Turtle drawing fractal with openGl - c

Question

I'm trying to do Koch snowflake for a computer graphics course. Searching on the web i've found that a sequence named Thue-morse can approximate the Koch snowflake by using a turtle drawing method.
Here is the code i have so far:
#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;
}
It worked quite well here.
But i don't understand how the turtleTurn function works. Someone can help me ?

Answer 1

This is the formula for a 2d rotation:
(mUx, mUy) contains the coordinates of the "heading vector" of the turtle, then what turtleTurn(float ang) does is turning this vector by an angle (ang).
If you want a nice explanation of this formula, in particular where the sine and cosine come from, you can take a look at the following page, that
has some drawings that will make it clearer:
https://www.siggraph.org/education/materials/HyperGraph/modeling/mod_tran/2drota.htm