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 /* DUMP MEMORY LEAKS */
42 /************************************** Defined Constants ***************************************/
43 /* Number of points to draw in the curves */
44 #define NUM_POINTS 512
46 /* Angle to rotate when the user presses an arrow key */
47 #define ROTATION_ANGLE 5.0
49 /* Amount to scale bu when the user presses PgUp or PgDn */
50 #define SCALE_FACTOR 0.8
53 /*************************************** Global Variables ***************************************/
54 /* Lorenz Attractor variables */
55 double s0 = 10.0, r0 = 28.0, b0 = 8.0/3.0 ; /* Default Lorenz attactor parameters */
56 double time_step = 0.03 ; /* Time step in the simulation */
57 double sigma = 10.0, r = 28.0, b = 8.0/3.0 ; /* Lorenz attactor parameters */
58 double red_position[NUM_POINTS][3] ; /* Path of the red point */
59 double grn_position[NUM_POINTS][3] ; /* Path of the green point */
60 int array_index ; /* Position in *_position arrays of most recent point */
61 double distance = 0.0 ; /* Distance between the two points */
64 double yaw = 0.0, pit = 0.0 ; /* Euler angles of the viewing rotation */
65 double scale = 1.0 ; /* Scale factor */
66 double xcen = 0.0, ycen = 0.0, zcen = 0.0 ; /* Coordinates of the point looked at */
68 int animate = 1 ; /* 0 - stop, 1 = go, 2 = single-step */
71 /******************************************* Functions ******************************************/
73 /* The Lorenz Attractor */
74 void calc_deriv ( double position[3], double deriv[3] )
76 /* Calculate the Lorenz attractor derivatives */
77 deriv[0] = sigma * ( position[1] - position[0] ) ;
78 deriv[1] = ( r + position[2] ) * position[0] - position[1] ;
79 deriv[2] = -position[0] * position[1] - b * position[2] ;
82 void advance_in_time ( double time_step, double position[3], double new_position[3] )
84 /* Move a point along the Lorenz attractor */
85 double deriv0[3], deriv1[3], deriv2[3], deriv3[3] ;
87 memcpy ( new_position, position, 3 * sizeof(double) ) ; /* Save the present values */
89 /* First pass in a Fourth-Order Runge-Kutta integration method */
90 calc_deriv ( position, deriv0 ) ;
91 for ( i = 0; i < 3; i++ )
92 new_position[i] = position[i] + 0.5 * time_step * deriv0[i] ;
95 calc_deriv ( new_position, deriv1 ) ;
96 for ( i = 0; i < 3; i++ )
97 new_position[i] = position[i] + 0.5 * time_step * deriv1[i] ;
100 calc_deriv ( position, deriv2 ) ;
101 for ( i = 0; i < 3; i++ )
102 new_position[i] = position[i] + time_step * deriv2[i] ;
105 calc_deriv ( new_position, deriv3 ) ;
106 for ( i = 0; i < 3; i++ )
107 new_position[i] = position[i] + 0.1666666666666666667 * time_step *
108 ( deriv0[i] + 2.0 * ( deriv1[i] + deriv2[i] ) + deriv3[i] ) ;
113 #define INPUT_LINE_LENGTH 80
115 void key_cb ( unsigned char key, int x, int y )
118 char inputline [ INPUT_LINE_LENGTH ] ;
122 case 'r' : case 'R' : /* Reset the simulation */
123 /* Reset the Lorenz parameters */
127 /* Set an initial position */
128 red_position[0][0] = (double)rand() / (double)RAND_MAX ;
129 red_position[0][1] = (double)rand() / (double)RAND_MAX ;
130 red_position[0][2] = (double)rand() / (double)RAND_MAX ;
131 grn_position[0][0] = (double)rand() / (double)RAND_MAX ;
132 grn_position[0][1] = (double)rand() / (double)RAND_MAX ;
133 grn_position[0][2] = (double)rand() / (double)RAND_MAX ;
135 /* Initialize the arrays */
136 for ( i = 1; i < NUM_POINTS; i++ )
138 memcpy ( red_position[i], red_position[0], 3 * sizeof(double) ) ;
139 memcpy ( grn_position[i], grn_position[0], 3 * sizeof(double) ) ;
144 case 'm' : case 'M' : /* Modify the Lorenz parameters */
145 printf ( "Please enter new value for <sigma> (default %f, currently %f): ", s0, sigma ) ;
146 fgets ( inputline, INPUT_LINE_LENGTH-1, stdin ) ;
147 sscanf ( inputline, "%lf", &sigma ) ;
149 printf ( "Please enter new value for <b> (default %f, currently %f): ", b0, b ) ;
150 fgets ( inputline, INPUT_LINE_LENGTH-1, stdin ) ;
151 sscanf ( inputline, "%lf", &b ) ;
153 printf ( "Please enter new value for <r> (default %f, currently %f): ", r0, r ) ;
154 fgets ( inputline, INPUT_LINE_LENGTH-1, stdin ) ;
155 sscanf ( inputline, "%lf", &r ) ;
159 case 's' : case 'S' : /* Stop the animation */
163 case 'g' : case 'G' : /* Start the animation */
167 case ' ' : /* Spacebar: Single step */
171 case 27 : /* Escape key */
172 glutLeaveMainLoop () ;
177 void special_cb ( int key, int x, int y )
181 case GLUT_KEY_UP : /* Rotate up a little */
182 glRotated ( ROTATION_ANGLE, 0.0, 1.0, 0.0 ) ;
185 case GLUT_KEY_DOWN : /* Rotate down a little */
186 glRotated ( -ROTATION_ANGLE, 0.0, 1.0, 0.0 ) ;
189 case GLUT_KEY_LEFT : /* Rotate left a little */
190 glRotated ( ROTATION_ANGLE, 0.0, 0.0, 1.0 ) ;
193 case GLUT_KEY_RIGHT : /* Rotate right a little */
194 glRotated ( -ROTATION_ANGLE, 0.0, 0.0, 1.0 ) ;
197 case GLUT_KEY_PAGE_UP : /* Zoom in a little */
198 glScaled ( 1.0 / SCALE_FACTOR, 1.0 / SCALE_FACTOR, 1.0 / SCALE_FACTOR ) ;
201 case GLUT_KEY_PAGE_DOWN : /* Zoom out a little */
202 glScaled ( SCALE_FACTOR, SCALE_FACTOR, SCALE_FACTOR ) ;
206 glutPostRedisplay () ;
209 void mouse_cb ( int button, int updown, int x, int y )
211 if ( updown == GLUT_DOWN )
213 double dist = 1.0e20 ; /* A very large number */
214 dist = 0.0 ; /* so we don't get "unused variable" compiler warning */
215 /* The idea here is that we go into "pick" mode and pick the nearest point
216 to the mouse click position. Unfortunately I don't have the time to implement
221 void draw_curve ( int index, double position [ NUM_POINTS ][3] )
225 glBegin ( GL_LINE_STRIP ) ;
228 i = ( i == NUM_POINTS-1 ) ? 0 : i + 1 ;
229 glVertex3dv ( position[i] ) ;
231 while ( i != index ) ;
236 void display_cb ( void )
240 glClear ( GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT ) ;
242 glColor3d ( 1.0, 1.0, 1.0 ) ; /* White */
244 glBegin ( GL_LINES ) ;
245 glVertex3d ( 0.0, 0.0, 0.0 ) ;
246 glVertex3d ( 2.0, 0.0, 0.0 ) ;
247 glVertex3d ( 0.0, 0.0, 0.0 ) ;
248 glVertex3d ( 0.0, 1.0, 0.0 ) ;
249 glVertex3d ( 0.0, 0.0, 0.0 ) ;
250 glVertex3d ( 0.0, 0.0, 1.0 ) ;
253 glColor3d ( 1.0, 0.0, 0.0 ) ; /* Red */
254 draw_curve ( array_index, red_position ) ;
256 glColor3d ( 0.0, 1.0, 0.0 ) ; /* Green */
257 draw_curve ( array_index, grn_position ) ;
259 /* Print the distance between the two points */
260 glColor3d ( 1.0, 1.0, 1.0 ) ; /* White */
261 sprintf ( string, "Distance: %10.6f", distance ) ;
262 glRasterPos2i ( 10, 10 ) ;
263 glutBitmapString ( GLUT_BITMAP_HELVETICA_12, (unsigned char*)string ) ;
268 void reshape_cb ( int width, int height )
271 glViewport ( 0, 0, width, height ) ;
272 glMatrixMode ( GL_PROJECTION ) ;
274 ar = (float) width / (float) height ;
275 glFrustum ( -ar, ar, -1.0, 1.0, 10.0, 100.0 ) ;
276 glMatrixMode ( GL_MODELVIEW ) ;
281 glTranslated ( xcen, ycen, zcen - 50.0 ) ;
285 void timer_cb ( int value )
287 /* Function called at regular intervals to update the positions of the points */
288 double deltax, deltay, deltaz ;
289 int new_index = array_index + 1 ;
291 /* Set the next timed callback */
292 glutTimerFunc ( 30, timer_cb, 0 ) ;
296 if ( new_index == NUM_POINTS ) new_index = 0 ;
297 advance_in_time ( time_step, red_position[array_index], red_position[new_index] ) ;
298 advance_in_time ( time_step, grn_position[array_index], grn_position[new_index] ) ;
299 array_index = new_index ;
301 deltax = red_position[array_index][0] - grn_position[array_index][0] ;
302 deltay = red_position[array_index][1] - grn_position[array_index][1] ;
303 deltaz = red_position[array_index][2] - grn_position[array_index][2] ;
304 distance = sqrt ( deltax * deltax + deltay * deltay + deltaz * deltaz ) ;
306 if ( animate == 2 ) animate = 0 ;
309 glutPostRedisplay () ;
314 /* The Main Program */
316 int main ( int argc, char *argv[] )
320 /* Initialize the random number generator */
323 /* Set up the OpenGL parameters */
324 glEnable ( GL_DEPTH_TEST ) ;
325 glClearColor ( 0.0, 0.0, 0.0, 0.0 ) ;
326 glClearDepth ( 1.0 ) ;
328 /* Initialize GLUT */
329 glutInitWindowSize ( 600, 600 ) ;
330 glutInit ( &pargc, argv ) ;
331 glutInitDisplayMode ( GLUT_RGB | GLUT_DOUBLE | GLUT_DEPTH ) ;
333 /* Create the window */
334 glutCreateWindow ( "Lorenz Attractor" ) ;
335 glutKeyboardFunc ( key_cb ) ;
336 glutMouseFunc ( mouse_cb ) ;
337 glutSpecialFunc ( special_cb ) ;
338 glutDisplayFunc ( display_cb ) ;
339 glutReshapeFunc ( reshape_cb ) ;
340 glutTimerFunc ( 30, timer_cb, 0 ) ;
342 /* Initialize the attractor: The easiest way is to call the keyboard callback with an
343 * argument of 'r' for Reset.
345 key_cb ( 'r', 0, 0 ) ;
347 /* Enter the GLUT main loop */
351 /* DUMP MEMORY LEAK INFORMATION */
352 _CrtDumpMemoryLeaks () ;