})^T$
          • +
          +

          In general, $(x, y, z, w) (w \ne 0)$ is the 3D point:

          +

          3D Transformations 3D 变换

          +

          Use $4\times 4$ matrices for affine transformations 使用 $4\times 4$ 齐次坐标表示:

          +

          先应用线性变换,再平移。

          +

          Lecture 04 Transformation Cont.

          Viewing (观测) transformation

          +
            +
          • View (视图) / Camera transformation
          • +
          • Projection (投影) transformation
          • +
          • Orthographic (正交) projection
          • +
          • Perspective (透视) projection
          • +
          +

          3D Transformations 3D 变换

          +

          Rotation around $x$-, $y$-, or $z$-axis 沿 $x$、$y$、$z$ 轴旋转:

          +

          png

          +

          Compose any 3D rotation from $\mathbf{R}_x$, $\mathbf{R}_y$, $\mathbf{R}_z$? 让 $\mathbf{R}_x$, $\mathbf{R}_y$, $\mathbf{R}_z$ 构成任意三维旋转?

          +
            +
          • So-called Euler angles 所谓的欧拉角
          • +
          • Often used in flight simulators: roll, pitch, yaw 常用于飞行模拟器:滚转、俯仰、偏航
          • +
          +

          png

          +

          Rodrigues’ Rotation Formula 罗德里格斯轮换公式

          +

          绕轴 $n$ 旋转角度 $\alpha$:

          +

          View / Camera Transformation 视图变换

          +
            +
          • What is view transformation? 什么是视图变换?
          • +
          • Think about how to take a photo 考虑拍照
              +
            • Find a good place and arrange people (model transformation) 摆好场景:模型变换
            • +
            • Find a good “angle” to put the camera (view transformation) 选好视角:视图变换
            • +
            • Cheese! (projection transformation) 拍照:投影变换
            • +
            +
          • +
          • How to perform view transformation? 如何进行视图变换?
              +
            • Define the camera first 首先定义一个相机
                +
              • Position 位置 $\vec{e}$
              • +
              • Look-at / gaze direction 注视/凝视方向 $\hat g$
              • +
              • Up direction 上方向 $\hat t$
              • +
              +
            • +
            +
          • +
          +

          png

          +
            +
          • Key observation 重点观察

            +
              +
            • If the camera and all objects move together, the “photo” will be the same 如果相机和所有物体一起移动,“照片”将是相同的
            • +
            +
          • +
          • How about that we always transform the camera to 不如我们总是把相机变成

            +
              +
            • The origin, up at $Y$, look at $-Z$ 处于原点,上方为 $Y$ 轴,看着 $-Z$ 轴方向
            • +
            • And transform the objects along with the camera 并随相机变换对象
            • +
            +
          • +
          • Transform the camera by $M_{view}$

            +
              +
            • So it’s located at the origin, up at $Y$, look at $-Z$ 所以它位于原点,向上看 $Y$,看 $-Z$
            • +
            +
          • +
          • $M_{view}$ in math? $M_{view}=R_{view}T_{view}$

            +
              +
            • Translates $e$ to origin

              +

              $T_{view}=\begin{bmatrix}1&0&0&-x_e\\0&1&0&-y_e\\0&0&1&-z_e\\0&0&0&1\end{bmatrix}$

              +
            • +
            • Rotates $g$ to $-Z$、Rotates $t$ to $Y$、Rotates ($g \times t$) To $X$

              +
                +
              • Difficult to write! 将 $g$ 旋转到 $-Z$ 轴、将 $t$ 旋转到 $Y$ 轴、将 ($g \times t$) 旋转到 $X$ 轴,描述成矩阵是很复杂的!

                +
              • +
              • 考虑求其逆变换

                +

                $R_{view}^{-1}=\begin{bmatrix}x_{\hat{g}\times\hat{t}}&x_{t}&x_{-g}&0\\y_{\hat{g}\times\hat{t}}&y_{t}&y_{-g}&0\\z_{\hat{g}\times\hat{t}}&z_{t}&z_{-g}&0\\0&0&0&1\end{bmatrix}$

                +
              • +
              • 由于旋转矩阵是正交矩阵,其逆为它的转置,因此易求得 $R_{view}$

                +

                $R_{view}=\begin{bmatrix}x_{\hat{g}\times\hat{t}}&y_{\hat{g}\times\hat{t}}&z_{\hat{g}\times\hat{t}}&0\\x_t&y_t&z_t&0\\x_{-g}&y_{-g}&z_{-g}&0\\0&0&0&1\end{bmatrix}$

                +
              • +
              +
            • +
            +
          • +
          • Summary 总结

            +
              +
            • Transform objects together with the camera 与摄影机一起变换对象
            • +
            • Until camera’s at the origin, up at $Y$, look at $-Z$ 直到相机移动到原点,以 $Y$ 轴为上方,视点在 $-Z$ 轴方向。
            • +
            +
          • +
          • Also known as ModelView Transformation 也称为 ModelView 变换

            +
          • +
          +

          Projection Transformation 投影变换

          +
            +
          • Projection in Computer Graphics 投影在计算机图形学
              +
            • 3D to 2D
            • +
            • Orthographic projection 正交投影
            • +
            • Perspective projection 透视投影
            • +
            +
          • +
          +

          png

          +

          Orthographic Projection 正交投影

          +
            +
          • A simple way of understanding 简单的理解方法

            +
              +
            • Camera located at origin, looking at $-Z$, up at $Y$ (looks familiar?) 相机移动到原点,以 $Y$ 轴为上方,视点在 $-Z$ 轴方向
            • +
            • Drop $Z$ coordinate 删除 $Z$ 坐标
            • +
            • Translate and scale the resulting rectangle to $[-1, 1]^2$ 将生成的矩形平移并缩放到 $[-1,1]^2$
            • +
            +
          • +
          • In General

            +
              +
            • We want to map a cuboid $[l, r]\times [b, t]\times [f, n]$ to the “canonical” cube $[-1, 1]^3$

              +

              我们想把长方体 $[l,r]\times[b,t]\times[f,n]$ 映射到“规范(正则、规范、标准)” 的立方体 $[-1,1]^3$

              +
            • +
            +
          • +
          • Slightly different orders (to the “simple way”)

            +
              +
            • Center cuboid by translating 通过平移使长方体居中
            • +
            • Scale into “canonical” cube 缩放为“规范”多维数据集
            • +
            +
          • +
          +

          $r,l,t,b,n,f$ 的定义如下所示:

          +

          png

          +
            +
          • Transformation matrix?

            +
              +
            • Translate (center to origin) first, then scale (length/width/height to 2) 首先平移(中心到原点),然后缩放(长度/宽度/高度到 2)

              +

              $M_{ortho}=\begin{bmatrix}\frac2{r-l}&0&0&0\\0&\frac2{t-b}&0&0\\0&0&\frac2{n-f}&0\\0&0&0&1\end{bmatrix}\begin{bmatrix}1&0&0&-\frac{r+l}2\\0&1&0&-\frac{t+b}2\\0&0&1&-\frac{n+f}2\\0&0&0&1\end{bmatrix}$

              +
            • +
            +
          • +
          • Caveat

            +
              +
            • Looking at / along $-Z$ is making near and far not intuitive ($n > f$) 观察/沿着 $-Z$ 轴使远近不直观($n>f$)
            • +
            • FYI: that’s why OpenGL (a Graphics API) uses left hand coords. 仅供参考:这就是 OpenGL(图形 API)使用左手坐标的原因。
            • +
            +
          • +
          +

          Perspective Projection 透视投影

          +
            +
          • Most common in Computer Graphics, art, visual system 最常见于计算机图形学、艺术、视觉系统
          • +
          • Further objects are smaller 其他物体较小
          • +
          • Parallel lines not parallel; converge to single point 平行线不平行;收敛到单点
          • +
          +

          png

          +
            +
          • How to do perspective projection 如何进行透视投影

            +
              +
            • First “squish” the frustum into a cuboid ($n \to n$, $f \to f$) ($M_{persp\to ortho}$)

              +

              首先将截头体“压扁”成长方体($n\to n$,$f\to f$)($M_{persp\to ortho}$)

              +
            • +
            • Do orthographic projection ($M_{ortho}$, already known!) 做正交投影($M_{ortho}$,已经知道了!)

              +
            • +
            +
          • +
          +

          png

          +
            +
          • In order to find a transformation 为了找到变换(推导出矩阵 $M_{persp\to ortho}$)

            +
              +
            • Recall the key idea: Find the relationship between transformed points $(x’, y’,z’)$ and the original points $(x, y, z)$

              +

              回想关键思想:找到转换之间的关系点 $(x’, y’,z’)$ 和原始点 $(x, y, z)$

              +

              $y^{\prime}=\frac nzy$

              +

              $x^{\prime}=\frac nzx$

              +
            • +
            +
          • +
          +

          png

          +
            +
          • In homogeneous coordinates, 齐次坐标下,

            +

            $\begin{pmatrix}x\\y\\z\\1\end{pmatrix}\Rightarrow\begin{pmatrix}nx/z\\ny/z\\\text{unknown}\\1\end{pmatrix}==\begin{pmatrix}nx\\ny\\\text{still unknown}\\z\end{pmatrix}$

            +
          • +
          +
            +
          • So the “squish” (persp to ortho) projection does this 所以“挤压”(透视到正交)投影可以做到这一点

            +

            $M_{persp\to ortho}^{(4\times4)}\begin{pmatrix}x\\y\\z\\1\end{pmatrix}=\begin{pmatrix}nx\\ny\\\text{unknown}\\z\end{pmatrix}$

            +
          • +
          • Already good enough to figure out part of $M_{persp\to ortho}$ 可以求得 $M_{persp\to ortho}$ 的一部分:

            +

            $M_{persp\to ortho}=\begin{pmatrix}n&0&0&0\\0&n&0&0\\?&?&?&?\\0&0&1&0\end{pmatrix}$

            +
          • +
          • Observation: the third row is responsible for $z’$ 观察:第三排负责 $z’$

            +
              +
            • Any point on the near plane will not change 近平面上的任何点都不会改变

              +
            • +
            • Any point’s z on the far plane will not change 远平面上的任何点的 $z$ 都不会改变

              +
            • +
            +
          • +
          • Any point on the near plane will not change 近平面上的任何点都不会改变

            +

            $M_{persp\to ortho}^{(4\times4)}\begin{pmatrix}x\\y\\z\\1\end{pmatrix}=\begin{pmatrix}nx\\ny\\\text{unknown}\\z\end{pmatrix}\overset{\text{replace z with n}}{\operatorname*{\to}}\begin{pmatrix}x\\y\\n\\1\end{pmatrix}\Rightarrow\begin{pmatrix}x\\y\\n\\1\end{pmatrix}==\begin{pmatrix}nx\\ny\\n^2\\n\end{pmatrix}$

            +
          • +
          • So the third row must be of the form $(0\ 0\ A\ B)$ 所以第三行的形式必须是$(0\ 0\ A\ B)$

            +

            $\begin{pmatrix}0&0&A&B\end{pmatrix}\begin{pmatrix}x\\y\\n\\1\end{pmatrix}=n^2$

            +
          • +
          • What do we have now?

            +

            $\begin{pmatrix}0&0&A&B\end{pmatrix}\begin{pmatrix}x\\y\\n\\1\end{pmatrix}=n^2\to An+B=n^2$

            +
          • +
          • Any point’s z on the far plane will not change 近平面上的任何点都不会改变

            +

            $\begin{pmatrix}0\\0\\f\\1\end{pmatrix}\Rightarrow\begin{pmatrix}0\\0\\f\\1\end{pmatrix}==\begin{pmatrix}0\\0\\f^2\\f\end{pmatrix}\quad\to\quad Af+B=f^2$

            +
          • +
          • Solve for $A$ and $B$

            +

            $\begin{aligned}An+B&=n^2\\Af+B&=f^2\end{aligned}\quad\to\quad\begin{aligned}A&=n+f\\B&=-nf\end{aligned}$

            +
          • +
          • Finally, every entry in $M_{persp\to ortho}$ is known!

            +
          • +
          • What’s next?

            +
              +
            • 完成正交投影($M_{ortho}$)
            • +
            • $\begin{aligned}M_{persp}=M_{ortho}M_{persp\to ortho}\end{aligned}$
            • +
            +
          • +
          +

          HW0

          环境配置

          +

          WSL2 下:

          +
          sudo apt-get install xfce4-terminal
          sudo apt-get install xfce4
          +

          配置环境变量:

          +
          echo "export DISPLAY=localhost:0">> ~/.bashrc
          +

          MobaXterm 下配置 wsl2:

          +

          png

          +

          开跑!

          +

          png

          +

          安装库:

          +
          sudo apt update
          sudo apt install g++ gdb cmake
          sudo apt install libopencv-dev libeigen3-dev
          sudo apt install libglu1-mesa-dev freeglut3-dev mesa-common-dev xorg-dev
          +

          Windows 下的 VSCode,安装 wsl 插件,点击左下角,连接 wsl。

          +

          image-20231027164805639

          +

          打开作业模板,有如下命令:

          +
            +
          • mkdir build: 创建名为 build 的文件夹。
          • +
          • cd build: 移动到 build 文件夹下。
          • +
          • cmake ..: 注意其中 ’..’ 表示上一级目录,若为 ’.’ 则表示当前目录。
          • +
          • make: 编译程序,错误提示会显示在终端中。
          • +
          • ./Transformation:若上一步无错误,则可运行程序(这里 Transformation 为可执行文件名,可参照 CMakeLists.txt 中修改)。
          • +
          +

          写一个 compile.sh 便于编译:

          +
          rm -rf build
          mkdir build
          cd build
          cmake ..
          make
          ./Transformation
          cd ../
          +

          修改权限:

          +
          chmod a+x compile.sh
          +

          开跑!

          +
          sh compile.sh
          +

          作业

          +

          示例代码

          +
          #include<cmath>
          #include<eigen3/Eigen/Core>
          #include<eigen3/Eigen/Dense>
          #include<iostream>

          int main(){

          // Basic Example of cpp
          std::cout << "Example of cpp \n";
          float a = 1.0, b = 2.0;
          std::cout << a << std::endl;
          std::cout << a/b << std::endl;
          std::cout << std::sqrt(b) << std::endl;
          std::cout << std::acos(-1) << std::endl;
          std::cout << std::sin(30.0/180.0*acos(-1)) << std::endl;

          // Example of vector
          std::cout << "Example of vector \n";
          // vector definition
          Eigen::Vector3f v(1.0f,2.0f,3.0f);
          Eigen::Vector3f w(1.0f,0.0f,0.0f);
          // vector output
          std::cout << "Example of output \n";
          std::cout << v << std::endl;
          // vector add
          std::cout << "Example of add \n";
          std::cout << v + w << std::endl;
          // vector scalar multiply
          std::cout << "Example of scalar multiply \n";
          std::cout << v * 3.0f << std::endl;
          std::cout << 2.0f * v << std::endl;

          // Example of matrix
          std::cout << "Example of matrix \n";
          // matrix definition
          Eigen::Matrix3f i,j;
          i << 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0;
          j << 2.0, 3.0, 1.0, 4.0, 6.0, 5.0, 9.0, 7.0, 8.0;
          // matrix output
          std::cout << "Example of output \n";
          std::cout << i << std::endl;
          // matrix add i + j
          // matrix scalar multiply i * 2.0
          // matrix multiply i * j
          // matrix multiply vector i * v

          return 0;
          }
          +
          +

          题目

          +

          给定一个点 $P=(2,1)$, 将该点绕原点先逆时针旋转 $45^\circ$,再平移
          $(1,2)$,计算出变换后点的坐标(要求用齐次坐标进行计算)。

          +
          +

          $P$ 以齐次坐标的表示形式为 $(2, 1, 1)^T$

          +

          旋转矩阵为 $R(\pi/4)=\begin{bmatrix}
          \cos \pi/4 & -\sin \pi/4 & 0 \\
          \sin \pi/4 & \cos \pi/4 & 0 \\
          0 & 0 & 1
          \end{bmatrix}$

          +

          平移矩阵 $T(1, 2)=\begin{bmatrix}
          1 & 0 & 1 \\
          0 & 1 & 2 \\
          0 & 0 & 1
          \end{bmatrix}$

          +

          最终坐标为 $TRP$。

          +
          #include<cmath>
          #include<eigen3/Eigen/Core>
          #include<eigen3/Eigen/Dense>
          #include<iostream>
          #define PI 3.1415926535
          using namespace std;
          using namespace Eigen;

          int main()
          {
          float theta = PI / 4.0f;
          Vector3f P(2.0f, 1.0f, 1.0f);
          Matrix3f R, T;
          R <<
          cos(theta), -sin(theta), 0,
          sin(theta), cos(theta), 0,
          0, 0, 1;
          T <<
          1, 0, 1,
          0, 1, 2,
          0, 0, 1;
          cout << T * R * P << endl;
          return 0;
          }
          +
          1.70711
          4.12132
          1
          +

          HW1

          diff --git a/search.xml b/search.xml index f94e063a72..f42572ddbe 100644 --- a/search.xml +++ b/search.xml @@ -8,7 +8,7 @@ /2023/10/25/CG-GAMES101-%E7%8E%B0%E4%BB%A3%E8%AE%A1%E7%AE%97%E6%9C%BA%E5%9B%BE%E5%BD%A2%E5%AD%A6%E5%85%A5%E9%97%A8-%E9%97%AB%E4%BB%A4%E7%90%AA%EF%BC%881%EF%BC%89/ -

          CG-GAMES101-现代计算机图形学入门-闫令琪(1)

          目录

          资源

          Lecture 01 Overview of Computer Graphics

          What is Computer Graphics? 图形学的应用:

          • Viedo Games 游戏

          • Movies 电影特效

          • Animations 动画

          • Design 设计

          • Visualization 可视化

          • Visual Reality 虚拟现实

          • Augmented Reality 增强现实
          • Digital Illustration 数字化描述

          • Simulation 模拟

          • Graphical User Interfaces, GUI 图形用户接口

          • Typography 字体


          Why study Computer Graphics?

          • Computer Graphics is AWESOME!

          Course Topics(mainly 4 parts)

          • Rasterization 光栅化

            • Project geometry primitives (3D triangles / polygons) onto the screen 将几何图元(三维三角形/多边形)投影到屏幕上
            • Break projected primitives into fragments (pixels) 将投影基元分解为片段(像素)
            • Gold standard in Video Games (Real-time Applications) 视频游戏(实时应用)的黄金标准
          • Curves and Meshes 曲线和模型

            • How to represent geometry in Computer Graphics 如何在计算机图形学中表示几何图形
          • Ray Tracing 光线追踪

            • Shoot rays from the camera though each pixel 通过每个像素从相机拍摄光线
              • Calculate intersection and shading 计算交点和着色
              • Continue to bounce the rays till they hit light sources 继续反射光线,直到光线碰到光源
              • Gold standard in Animations / Movies (Offline Applications) 动画/电影黄金标准(离线应用程序)
          • Animation / Simulation 动画 / 模拟

            • Key frame Animation 关键帧动画
            • Mass-spring System 质量弹簧系统

          计算机图形学、计算机视觉、数字图像处理的区别:

          png

          Lecture 02 Review of Linear Algebra

          A Swift and Brutal Introduction to Linear Algebra! 一份对线性代数的迅速直接的介绍!

          Graphics’ Dependencies 图形相关的依赖项

          • Basic mathematics 基础数学

            • Linear algebra, calculus, statistics 线性代数、微积分、统计学
          • Basic physics 基础物理学

            • Optics, Mechanics 光学、力学
          • Misc 杂项

            • Signal processing 信号处理

            • Numerical analysis 数值分析

          • And a bit of aesthetics 还有一点美学

          This Course

          • More dependent on Linear Algebra 更多依赖于线性代数
            • Vectors (dot products, cross products, …) 向量(点积、叉积等)
            • Matrices (matrix-matrix, matrix-vector mult., …) 矩阵(矩阵相乘、矩阵与向量相乘等)
          • For example,
            • A point is a vector? 一个点是一个向量吗?
            • An operation like translating or rotating objects can be matrix-vector multiplication 类似平移或旋转物体的操作可以通过矩阵与向量相乘实现

          Vectors

          • 通常被写作 $\vec{a}$ 或 $\boldsymbol{a}$

          • 向量的长度被写作 $\left|\left|\vec{a}\right|\right|$

          • 向量的归一化,方向不变,长度设为 1:$\hat{a}=\vec{a}/|\vec{a}|$

          Cartesian Coordinates 笛卡尔坐标系

          Dot (scalar) Product 向量点乘

          对于 Unit Vectors 单位向量:

          性质:

          交换律:$\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$

          分配律:$\vec{a}\cdot(\vec{b}+\vec{c})=\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}$

          结合律:$(k\vec{a})\cdot\vec{b}=\vec{a}\cdot(k\vec{b})=k(\vec{a}\cdot\vec{b})$

          Dot Product in Cartesian Coordinates 笛卡尔坐标系下的向量点乘

          Component-wise multiplication, then adding up 分量逐个相乘,然后求和

          • In 2D

            • $\vec{a}\cdot\vec{b}=\begin{pmatrix}x_a\\y_a\end{pmatrix}\cdot\begin{pmatrix}x_b\\y_b\end{pmatrix}=x_ax_b+y_ay_b$
          • In 3D

            • $\vec a\cdot\vec b=\begin{pmatrix}x_a\\y_a\\z_a\end{pmatrix}\cdot\begin{pmatrix}x_b\\y_b\\z_b\end{pmatrix}=x_ax_b+y_ay_b+z_az_b$

          Dot Product in Graphics 向量点乘在图形学中的应用

          • Find angle between two vectors (e.g. cosine of angle between light source and surface) 寻找两个向量之间的夹角,例如光源和表面之间的夹角的余弦值。
          • Finding projection of one vector on another 找到一个向量在另一个向量上的投影。

            • $\vec{b}_{\perp}$:$\vec{b}$ 在 $\vec{a}$ 方向的投影。
              • $\vec{b}_{\perp}$ 必须与 $\vec{a}$ 或 $\hat{a}$ 平行。
                • $\vec{b}_{\perp}=k\hat{a}, k=|\vec{b}_\perp|=|\vec{b}|\cos\theta $
          • Measure how close two directions are 测量两个方向之间的接近程度

          • Decompose a vector 分解一个向量
          • Determine forward / backward 确定正向/反向

          Cross (vector) Product 矢量积

          png

          • Cross product is orthogonal to two initial vectors 叉乘结果与两个初始向量垂直
          • Direction determined by right-hand rule 方向由右手定则确定
          • Useful in constructing coordinate systems (later) 在构建坐标系时非常有用

          性质:

          对于右手坐标系:

          不满足交换律:$\vec{a}\times\vec{b}=-\vec{b}\times\vec{a}$

          跟自身叉乘为零向量:$\vec{a}\times\vec{a}=\vec{0}$

          满足分配律:

          $\vec{a}\times(\vec{b}+\vec{c})=\vec{a}\times\vec{b}+\vec{a}\times\vec{c}$

          $\vec{a}\times(k\vec{b})=k(\vec{a}\times\vec{b})$

          Cross Product: Cartesian Formula? 矢量积在笛卡尔坐标系中

          $A^*$ 是 $\vec{a}$ 的伴随矩阵(伴随矩阵 - 维基百科,自由的百科全书 (wikipedia.org)

          Cross Product in Graphics 矢量积在图形学中的应用

          • Determine left / right 判别左右
          • Determine inside / outside 判别内外

          Orthonormal Bases / Coordinate Frames 正交基 / 坐标系

          构建右手坐标系:

          Matrix 矩阵

          Matrix-Matrix Multiplication 矩阵乘法

          一般不满足交换律(一般不满足 $AB\= BA$)

          满足结合律和分配律:

          • $(AB)C=A(BC)$
          • $A(B+C)=AB+AC$
          • $(A+B)C=AC+BC$

          Identity Matrix 单位矩阵

          Vector multiplication in Matrix form 向量与矩阵相乘

          • Dot product 点乘
          • Cross product 叉乘

          HW0

          image-20231026155230885

          ]]>
          +

          CG-GAMES101-现代计算机图形学入门-闫令琪(1)

          目录

          资源

          Lecture 01 Overview of Computer Graphics

          What is Computer Graphics? 图形学的应用:

          • Viedo Games 游戏

          • Movies 电影特效

          • Animations 动画

          • Design 设计

          • Visualization 可视化

          • Visual Reality 虚拟现实

          • Augmented Reality 增强现实
          • Digital Illustration 数字化描述

          • Simulation 模拟

          • Graphical User Interfaces, GUI 图形用户接口

          • Typography 字体


          Why study Computer Graphics?

          • Computer Graphics is AWESOME!

          Course Topics(mainly 4 parts)

          • Rasterization 光栅化

            • Project geometry primitives (3D triangles / polygons) onto the screen 将几何图元(三维三角形/多边形)投影到屏幕上
            • Break projected primitives into fragments (pixels) 将投影基元分解为片段(像素)
            • Gold standard in Video Games (Real-time Applications) 视频游戏(实时应用)的黄金标准
          • Curves and Meshes 曲线和模型

            • How to represent geometry in Computer Graphics 如何在计算机图形学中表示几何图形
          • Ray Tracing 光线追踪

            • Shoot rays from the camera though each pixel 通过每个像素从相机拍摄光线
              • Calculate intersection and shading 计算交点和着色
              • Continue to bounce the rays till they hit light sources 继续反射光线,直到光线碰到光源
              • Gold standard in Animations / Movies (Offline Applications) 动画/电影黄金标准(离线应用程序)
          • Animation / Simulation 动画 / 模拟

            • Key frame Animation 关键帧动画
            • Mass-spring System 质量弹簧系统

          计算机图形学、计算机视觉、数字图像处理的区别:

          png

          Lecture 02 Review of Linear Algebra

          A Swift and Brutal Introduction to Linear Algebra! 一份对线性代数的迅速直接的介绍!

          Graphics’ Dependencies 图形相关的依赖项

          • Basic mathematics 基础数学

            • Linear algebra, calculus, statistics 线性代数、微积分、统计学
          • Basic physics 基础物理学

            • Optics, Mechanics 光学、力学
          • Misc 杂项

            • Signal processing 信号处理

            • Numerical analysis 数值分析

          • And a bit of aesthetics 还有一点美学

          This Course

          • More dependent on Linear Algebra 更多依赖于线性代数
            • Vectors (dot products, cross products, …) 向量(点积、叉积等)
            • Matrices (matrix-matrix, matrix-vector mult., …) 矩阵(矩阵相乘、矩阵与向量相乘等)
          • For example,
            • A point is a vector? 一个点是一个向量吗?
            • An operation like translating or rotating objects can be matrix-vector multiplication 类似平移或旋转物体的操作可以通过矩阵与向量相乘实现

          Vectors

          • 通常被写作 $\vec{a}$ 或 $\boldsymbol{a}$

          • 向量的长度被写作 $\left|\left|\vec{a}\right|\right|$

          • 向量的归一化,方向不变,长度设为 1:$\hat{a}=\vec{a}/|\vec{a}|$

          Cartesian Coordinates 笛卡尔坐标系

          Dot (scalar) Product 向量点乘

          对于 Unit Vectors 单位向量:

          性质:

          交换律:$\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$

          分配律:$\vec{a}\cdot(\vec{b}+\vec{c})=\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}$

          结合律:$(k\vec{a})\cdot\vec{b}=\vec{a}\cdot(k\vec{b})=k(\vec{a}\cdot\vec{b})$

          Dot Product in Cartesian Coordinates 笛卡尔坐标系下的向量点乘

          Component-wise multiplication, then adding up 分量逐个相乘,然后求和

          • In 2D

            • $\vec{a}\cdot\vec{b}=\begin{pmatrix}x_a\\y_a\end{pmatrix}\cdot\begin{pmatrix}x_b\\y_b\end{pmatrix}=x_ax_b+y_ay_b$
          • In 3D

            • $\vec a\cdot\vec b=\begin{pmatrix}x_a\\y_a\\z_a\end{pmatrix}\cdot\begin{pmatrix}x_b\\y_b\\z_b\end{pmatrix}=x_ax_b+y_ay_b+z_az_b$

          Dot Product in Graphics 向量点乘在图形学中的应用

          • Find angle between two vectors (e.g. cosine of angle between light source and surface) 寻找两个向量之间的夹角,例如光源和表面之间的夹角的余弦值。
          • Finding projection of one vector on another 找到一个向量在另一个向量上的投影。

            • $\vec{b}_{\perp}$:$\vec{b}$ 在 $\vec{a}$ 方向的投影。
              • $\vec{b}_{\perp}$ 必须与 $\vec{a}$ 或 $\hat{a}$ 平行。
                • $\vec{b}_{\perp}=k\hat{a}, k=|\vec{b}_\perp|=|\vec{b}|\cos\theta $
          • Measure how close two directions are 测量两个方向之间的接近程度

          • Decompose a vector 分解一个向量
          • Determine forward / backward 确定正向/反向

          Cross (vector) Product 矢量积

          png

          • Cross product is orthogonal to two initial vectors 叉乘结果与两个初始向量垂直
          • Direction determined by right-hand rule 方向由右手定则确定
          • Useful in constructing coordinate systems (later) 在构建坐标系时非常有用

          性质:

          对于右手坐标系:

          不满足交换律:$\vec{a}\times\vec{b}=-\vec{b}\times\vec{a}$

          跟自身叉乘为零向量:$\vec{a}\times\vec{a}=\vec{0}$

          满足分配律:

          $\vec{a}\times(\vec{b}+\vec{c})=\vec{a}\times\vec{b}+\vec{a}\times\vec{c}$

          $\vec{a}\times(k\vec{b})=k(\vec{a}\times\vec{b})$

          Cross Product: Cartesian Formula? 矢量积在笛卡尔坐标系中

          $A^*$ 是 $\vec{a}$ 的伴随矩阵(伴随矩阵 - 维基百科,自由的百科全书 (wikipedia.org)

          Cross Product in Graphics 矢量积在图形学中的应用

          • Determine left / right 判别左右
          • Determine inside / outside 判别内外

          Orthonormal Bases / Coordinate Frames 正交基 / 坐标系

          构建右手坐标系:

          Matrix 矩阵

          Matrix-Matrix Multiplication 矩阵乘法

          一般不满足交换律(一般不满足 $AB\ne BA$)

          满足结合律和分配律:

          • $(AB)C=A(BC)$
          • $A(B+C)=AB+AC$
          • $(A+B)C=AC+BC$

          Identity Matrix 单位矩阵

          Vector multiplication in Matrix form 向量与矩阵相乘

          • Dot product 点乘
          • Cross product 叉乘

          Lecture 03 Transformation

          2D Transform 二维变换

          Scale Transform 缩放

          等比缩放:

          用矩阵乘法表示:

          Scale (Non-Uniform) 任意缩放

          Reflection Matrix 镜像

          Horizontal reflection 水平镜像:

          用矩阵乘法表示:

          Shear Matrix 斜切:

          png

          Hints:

          • Horizontal shift is $0$ at $y=0$
          • Horizontal shift is $a$ at $y=1$
          • Vertical shift is always $0$

          Rotation Matrix 旋转

          旋转相反的角等于矩阵的逆,等于矩阵的转置:

          旋转矩阵是正交矩阵

          Linear Transforms = Matrices 线性变换 = 矩阵

          简写为:

          Translation 平移

          Why Homogeneous Coordinates 为什么引入齐次坐标

          • Translation cannot be represented in matrix form 平移变换不能用矩阵来表示

          因此,平移变换不是线性变换!

          • But we don’t want translation to be a special case 但我们不希望平移变换成为特例

          • Is there a unified way to represent all transformations? (and what’s the cost?) 用统一的方式描述所有变换?(代价是什么?)

          Solution: Homogenous Coordinates 解决办法:齐次坐标

          Add a third coordinate (w-coordinate) 增加一个维度

          • 2D point = $(x, y, {\color{Red}1})^T$
          • 2D vector = $(x, y, {\color{Red}0})^T$

          Matrix representation of translations 用矩阵描述所有变换

          Valid operation if w-coordinate of result is 1 or 0 所增加维度的值不是 1 就是 0,这是因为:

          • vector + vector = vector
          • point – point = vector
          • point + vector = point
          • point + point = 它们的中点

          因此定义,在齐次坐标中:

          Affine Transformation 仿射变换

          Affine map = linear map + translation 仿射变换 = 线性变换 + 平移

          Using homogenous coordinates: 使用齐次坐标

          2D Transformations 2D 变换

          Scale 缩放

          Rotation 旋转

          Translation 平移

          Inverse Transform 逆变换

          Composing Transforms 复合变换

          Sequence of affine transforms $A_1, A_2, A_3, …$ 对于仿射变换序列 $A_1, A_2, A_3, …$

          • Compose by matrix multiplication 通过矩阵乘法进行合成
            • Very important for performance! 对性能非常重要!

          预乘 $n$ 个矩阵以获得表示组合变换的单个矩阵。

          Decomposing Complex Transforms 分解复合变换

          png

          How to rotate around a given point $c$? 如何绕空间任意一点 $c$ 旋转?

          1. Translate center to origin 平移回原点
          2. Rotate 旋转
          3. Translate back 平移回去

          3D Transforms

          Use homogeneous coordinates again 再次使用齐次坐标:

          • 3D point = $(x, y, z, {\color{Red}1})^T$
          • 3D vector = $(x, y, z, {\color{Red}0})^T$

          In general, $(x, y, z, w) (w \ne 0)$ is the 3D point:

          3D Transformations 3D 变换

          Use $4\times 4$ matrices for affine transformations 使用 $4\times 4$ 齐次坐标表示:

          先应用线性变换,再平移。

          Lecture 04 Transformation Cont.

          Viewing (观测) transformation

          • View (视图) / Camera transformation
          • Projection (投影) transformation
          • Orthographic (正交) projection
          • Perspective (透视) projection

          3D Transformations 3D 变换

          Rotation around $x$-, $y$-, or $z$-axis 沿 $x$、$y$、$z$ 轴旋转:

          png

          Compose any 3D rotation from $\mathbf{R}_x$, $\mathbf{R}_y$, $\mathbf{R}_z$? 让 $\mathbf{R}_x$, $\mathbf{R}_y$, $\mathbf{R}_z$ 构成任意三维旋转?

          • So-called Euler angles 所谓的欧拉角
          • Often used in flight simulators: roll, pitch, yaw 常用于飞行模拟器:滚转、俯仰、偏航

          png

          Rodrigues’ Rotation Formula 罗德里格斯轮换公式

          绕轴 $n$ 旋转角度 $\alpha$:

          View / Camera Transformation 视图变换

          • What is view transformation? 什么是视图变换?
          • Think about how to take a photo 考虑拍照
            • Find a good place and arrange people (model transformation) 摆好场景:模型变换
            • Find a good “angle” to put the camera (view transformation) 选好视角:视图变换
            • Cheese! (projection transformation) 拍照:投影变换
          • How to perform view transformation? 如何进行视图变换?
            • Define the camera first 首先定义一个相机
              • Position 位置 $\vec{e}$
              • Look-at / gaze direction 注视/凝视方向 $\hat g$
              • Up direction 上方向 $\hat t$

          png

          • Key observation 重点观察

            • If the camera and all objects move together, the “photo” will be the same 如果相机和所有物体一起移动,“照片”将是相同的
          • How about that we always transform the camera to 不如我们总是把相机变成

            • The origin, up at $Y$, look at $-Z$ 处于原点,上方为 $Y$ 轴,看着 $-Z$ 轴方向
            • And transform the objects along with the camera 并随相机变换对象
          • Transform the camera by $M_{view}$

            • So it’s located at the origin, up at $Y$, look at $-Z$ 所以它位于原点,向上看 $Y$,看 $-Z$
          • $M_{view}$ in math? $M_{view}=R_{view}T_{view}$

            • Translates $e$ to origin

              $T_{view}=\begin{bmatrix}1&0&0&-x_e\\0&1&0&-y_e\\0&0&1&-z_e\\0&0&0&1\end{bmatrix}$

            • Rotates $g$ to $-Z$、Rotates $t$ to $Y$、Rotates ($g \times t$) To $X$

              • Difficult to write! 将 $g$ 旋转到 $-Z$ 轴、将 $t$ 旋转到 $Y$ 轴、将 ($g \times t$) 旋转到 $X$ 轴,描述成矩阵是很复杂的!

              • 考虑求其逆变换

                $R_{view}^{-1}=\begin{bmatrix}x_{\hat{g}\times\hat{t}}&x_{t}&x_{-g}&0\\y_{\hat{g}\times\hat{t}}&y_{t}&y_{-g}&0\\z_{\hat{g}\times\hat{t}}&z_{t}&z_{-g}&0\\0&0&0&1\end{bmatrix}$

              • 由于旋转矩阵是正交矩阵,其逆为它的转置,因此易求得 $R_{view}$

                $R_{view}=\begin{bmatrix}x_{\hat{g}\times\hat{t}}&y_{\hat{g}\times\hat{t}}&z_{\hat{g}\times\hat{t}}&0\\x_t&y_t&z_t&0\\x_{-g}&y_{-g}&z_{-g}&0\\0&0&0&1\end{bmatrix}$

          • Summary 总结

            • Transform objects together with the camera 与摄影机一起变换对象
            • Until camera’s at the origin, up at $Y$, look at $-Z$ 直到相机移动到原点,以 $Y$ 轴为上方,视点在 $-Z$ 轴方向。
          • Also known as ModelView Transformation 也称为 ModelView 变换

          Projection Transformation 投影变换

          • Projection in Computer Graphics 投影在计算机图形学
            • 3D to 2D
            • Orthographic projection 正交投影
            • Perspective projection 透视投影

          png

          Orthographic Projection 正交投影

          • A simple way of understanding 简单的理解方法

            • Camera located at origin, looking at $-Z$, up at $Y$ (looks familiar?) 相机移动到原点,以 $Y$ 轴为上方,视点在 $-Z$ 轴方向
            • Drop $Z$ coordinate 删除 $Z$ 坐标
            • Translate and scale the resulting rectangle to $[-1, 1]^2$ 将生成的矩形平移并缩放到 $[-1,1]^2$
          • In General

            • We want to map a cuboid $[l, r]\times [b, t]\times [f, n]$ to the “canonical” cube $[-1, 1]^3$

              我们想把长方体 $[l,r]\times[b,t]\times[f,n]$ 映射到“规范(正则、规范、标准)” 的立方体 $[-1,1]^3$

          • Slightly different orders (to the “simple way”)

            • Center cuboid by translating 通过平移使长方体居中
            • Scale into “canonical” cube 缩放为“规范”多维数据集

          $r,l,t,b,n,f$ 的定义如下所示:

          png

          • Transformation matrix?

            • Translate (center to origin) first, then scale (length/width/height to 2) 首先平移(中心到原点),然后缩放(长度/宽度/高度到 2)

              $M_{ortho}=\begin{bmatrix}\frac2{r-l}&0&0&0\\0&\frac2{t-b}&0&0\\0&0&\frac2{n-f}&0\\0&0&0&1\end{bmatrix}\begin{bmatrix}1&0&0&-\frac{r+l}2\\0&1&0&-\frac{t+b}2\\0&0&1&-\frac{n+f}2\\0&0&0&1\end{bmatrix}$

          • Caveat

            • Looking at / along $-Z$ is making near and far not intuitive ($n > f$) 观察/沿着 $-Z$ 轴使远近不直观($n>f$)
            • FYI: that’s why OpenGL (a Graphics API) uses left hand coords. 仅供参考:这就是 OpenGL(图形 API)使用左手坐标的原因。

          Perspective Projection 透视投影

          • Most common in Computer Graphics, art, visual system 最常见于计算机图形学、艺术、视觉系统
          • Further objects are smaller 其他物体较小
          • Parallel lines not parallel; converge to single point 平行线不平行;收敛到单点

          png

          • How to do perspective projection 如何进行透视投影

            • First “squish” the frustum into a cuboid ($n \to n$, $f \to f$) ($M_{persp\to ortho}$)

              首先将截头体“压扁”成长方体($n\to n$,$f\to f$)($M_{persp\to ortho}$)

            • Do orthographic projection ($M_{ortho}$, already known!) 做正交投影($M_{ortho}$,已经知道了!)

          png

          • In order to find a transformation 为了找到变换(推导出矩阵 $M_{persp\to ortho}$)

            • Recall the key idea: Find the relationship between transformed points $(x’, y’,z’)$ and the original points $(x, y, z)$

              回想关键思想:找到转换之间的关系点 $(x’, y’,z’)$ 和原始点 $(x, y, z)$

              $y^{\prime}=\frac nzy$

              $x^{\prime}=\frac nzx$

          png

          • In homogeneous coordinates, 齐次坐标下,

            $\begin{pmatrix}x\\y\\z\\1\end{pmatrix}\Rightarrow\begin{pmatrix}nx/z\\ny/z\\\text{unknown}\\1\end{pmatrix}==\begin{pmatrix}nx\\ny\\\text{still unknown}\\z\end{pmatrix}$

          • So the “squish” (persp to ortho) projection does this 所以“挤压”(透视到正交)投影可以做到这一点

            $M_{persp\to ortho}^{(4\times4)}\begin{pmatrix}x\\y\\z\\1\end{pmatrix}=\begin{pmatrix}nx\\ny\\\text{unknown}\\z\end{pmatrix}$

          • Already good enough to figure out part of $M_{persp\to ortho}$ 可以求得 $M_{persp\to ortho}$ 的一部分:

            $M_{persp\to ortho}=\begin{pmatrix}n&0&0&0\\0&n&0&0\\?&?&?&?\\0&0&1&0\end{pmatrix}$

          • Observation: the third row is responsible for $z’$ 观察:第三排负责 $z’$

            • Any point on the near plane will not change 近平面上的任何点都不会改变

            • Any point’s z on the far plane will not change 远平面上的任何点的 $z$ 都不会改变

          • Any point on the near plane will not change 近平面上的任何点都不会改变

            $M_{persp\to ortho}^{(4\times4)}\begin{pmatrix}x\\y\\z\\1\end{pmatrix}=\begin{pmatrix}nx\\ny\\\text{unknown}\\z\end{pmatrix}\overset{\text{replace z with n}}{\operatorname*{\to}}\begin{pmatrix}x\\y\\n\\1\end{pmatrix}\Rightarrow\begin{pmatrix}x\\y\\n\\1\end{pmatrix}==\begin{pmatrix}nx\\ny\\n^2\\n\end{pmatrix}$

          • So the third row must be of the form $(0\ 0\ A\ B)$ 所以第三行的形式必须是$(0\ 0\ A\ B)$

            $\begin{pmatrix}0&0&A&B\end{pmatrix}\begin{pmatrix}x\\y\\n\\1\end{pmatrix}=n^2$

          • What do we have now?

            $\begin{pmatrix}0&0&A&B\end{pmatrix}\begin{pmatrix}x\\y\\n\\1\end{pmatrix}=n^2\to An+B=n^2$

          • Any point’s z on the far plane will not change 近平面上的任何点都不会改变

            $\begin{pmatrix}0\\0\\f\\1\end{pmatrix}\Rightarrow\begin{pmatrix}0\\0\\f\\1\end{pmatrix}==\begin{pmatrix}0\\0\\f^2\\f\end{pmatrix}\quad\to\quad Af+B=f^2$

          • Solve for $A$ and $B$

            $\begin{aligned}An+B&=n^2\\Af+B&=f^2\end{aligned}\quad\to\quad\begin{aligned}A&=n+f\\B&=-nf\end{aligned}$

          • Finally, every entry in $M_{persp\to ortho}$ is known!

          • What’s next?

            • 完成正交投影($M_{ortho}$)
            • $\begin{aligned}M_{persp}=M_{ortho}M_{persp\to ortho}\end{aligned}$

          HW0

          环境配置

          WSL2 下:

          sudo apt-get install xfce4-terminal
          sudo apt-get install xfce4

          配置环境变量:

          echo "export DISPLAY=localhost:0">> ~/.bashrc

          MobaXterm 下配置 wsl2:

          png

          开跑!

          png

          安装库:

          sudo apt update
          sudo apt install g++ gdb cmake
          sudo apt install libopencv-dev libeigen3-dev
          sudo apt install libglu1-mesa-dev freeglut3-dev mesa-common-dev xorg-dev

          Windows 下的 VSCode,安装 wsl 插件,点击左下角,连接 wsl。

          image-20231027164805639

          打开作业模板,有如下命令:

          • mkdir build: 创建名为 build 的文件夹。
          • cd build: 移动到 build 文件夹下。
          • cmake ..: 注意其中 ’..’ 表示上一级目录,若为 ’.’ 则表示当前目录。
          • make: 编译程序,错误提示会显示在终端中。
          • ./Transformation:若上一步无错误,则可运行程序(这里 Transformation 为可执行文件名,可参照 CMakeLists.txt 中修改)。

          写一个 compile.sh 便于编译:

          rm -rf build
          mkdir build
          cd build
          cmake ..
          make
          ./Transformation
          cd ../

          修改权限:

          chmod a+x compile.sh

          开跑!

          sh compile.sh

          作业

          示例代码

          #include<cmath>
          #include<eigen3/Eigen/Core>
          #include<eigen3/Eigen/Dense>
          #include<iostream>

          int main(){

          // Basic Example of cpp
          std::cout << "Example of cpp \n";
          float a = 1.0, b = 2.0;
          std::cout << a << std::endl;
          std::cout << a/b << std::endl;
          std::cout << std::sqrt(b) << std::endl;
          std::cout << std::acos(-1) << std::endl;
          std::cout << std::sin(30.0/180.0*acos(-1)) << std::endl;

          // Example of vector
          std::cout << "Example of vector \n";
          // vector definition
          Eigen::Vector3f v(1.0f,2.0f,3.0f);
          Eigen::Vector3f w(1.0f,0.0f,0.0f);
          // vector output
          std::cout << "Example of output \n";
          std::cout << v << std::endl;
          // vector add
          std::cout << "Example of add \n";
          std::cout << v + w << std::endl;
          // vector scalar multiply
          std::cout << "Example of scalar multiply \n";
          std::cout << v * 3.0f << std::endl;
          std::cout << 2.0f * v << std::endl;

          // Example of matrix
          std::cout << "Example of matrix \n";
          // matrix definition
          Eigen::Matrix3f i,j;
          i << 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0;
          j << 2.0, 3.0, 1.0, 4.0, 6.0, 5.0, 9.0, 7.0, 8.0;
          // matrix output
          std::cout << "Example of output \n";
          std::cout << i << std::endl;
          // matrix add i + j
          // matrix scalar multiply i * 2.0
          // matrix multiply i * j
          // matrix multiply vector i * v

          return 0;
          }

          题目

          给定一个点 $P=(2,1)$, 将该点绕原点先逆时针旋转 $45^\circ$,再平移
          $(1,2)$,计算出变换后点的坐标(要求用齐次坐标进行计算)。


          $P$ 以齐次坐标的表示形式为 $(2, 1, 1)^T$

          旋转矩阵为 $R(\pi/4)=\begin{bmatrix}
          \cos \pi/4 & -\sin \pi/4 & 0 \\
          \sin \pi/4 & \cos \pi/4 & 0 \\
          0 & 0 & 1
          \end{bmatrix}$

          平移矩阵 $T(1, 2)=\begin{bmatrix}
          1 & 0 & 1 \\
          0 & 1 & 2 \\
          0 & 0 & 1
          \end{bmatrix}$

          最终坐标为 $TRP$。

          #include<cmath>
          #include<eigen3/Eigen/Core>
          #include<eigen3/Eigen/Dense>
          #include<iostream>
          #define PI 3.1415926535
          using namespace std;
          using namespace Eigen;

          int main()
          {
          float theta = PI / 4.0f;
          Vector3f P(2.0f, 1.0f, 1.0f);
          Matrix3f R, T;
          R <<
          cos(theta), -sin(theta), 0,
          sin(theta), cos(theta), 0,
          0, 0, 1;
          T <<
          1, 0, 1,
          0, 1, 2,
          0, 0, 1;
          cout << T * R * P << endl;
          return 0;
          }
          1.70711
          4.12132
          1

          HW1

          ]]>