[Study Notes] Computer Graphics (1)

Computer Graphics Study Notes (1)

What is Computer Graphics

The main research content of computer graphics is to study how to represent graphics in computers, as well as the related principles and algorithms of computing, processing and displaying graphics using computers.

What is a good picture?

A simple criterion for judging from a technical level: directly look at whether the picture is "bright" enough. It reflects whether the global illumination in graphics is done well. If the lighting is done well, the picture will be bright, otherwise it will be dark.

Linear Algebra

vector

The two basic contents of a vector, direction and length.

1. Unit vector

A vector of length one is called a unit vector.

$\hat{a}$ is the unit vector
$$\hat{a}=\frac{\vec{a}}{|\vec{a}|}$$

2. Sum of vectors

$\vec{a}$为$(x_a,y_a)$

$\vec{b}$is $(x_b,y_b)$

$$\vec{a}+\vec{b}=(x_a+x_b,y_a+y_b)$$

3. Vector dot multiplication

$\vec{a}$为$(x_a,y_a,z_a)$
$\vec{b}$为$(x_b,y_b,z_b)$

$\vec{a}$and $\vec{b}$The dot product is expressed as:
$$\vec{a}\cdot \vec{b}=cos\theta |\vec{a}||\vec{b}|if$$ a ⃗ \vec{a
}$\vec{a}$and $\vec{b}$Both are unit vectors, namely $\hat{a}$ $\hat{b}$
$$cos\theta =\hat{a}\cdot \hat{b}$$
$\vec{a}$and $\vec{b}$点乘还可以表示为：
$$\vec{a}\cdot \vec{b}={\vec{a}}^T\vec{b}=\left(\begin{array}{ccc}{x}_a& y_a& z_a\end{array}\right)\left(\begin{array}{c}{x}_b\\ y_b\\ z_b\end{array}\right)=x_a{x}_b+y_a{y}_b+z_a{z}_b$$
The role of dot multiplication

1. find angle

   The two vectors are normalized to a unit vector, and the dot product of the unit vector is the cosine of the angle.

   Right now:

$$cos\theta =\hat{a}\cdot \hat{b}$$
​ Add the cosine law and derivation process:

​ 余弦定理公式：
$$cos\theta =\frac{|\overrightarrow{AC}{|}^2+|\overrightarrow{AB}{|}^2-|\overrightarrow{BC}{|}^2}{2|\overrightarrow{AC}||\overrightarrow{AB}|}$$
​ Cosine law derivation process:

$$\begin{gathered} \overrightarrow{BC}=\overrightarrow{AC}-\overrightarrow{AB} \\ {\overrightarrow{BC}}^2=(\overrightarrow{AC}-\overrightarrow{AB}{)}^2 \\ {\overrightarrow{BC}}^T\overrightarrow{BC}=(\overrightarrow{AC}-\overrightarrow{AB}{)}^T(\overrightarrow{AC}-\overrightarrow{AB}) \\ |\overrightarrow{BC}{|}^2={\overrightarrow{AC}}^T\overrightarrow{AC}-2\overrightarrow{AC}\cdot \overrightarrow{AB}+{\overrightarrow{AB}}^T\overrightarrow{AB} \\ |\overrightarrow{BC}{|}^2=|\overrightarrow{AC}{|}^2-2\overrightarrow{AC}\cdot \overrightarrow{AB}+|\overrightarrow{AB}{|}^2 \\ |\overrightarrow{BC}{|}^2=|\overrightarrow{AC}{|}^2-2cos\theta |\overrightarrow{AC}||\overrightarrow{AB}|+|\overrightarrow{AB}{|}^2 \\ cos\theta =\frac{|\overrightarrow{AC}{|}^2+|\overrightarrow{AB}{|}^2-|\overrightarrow{BC}{|}^2}{2|\overrightarrow{AC}||\overrightarrow{AB}|} \end{gathered}$$
​ The law of cosines can also be derived by dot product of vectors

2. Find the projection

   

   求$\overrightarrow{AB}$to $\overrightarrow{AC}$The projection $\overrightarrow{AD}$

   $\widehat{AC}$ is$\overrightarrow{AB}$the unit vector of

   $|\overrightarrow{AD}|=|\overrightarrow{AB}|cos\theta$

   $\overrightarrow{AD}=\widehat{AC}|\overrightarrow{AD}|$

3. approach and front

   The distribution of cos values ​​from 0° to 180°

   [External link picture transfer failed, the source site may have an anti-theft link mechanism, it is recommended to save the picture and upload it directly (img-egZWC9gH-1681284850478)(https://songimghost.oss-cn-nanjing.aliyuncs.com/img/desmos -graph1.png)]

   ⃗ $\overrightarrow{AB}$The closer to $\overrightarrow{AC}$, when θ is about small, $The\; closer\; the\; cos\theta$ is to 1

   ⃗ $\overrightarrow{AB}$和$\overrightarrow{AC}$When the direction is opposite, θ is greater than 90°, $cos\theta$ is a negative value, and the closer it is to -1

   

Four, vector cross multiplication

In a right-handed coordinate system: $\overrightarrow{AD}=\overrightarrow{AB}\times \overrightarrow{AC}$

$\overrightarrow{AD}$Perpendicular to $\overrightarrow{AC}$
$\overrightarrow{AD}$perpendicular to $\overrightarrow{AB}$

$|\overrightarrow{AD}|=sin\theta |\overrightarrow{AB}||\overrightarrow{AC}|$

official:

$\vec{a}=(x_a,y_a,z_a)$

$\vec{b}=(x_b,y_b,z_b)$

$\vec{a}\times \vec{b}=\left(\begin{array}{c}0,-z_a,y_a\\ z_a,0,-x_a\\ -y_a,x_a,0\end{array}\right)\left(\begin{array}{c}{x}_b\\ y_b\\ z_b\end{array}\right)=\left(\begin{array}{c}{y}_a{z}_b-y_b{z}_a\\ z_a{x}_b-x_a{z}_b\\ x_a{y}_b-y_a{x}_b\end{array}\right)$

The role of cross product

1. Judging left and right

   

   In a left-handed coordinate system, $\vec{b}\times \vec{a}$And the z axis is a positive value, indicating that $\vec{b}$在$\vec{a}$clockwise direction.

2. Judging inside and outside

   

   $\because (\overrightarrow{CA}\times \overrightarrow{CD})\cdot (\overrightarrow{AB}\times \overrightarrow{AD})$ has a positive z-axis

   $\because (\overrightarrow{AB}\times \overrightarrow{AD})\cdot (\overrightarrow{BC}\times \overrightarrow{BD})$ has a positive z-axis

   $\therefore$ Point D is in the middle of ABC

Coordinate system definition

Coordinate decomposition
$$\begin{gathered} |\vec{u}|=|\vec{v}|=|\vec{w}|=1 \\ All\; three\; vectors\; are\; unit\; vectors \\ \vec{u}\cdot \vec{v}=\vec{v}\cdot \vec{w}=\vec{w}\cdot \vec{u}=0 \\ The\; angle\; between\; two\; of\; the\; three\; vectors\; is90° \\ \vec{w}=\vec{u}\times \vec{v} \\ vector\vec{w}perpendicular\; to\; the\; vector\vec{u}and\vec{v}composed\; of\; planes \\ \vec{p}=\left(\begin{array}{c}(\vec{p}\cdot \vec{u})\vec{u}\\ (\vec{p}\cdot \vec{v})\vec{v}\\ (\vec{p}\cdot \vec{w})\vec{w}\end{array}\right) \\ The\; component\; of\; the\; world\; coordinate\; system\; vector\; onthe\; uvwcoordinate \end{gathered}$$

matrix

identity matrix

$$\left(\begin{array}{c}1,0,0,0\\ 0,1,0,0\\ 0,0,1,0\\ 0,0,0,1\end{array}\right)$$

matrix product

$$\begin{gathered} \left(\begin{array}{c}{a}_{00},a_{01},a_{02},a_{03}\\ a_{10},a_{11}.a_{12},a_{13}\\ a_{20},a_{21},a_{22},a_{23}\\ a_{30},a_{31},a_{32},a_{33}\end{array}\right)\times \left(\begin{array}{c}{b}_{00},b_{01},b_{02},b_{03}\\ b_{10},b_{11}.b_{12},b_{13}\\ b_{20},b_{21},b_{22},b_{23}\\ b_{30},b_{31},b_{32},b_{33}\end{array}\right)=\left(\begin{array}{c}{c}_{00},c_{01},c_{02},c_{03}\\ c_{10},c_{11}.c_{12},c_{13}\\ c_{20},c_{21},c_{22},c_{23}\\ c_{30},c_{31},c_{32},c_{33}\end{array}\right) \\ \left(\begin{array}{c}{b}_{00},b_{01},b_{02},b_{03}\\ b_{10},b_{11}.b_{12},b_{13}\\ b_{20},b_{21},b_{22},b_{23}\\ b_{30},b_{31},b_{32},b_{33}\end{array}\right) \\ \left(\begin{array}{c}{a}_{00},a_{01},a_{02},a_{03}\\ a_{10},a_{11}.a_{12},a_{13}\\ a_{20},a_{21},a_{22},a_{23}\\ a_{30},a_{31},a_{32},a_{33}\end{array}\right)\left(\begin{array}{c}{c}_{00},c_{01},c_{02},c_{03}\\ c_{10},c_{11}.c_{12},c_{13}\\ c_{20},c_{21},c_{22},c_{23}\\ c_{30},c_{31},c_{32},c_{33}\end{array}\right) \\ {c}_{00}=a_{00}{b}_{00}+a_{01}{b}_{10}+a_{02}{b}_{20}+a_{03}{b}_{30} \\ {c}_{01}=a_{00}{b}_{01}+a_{01}{b}_{11}+a_{02}{b}_{21}+a_{03}{b}_{31} \\ . \\ . \end{gathered}$$

c c xxof the$c\; matrix$line$x$$y$列$c_{xy}$equal to aaThe xxth$of\; a$ matrix$row\; x$ andbbThe yyth$of\; the\; b$ matrix$The\; y$ columns are multiplied and added.

matrix multiplication vector

$$\left(\begin{array}{c}{a}_{00},a_{01},a_{02},a_{03}\\ a_{10},a_{11}.a_{12},a_{13}\\ a_{20},a_{21},a_{22},a_{23}\\ a_{30},a_{31},a_{32},a_{33}\end{array}\right)\left(\begin{array}{c}{b}_{00}\\ b_{10}\\ b_{20}\\ 0\end{array}\right)=\left(\begin{array}{c}{a}_{00}{b}_{00}+a_{01}{b}_{10}+a_{02}{b}_{20}+0\\ a_{10}{b}_{00}+a_{11}{b}_{10}+a_{12}{b}_{20}+0\\ a_{20}{b}_{00}+a_{21}{b}_{10}+a_{22}{b}_{20}+0\\ a_{30}{b}_{00}+a_{31}{b}_{10}+a_{32}{b}_{20}+0\end{array}\right)$$

matrix transformation

scaling matrix
$$\left(\begin{array}{c}{S}_{00},0,0,0\\ 0,S_{11},0,0\\ 0,0,S_{22},0\\ 0,0,0,1\end{array}\right)$$
Translation matrix
$$\left(\begin{array}{c}0,0,0,T_{04}\\ 0,0,0,T_{14}\\ 0,0,0,T_{24}\\ 0,0,0,1\end{array}\right)$$
rotation matrix

Matrix transpose

$${\left(\begin{array}{c}1,2\\ 3,4\\ 5,6\\ 7,8\end{array}\right)}^T=\left(\begin{array}{c}1,3,5,7\\ 2,4,6,8\end{array}\right)$$