CPT205 W2

This’s the note of CPT205 W2.

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The topics for this week:

1. Computer representation of objects
2. Cartesian co-ordinate system
3. Points, lines and angles
4. Trigonometry 三角函数
5. Vectors (unit vector) and vector calculations (addition, subtraction, scaling, dot product, cross product) 矢量和矢量计算
6. Matrices (dimension, transpose, square/symmetric/identity, inverse) and matrix calculations (addition, subtraction, multiplication) 矩阵和矩阵计算

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因为矩阵的知识快忘光了，所以矩阵的部分写的比较多。

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Lecture

Computer representation of objects

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Cartesian co-ordinate system

都是笛卡尔平面直角坐标系的知识，不会的建议重开。

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Points, lines and angles

同上。

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Trigonometry

同上。

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Vectors

同上。

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Matrices

• Techniques for applying transformations use matrices. 应用变换的技术使用矩阵。
• A matrix is simply a set of numbers arranged in a rectangular format. 矩阵是一组按矩形形式排列的数字。
• Each number is known as an element. 每个数字都是一个元素。
• Capital letters are used to represent a matrix. 用大写字母表示矩阵。
• Bold letters when printed (M), or underlined when written. 打印时加粗或下划线。
• A matrix has dimensions that refer to the number of rows and the number of columns it has. 矩阵的维数指的是它的行数和列数

Dimensions of matrices

The dimensions of Matrix are (x * y). x is the number of rows in matrix and y is the columns.
$$
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
\end{bmatrix}
$$
The dimensions of this matrix are (2*3).

Transpose matrix

When a matrix is rewritten so that its rows and columns are interchanged, then the resulting matrix is called the transpose of the original. 当一个矩阵被改写使它的行和列互换时，得到的矩阵叫做原矩阵的转置。
The original A:
$$
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
\end{bmatrix}
$$
The transpose matrix of A:
$$
\begin{bmatrix}
1 & 4 \\
2 & 5 \\
3 & 6 \\
\end{bmatrix}
$$

Square and symmetric matrices

• A square matrix is matrix where the number of rows equals the number of columns. 行列数量相同的是方阵。
• A symmetric matrix is a square matrix where the rows and columns are such that its transponse is the same as the original matrix. 对角线对称的方阵是对称矩阵。

Identity matrices

An identity matrix, I is a square matrix with zeros everywhere except its diagonal elements which have a value of 1. 单位矩阵就是对角线是1，剩下的都是0的方阵。

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Adding matrices

• Matrices A and B may be added if they have the same dimensions.
• That is, the corresponding elements may be added to yield a resulting matrix.
• The sum is commutative, i.e. A + B = B + A 加法交换律
  

Subtracting matrices

Matrix B may be subtracted from matrix A if they have the same dimensions, i.e. the corresponding elements of B may be subtracted from those of A to yield a resulting matrix.

The result is not commutative. Reversing the order of the matrices yields different results, i.e. A - B ≠ B -A ⚠️减法🈚️交换律!

Multiplying matrices

• By a constant
  
• By a matrix - The rule for multiplying one matrix to another is simple: if the number of columns in the first matrix is the same as the number of rows in the second matrix, the multiplication can be done. 也就是必须要是(x * y)&(y * z)，计算结果是(x * z)。
  
  

Matrix multiplication is not commutative. Reversing the order of the matrices yields different results. ⚠️矩阵的乘法也是没有交换律的！交换相乘的两个矩阵的位置会产生不同的结果。

Inverse matrices

If two matrices A and B, when multiplieid together, results in an indentity matrix I, then matrix A is the inverse of matrix B and vice versa, i.e.
$$A * B = B * A = I$$
$$A = B^{-1} and B = A^{-1}$$
两个矩阵相乘是一个单位矩阵那这两个矩阵就是逆矩阵。

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Lab

The tutorial explains the C/C++ basics that will be used in CPT205 Computer Graphics.
跟着lab做问题不大，注意一个cpp的project中只有可以有一个main方法，这是和py不一样的地方。
学校给的那个lab代码需要引入math才可以正常操作。
引入库的时候需要手动输入，复制粘贴的时候会出现无法读取的问题 #include <GL/freeglut.h>

#define FREEGLUT_STATIC 

#include <GL/freeglut.h> 
#include <math.h>
void define_to_OpenGL(); 
 
/////////////////////////////////// 
int main(int argc, char** argv) 
{ 
    glutInit(&argc,argv); 
    // Task 2
    glutInitWindowSize(600, 400); // 整个window的大小
    glutInitWindowPosition(50, 50); // window的左上角对于屏幕左上角的坐标
    glutCreateWindow("Graphics Primitives"); // 初始化一个窗口
    glutDisplayFunc(define_to_OpenGL); 
    glutMainLoop(); // 实现循环
} 
/////////////////////////////////// 
void define_to_OpenGL() 
{ 
    glClearColor(1,1,1,1); 
    glClear(GL_COLOR_BUFFER_BIT); 

    // The stuff to appear on screen goes here 
    // Task 2 Set the Dimensions
    glMatrixMode(GL_PROJECTION); 
    glLoadIdentity(); 
    gluOrtho2D(-100,500,-200,200); // window的原点设定，因为window是600*400，所以这里的x1x2y1y2分别代表了左下角的点和右上角的点。

    // Task 3 Draw the Axes
    glLineWidth(1.0); 
    glColor3f(1.0,0.0,0.0); 
    glBegin(GL_LINES); 
        glVertex2f(0.0,0.0);  // start location 
        glVertex2f(450.0,0.0);  // end location 
        glVertex2f(0.0,-150.0);  // start location 
        glVertex2f(0.0,150.0);  // end location 
    glEnd(); 

    // Task 4 Draw a Dot at the Origin
    glPointSize(10); 
    glColor3f(1.0,0.0,1.0); 
    glBegin(GL_POINTS); 
    glVertex2f(0.0,0.0); 
    glEnd(); 

    // Task 5 Plot a Sine Wave
    // draw a sine wave 
    int i;    
    float x,y; 
    glColor3f(0.0,0.0,1.0); 
    glPointSize(1); 
    glBegin(GL_POINTS); 
    for(i=0;i<361;i=i+1) 
    { 
        x = (float)i;  
        y = 100.0 * sin(i*(2*3.14/360.0)); //角度360转弧度2Π，sin函数里面放的是弧度
        glVertex2f(x,y); 
    } 
    glEnd(); 

    // Tasks 6 Draw a Triangle
    // Tasks 7 Draw A Multi-coloured Triangle
    // Tasks 8 Draw a Single-coloured Triangle 
    glShadeModel(GL_FLAT); //turn off the smoothing capability
    glBegin(GL_TRIANGLES);
    glColor3f(1, 0, 0);
    glVertex2f( -50, 50 ); 
    glColor3f(0, 1, 0);
    glVertex2f( -50, 0 ); 
    glColor3f(0, 0, 1);
    glVertex2f( 0, 0 ); 
    glEnd(); 

    glFlush(); // 这两个是让图像显示到屏幕上
}

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References

1. XJTLU CPT205 slides (Week2)