Question

我正在尝试为计算机图形学课程做Koch雪花。在网上搜索我发现一个名为Thue-morse的序列可以通过使用乌龟绘图方法来近似Koch雪花。

这是我到目前为止的代码：

#include <GLUT/glut.h>
#include <math.h>
#include <string.h>

//screen size
#define WIDTH 1024
#define HEIGHT 800

float x, y,mUx,mUy;

//init the turtle environment
void turtleInit(){

    x = WIDTH/2; // this is the starting point for the x
    y = HEIGHT/2; // this is the starting point for the y

    mUx = 1;
    mUy = 0;
}

//move the turtle ds units
void turtleMove(float ds){
    x += mUx * ds;
    y += mUy * ds;
}

//turn left by "ang" radians if positive and right if negative.
void turtleTurn(float ang){
    float ux = mUx;
    float uy = mUy;

    mUx = ux * cos(ang) - uy * sin(ang);
    mUy = uy * cos(ang) + ux * sin(ang);

}

//thue morse sequence used to approximate the Koch snowflake
char thue_memoization[10000000];
int thueMorseRecurrenceRelation(int i){

    if( thue_memoization[i] != -1 )
        return thue_memoization[i];

    if ( i % 2 != 0 )
        return thue_memoization[i] = 1 - thueMorseRecurrenceRelation(i / 2);

    else
        return thue_memoization[i] = thueMorseRecurrenceRelation(i / 2);
}

void display( void ){
    glClearColor(0, 0, 0, 0);
    glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);

    glMatrixMode(GL_PROJECTION);
    glLoadIdentity();

    glOrtho(-WIDTH, WIDTH, -HEIGHT, HEIGHT, -50, 50);

    glMatrixMode(GL_MODELVIEW);
    glLoadIdentity();


    glBegin(GL_POINTS);

    glColor3f(0, 0, 1);

    turtleInit();

    for (int i = 0; i < 1000000; ++i) {
        const static float p = 1;//turtle's step

        if ( thueMorseRecurrenceRelation(i) )
            turtleTurn(M_PI/3.0);

        turtleMove(p);

        glVertex2f(x, y);

    }

    glEnd();

    glFlush();
}


int main(int argc,char **argv){

    memset(thue_memoization,-1,sizeof(thue_memoization));
    thue_memoization[0] = 0; //stop condition for the recurrence relation

    glutInit(&argc, argv);
    glutInitDisplayMode(GLUT_RGB | GLUT_SINGLE);
    glutInitWindowPosition(0,0);
    glutInitWindowSize(WIDTH, HEIGHT);

    glutCreateWindow("Koch snowflake. The winter is comming ...");
    glutDisplayFunc(display);

    glutMainLoop();

    return 0;
}

这里工作得很好。但我不明白 turtleTurn 功能是如何工作的。有人可以帮帮我吗？

Answer 1

这是2d轮换的公式：（mUx，mUy）包含乌龟的“标题向量”的坐标，然后turtleTurn（float ang）的作用是将该向量转动一个角度（ang）。

如果你想对这个公式有一个很好的解释，特别是在正弦和余弦来自的地方，你可以看看下面的页面，有一些图纸可以让它更清晰：

https://www.siggraph.org/education/materials/HyperGraph/modeling/mod_tran/2drota.htm

与openGl的乌龟图画分数维

1 个答案: