2 * Lorenz Strange Attractor
4 * Written by John F. Fay in honor of the "freeglut" 2.0.0 release in July 2003
7 * This program starts with two particles right next to each other. The particles
8 * move through a three-dimensional phase space governed by the following equations:
9 * dx/dt = sigma * ( y - x )
10 * dy/dt = r * x - y + x * z
11 * dz/dt = x * y + b * z
12 * These are the Lorenz equations and define the "Lorenz Attractor." Any two particles
13 * arbitrarily close together will move apart as time increases, but their tracks are
14 * confined within a region of the space.
17 * Arrow keys: Rotate the view
18 * PgUp, PgDn: Zoom in and out
19 * Mouse click: Center on the nearest point on a particle trajectory
21 * 'r'/'R': Reset the simulation
22 * 'm'/'M': Modify the Lorenz parameters (in the text window)
23 * 's'/'S': Stop (the advancement in time)
25 * <spacebar>: Single-step
35 #include <GL/freeglut.h>
37 #include <crtdbg.h> // DUMP MEMORY LEAKS
41 /************************************** Defined Constants ***************************************/
42 /* Number of points to draw in the curves */
43 #define NUM_POINTS 512
45 /* Angle to rotate when the user presses an arrow key */
46 #define ROTATION_ANGLE 5.0
48 /* Amount to scale bu when the user presses PgUp or PgDn */
49 #define SCALE_FACTOR 0.8
52 /*************************************** Global Variables ***************************************/
53 /* Lorenz Attractor variables */
54 double s0 = 10.0, r0 = 28.0, b0 = 8.0/3.0 ; /* Default Lorenz attactor parameters */
55 double time_step = 0.03 ; /* Time step in the simulation */
56 double sigma = 10.0, r = 28.0, b = 8.0/3.0 ; /* Lorenz attactor parameters */
57 double red_position[NUM_POINTS][3] ; /* Path of the red point */
58 double grn_position[NUM_POINTS][3] ; /* Path of the green point */
59 int array_index ; /* Position in *_position arrays of most recent point */
60 double distance = 0.0 ; /* Distance between the two points */
63 double yaw = 0.0, pit = 0.0 ; /* Euler angles of the viewing rotation */
64 double scale = 1.0 ; /* Scale factor */
65 double xcen = 0.0, ycen = 0.0, zcen = 0.0 ; /* Coordinates of the point looked at */
67 int animate = 1 ; /* 0 - stop, 1 = go, 2 = single-step */
70 /******************************************* Functions ******************************************/
72 /* The Lorenz Attractor */
73 void calc_deriv ( double position[3], double deriv[3] )
75 /* Calculate the Lorenz attractor derivatives */
76 deriv[0] = sigma * ( position[1] - position[0] ) ;
77 deriv[1] = ( r + position[2] ) * position[0] - position[1] ;
78 deriv[2] = -position[0] * position[1] - b * position[2] ;
81 void advance_in_time ( double time_step, double position[3], double new_position[3] )
83 /* Move a point along the Lorenz attractor */
84 double deriv0[3], deriv1[3], deriv2[3], deriv3[3] ;
86 memcpy ( new_position, position, 3 * sizeof(double) ) ; /* Save the present values */
88 /* First pass in a Fourth-Order Runge-Kutta integration method */
89 calc_deriv ( position, deriv0 ) ;
90 for ( i = 0; i < 3; i++ )
91 new_position[i] = position[i] + 0.5 * time_step * deriv0[i] ;
94 calc_deriv ( new_position, deriv1 ) ;
95 for ( i = 0; i < 3; i++ )
96 new_position[i] = position[i] + 0.5 * time_step * deriv1[i] ;
99 calc_deriv ( position, deriv2 ) ;
100 for ( i = 0; i < 3; i++ )
101 new_position[i] = position[i] + time_step * deriv2[i] ;
104 calc_deriv ( new_position, deriv3 ) ;
105 for ( i = 0; i < 3; i++ )
106 new_position[i] = position[i] + 0.1666666666666666667 * time_step *
107 ( deriv0[i] + 2.0 * ( deriv1[i] + deriv2[i] ) + deriv3[i] ) ;
112 #define INPUT_LINE_LENGTH 80
114 void key_cb ( unsigned char key, int x, int y )
117 char inputline [ INPUT_LINE_LENGTH ] ;
121 case 'r' : case 'R' : /* Reset the simulation */
122 /* Reset the Lorenz parameters */
126 /* Set an initial position */
127 red_position[0][0] = (double)rand() / (double)RAND_MAX ;
128 red_position[0][1] = (double)rand() / (double)RAND_MAX ;
129 red_position[0][2] = (double)rand() / (double)RAND_MAX ;
130 grn_position[0][0] = (double)rand() / (double)RAND_MAX ;
131 grn_position[0][1] = (double)rand() / (double)RAND_MAX ;
132 grn_position[0][2] = (double)rand() / (double)RAND_MAX ;
134 /* Initialize the arrays */
135 for ( i = 1; i < NUM_POINTS; i++ )
137 memcpy ( red_position[i], red_position[0], 3 * sizeof(double) ) ;
138 memcpy ( grn_position[i], grn_position[0], 3 * sizeof(double) ) ;
143 case 'm' : case 'M' : /* Modify the Lorenz parameters */
144 printf ( "Please enter new value for <sigma> (default %lf, currently %lf): ", s0, sigma ) ;
145 fgets ( inputline, INPUT_LINE_LENGTH-1, stdin ) ;
146 sscanf ( inputline, "%lf", &sigma ) ;
148 printf ( "Please enter new value for <b> (default %lf, currently %lf): ", b0, b ) ;
149 fgets ( inputline, INPUT_LINE_LENGTH-1, stdin ) ;
150 sscanf ( inputline, "%lf", &b ) ;
152 printf ( "Please enter new value for <r> (default %lf, currently %lf): ", r0, r ) ;
153 fgets ( inputline, INPUT_LINE_LENGTH-1, stdin ) ;
154 sscanf ( inputline, "%lf", &r ) ;
158 case 's' : case 'S' : /* Stop the animation */
162 case 'g' : case 'G' : /* Start the animation */
166 case ' ' : /* Spacebar: Single step */
170 case 27 : /* Escape key */
171 glutLeaveMainLoop () ;
176 void special_cb ( int key, int x, int y )
180 case GLUT_KEY_UP : /* Rotate up a little */
181 glRotated ( ROTATION_ANGLE, 0.0, 1.0, 0.0 ) ;
184 case GLUT_KEY_DOWN : /* Rotate down a little */
185 glRotated ( -ROTATION_ANGLE, 0.0, 1.0, 0.0 ) ;
188 case GLUT_KEY_LEFT : /* Rotate left a little */
189 glRotated ( ROTATION_ANGLE, 0.0, 0.0, 1.0 ) ;
192 case GLUT_KEY_RIGHT : /* Rotate right a little */
193 glRotated ( -ROTATION_ANGLE, 0.0, 0.0, 1.0 ) ;
196 case GLUT_KEY_PAGE_UP : /* Zoom in a little */
197 glScaled ( 1.0 / SCALE_FACTOR, 1.0 / SCALE_FACTOR, 1.0 / SCALE_FACTOR ) ;
200 case GLUT_KEY_PAGE_DOWN : /* Zoom out a little */
201 glScaled ( SCALE_FACTOR, SCALE_FACTOR, SCALE_FACTOR ) ;
205 glutPostRedisplay () ;
208 void mouse_cb ( int button, int updown, int x, int y )
210 if ( updown == GLUT_DOWN )
212 double dist = 1.0e20 ; /* A very large number */
213 dist = 0.0 ; /* so we don't get "unused variable" compiler warning */
214 /* The idea here is that we go into "pick" mode and pick the nearest point
215 to the mouse click position. Unfortunately I don't have the time to implement
220 void draw_curve ( int index, double position [ NUM_POINTS ][3] )
224 glBegin ( GL_LINE_STRIP ) ;
227 i = ( i == NUM_POINTS-1 ) ? 0 : i + 1 ;
228 glVertex3dv ( position[i] ) ;
230 while ( i != index ) ;
235 void display_cb ( void )
239 glClear ( GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT ) ;
241 glColor3d ( 1.0, 1.0, 1.0 ) ; /* White */
243 glBegin ( GL_LINES ) ;
244 glVertex3d ( 0.0, 0.0, 0.0 ) ;
245 glVertex3d ( 2.0, 0.0, 0.0 ) ;
246 glVertex3d ( 0.0, 0.0, 0.0 ) ;
247 glVertex3d ( 0.0, 1.0, 0.0 ) ;
248 glVertex3d ( 0.0, 0.0, 0.0 ) ;
249 glVertex3d ( 0.0, 0.0, 1.0 ) ;
252 glColor3d ( 1.0, 0.0, 0.0 ) ; /* Red */
253 draw_curve ( array_index, red_position ) ;
255 glColor3d ( 0.0, 1.0, 0.0 ) ; /* Green */
256 draw_curve ( array_index, grn_position ) ;
258 /* Print the distance between the two points */
259 glColor3d ( 1.0, 1.0, 1.0 ) ; /* White */
260 sprintf ( string, "Distance: %10.6lf", distance ) ;
261 glRasterPos2i ( 10, 10 ) ;
262 glutBitmapString ( GLUT_BITMAP_HELVETICA_12, string ) ;
267 void reshape_cb ( int width, int height )
270 glViewport ( 0, 0, width, height ) ;
271 glMatrixMode ( GL_PROJECTION ) ;
273 ar = (float) width / (float) height ;
274 glFrustum ( -ar, ar, -1.0, 1.0, 10.0, 100.0 ) ;
275 glMatrixMode ( GL_MODELVIEW ) ;
280 glTranslated ( xcen, ycen, zcen - 50.0 ) ;
284 void timer_cb ( int value )
286 /* Function called at regular intervals to update the positions of the points */
287 double deltax, deltay, deltaz ;
288 int new_index = array_index + 1 ;
290 /* Set the next timed callback */
291 glutTimerFunc ( 30, timer_cb, 0 ) ;
295 if ( new_index == NUM_POINTS ) new_index = 0 ;
296 advance_in_time ( time_step, red_position[array_index], red_position[new_index] ) ;
297 advance_in_time ( time_step, grn_position[array_index], grn_position[new_index] ) ;
298 array_index = new_index ;
300 deltax = red_position[array_index][0] - grn_position[array_index][0] ;
301 deltay = red_position[array_index][1] - grn_position[array_index][1] ;
302 deltaz = red_position[array_index][2] - grn_position[array_index][2] ;
303 distance = sqrt ( deltax * deltax + deltay * deltay + deltaz * deltaz ) ;
305 if ( animate == 2 ) animate = 0 ;
308 glutPostRedisplay () ;
313 /* The Main Program */
315 int main ( int argc, char *argv[] )
319 /* Initialize the random number generator */
322 /* Set up the OpenGL parameters */
323 glEnable ( GL_DEPTH_TEST ) ;
324 glClearColor ( 0.0, 0.0, 0.0, 0.0 ) ;
325 glClearDepth ( 1.0 ) ;
327 /* Initialize GLUT */
328 glutInitWindowSize ( 600, 600 ) ;
329 glutInit ( &pargc, argv ) ;
330 glutInitDisplayMode ( GLUT_RGB | GLUT_DOUBLE | GLUT_DEPTH ) ;
332 /* Create the window */
333 glutCreateWindow ( "Lorenz Attractor" ) ;
334 glutKeyboardFunc ( key_cb ) ;
335 glutMouseFunc ( mouse_cb ) ;
336 glutSpecialFunc ( special_cb ) ;
337 glutDisplayFunc ( display_cb ) ;
338 glutReshapeFunc ( reshape_cb ) ;
339 glutTimerFunc ( 30, timer_cb, 0 ) ;
341 /* Initialize the attractor: The easiest way is to call the keyboard callback with an
342 * argument of 'r' for Reset.
344 key_cb ( 'r', 0, 0 ) ;
346 /* Enter the GLUT main loop */
350 _CrtDumpMemoryLeaks () ; // DUMP MEMORY LEAK INFORMATION