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>
38 /************************************** Defined Constants ***************************************/
39 /* Number of points to draw in the curves */
40 #define NUM_POINTS 512
42 /* Angle to rotate when the user presses an arrow key */
43 #define ROTATION_ANGLE 5.0
45 /* Amount to scale bu when the user presses PgUp or PgDn */
46 #define SCALE_FACTOR 0.8
49 /*************************************** Global Variables ***************************************/
50 /* Lorenz Attractor variables */
51 double s0 = 10.0, r0 = 28.0, b0 = 8.0/3.0 ; /* Default Lorenz attactor parameters */
52 double time_step = 0.03 ; /* Time step in the simulation */
53 double sigma = 10.0, r = 28.0, b = 8.0/3.0 ; /* Lorenz attactor parameters */
54 double red_position[NUM_POINTS][3] ; /* Path of the red point */
55 double grn_position[NUM_POINTS][3] ; /* Path of the green point */
56 int array_index ; /* Position in *_position arrays of most recent point */
57 double distance = 0.0 ; /* Distance between the two points */
60 double yaw = 0.0, pit = 0.0 ; /* Euler angles of the viewing rotation */
61 double scale = 1.0 ; /* Scale factor */
62 double xcen = 0.0, ycen = 0.0, zcen = 0.0 ; /* Coordinates of the point looked at */
64 int animate = 1 ; /* 0 - stop, 1 = go, 2 = single-step */
67 /******************************************* Functions ******************************************/
69 /* The Lorenz Attractor */
70 void calc_deriv ( double position[3], double deriv[3] )
72 /* Calculate the Lorenz attractor derivatives */
73 deriv[0] = sigma * ( position[1] - position[0] ) ;
74 deriv[1] = ( r + position[2] ) * position[0] - position[1] ;
75 deriv[2] = -position[0] * position[1] - b * position[2] ;
78 void advance_in_time ( double time_step, double position[3], double new_position[3] )
80 /* Move a point along the Lorenz attractor */
81 double deriv0[3], deriv1[3], deriv2[3], deriv3[3] ;
83 memcpy ( new_position, position, 3 * sizeof(double) ) ; /* Save the present values */
85 /* First pass in a Fourth-Order Runge-Kutta integration method */
86 calc_deriv ( position, deriv0 ) ;
87 for ( i = 0; i < 3; i++ )
88 new_position[i] = position[i] + 0.5 * time_step * deriv0[i] ;
91 calc_deriv ( new_position, deriv1 ) ;
92 for ( i = 0; i < 3; i++ )
93 new_position[i] = position[i] + 0.5 * time_step * deriv1[i] ;
96 calc_deriv ( position, deriv2 ) ;
97 for ( i = 0; i < 3; i++ )
98 new_position[i] = position[i] + time_step * deriv2[i] ;
101 calc_deriv ( new_position, deriv3 ) ;
102 for ( i = 0; i < 3; i++ )
103 new_position[i] = position[i] + 0.1666666666666666667 * time_step *
104 ( deriv0[i] + 2.0 * ( deriv1[i] + deriv2[i] ) + deriv3[i] ) ;
109 #define INPUT_LINE_LENGTH 80
111 void key_cb ( unsigned char key, int x, int y )
114 char inputline [ INPUT_LINE_LENGTH ] ;
118 case 'r' : case 'R' : /* Reset the simulation */
119 /* Reset the Lorenz parameters */
123 /* Set an initial position */
124 red_position[0][0] = (double)rand() / (double)RAND_MAX ;
125 red_position[0][1] = (double)rand() / (double)RAND_MAX ;
126 red_position[0][2] = (double)rand() / (double)RAND_MAX ;
127 grn_position[0][0] = (double)rand() / (double)RAND_MAX ;
128 grn_position[0][1] = (double)rand() / (double)RAND_MAX ;
129 grn_position[0][2] = (double)rand() / (double)RAND_MAX ;
131 /* Initialize the arrays */
132 for ( i = 1; i < NUM_POINTS; i++ )
134 memcpy ( red_position[i], red_position[0], 3 * sizeof(double) ) ;
135 memcpy ( grn_position[i], grn_position[0], 3 * sizeof(double) ) ;
140 case 'm' : case 'M' : /* Modify the Lorenz parameters */
141 printf ( "Please enter new value for <sigma> (default %lf, currently %lf): ", s0, sigma ) ;
142 fgets ( inputline, INPUT_LINE_LENGTH-1, stdin ) ;
143 sscanf ( inputline, "%lf", &sigma ) ;
145 printf ( "Please enter new value for <b> (default %lf, currently %lf): ", b0, b ) ;
146 fgets ( inputline, INPUT_LINE_LENGTH-1, stdin ) ;
147 sscanf ( inputline, "%lf", &b ) ;
149 printf ( "Please enter new value for <r> (default %lf, currently %lf): ", r0, r ) ;
150 fgets ( inputline, INPUT_LINE_LENGTH-1, stdin ) ;
151 sscanf ( inputline, "%lf", &r ) ;
155 case 's' : case 'S' : /* Stop the animation */
159 case 'g' : case 'G' : /* Start the animation */
163 case ' ' : /* Spacebar: Single step */
167 case 27 : /* Escape key */
168 glutLeaveMainLoop () ;
173 void special_cb ( int key, int x, int y )
177 case GLUT_KEY_UP : /* Rotate up a little */
178 glRotated ( ROTATION_ANGLE, 0.0, 1.0, 0.0 ) ;
181 case GLUT_KEY_DOWN : /* Rotate down a little */
182 glRotated ( -ROTATION_ANGLE, 0.0, 1.0, 0.0 ) ;
185 case GLUT_KEY_LEFT : /* Rotate left a little */
186 glRotated ( ROTATION_ANGLE, 0.0, 0.0, 1.0 ) ;
189 case GLUT_KEY_RIGHT : /* Rotate right a little */
190 glRotated ( -ROTATION_ANGLE, 0.0, 0.0, 1.0 ) ;
193 case GLUT_KEY_PAGE_UP : /* Zoom in a little */
194 glScaled ( 1.0 / SCALE_FACTOR, 1.0 / SCALE_FACTOR, 1.0 / SCALE_FACTOR ) ;
197 case GLUT_KEY_PAGE_DOWN : /* Zoom out a little */
198 glScaled ( SCALE_FACTOR, SCALE_FACTOR, SCALE_FACTOR ) ;
202 glutPostRedisplay () ;
205 void mouse_cb ( int button, int updown, int x, int y )
207 if ( updown == GLUT_DOWN )
209 double dist = 1.0e20 ; /* A very large number */
210 dist = 0.0 ; /* so we don't get "unused variable" compiler warning */
211 /* The idea here is that we go into "pick" mode and pick the nearest point
212 to the mouse click position. Unfortunately I don't have the time to implement
217 void draw_curve ( int index, double position [ NUM_POINTS ][3] )
221 glBegin ( GL_LINE_STRIP ) ;
224 i = ( i == NUM_POINTS-1 ) ? 0 : i + 1 ;
225 glVertex3dv ( position[i] ) ;
227 while ( i != index ) ;
232 void display_cb ( void )
236 glClear ( GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT ) ;
238 glColor3d ( 1.0, 1.0, 1.0 ) ; /* White */
240 glBegin ( GL_LINES ) ;
241 glVertex3d ( 0.0, 0.0, 0.0 ) ;
242 glVertex3d ( 2.0, 0.0, 0.0 ) ;
243 glVertex3d ( 0.0, 0.0, 0.0 ) ;
244 glVertex3d ( 0.0, 1.0, 0.0 ) ;
245 glVertex3d ( 0.0, 0.0, 0.0 ) ;
246 glVertex3d ( 0.0, 0.0, 1.0 ) ;
249 glColor3d ( 1.0, 0.0, 0.0 ) ; /* Red */
250 draw_curve ( array_index, red_position ) ;
252 glColor3d ( 0.0, 1.0, 0.0 ) ; /* Green */
253 draw_curve ( array_index, grn_position ) ;
255 /* Print the distance between the two points */
256 glColor3d ( 1.0, 1.0, 1.0 ) ; /* White */
257 sprintf ( string, "Distance: %10.6lf", distance ) ;
258 glRasterPos2i ( 10, 10 ) ;
259 glutBitmapString ( GLUT_BITMAP_HELVETICA_12, string ) ;
264 void reshape_cb ( int width, int height )
267 glViewport ( 0, 0, width, height ) ;
268 glMatrixMode ( GL_PROJECTION ) ;
270 ar = (float) width / (float) height ;
271 glFrustum ( -ar, ar, -1.0, 1.0, 10.0, 100.0 ) ;
272 glMatrixMode ( GL_MODELVIEW ) ;
277 glTranslated ( xcen, ycen, zcen - 50.0 ) ;
281 void timer_cb ( int value )
283 /* Function called at regular intervals to update the positions of the points */
284 double deltax, deltay, deltaz ;
285 int new_index = array_index + 1 ;
287 /* Set the next timed callback */
288 glutTimerFunc ( 30, timer_cb, 0 ) ;
292 if ( new_index == NUM_POINTS ) new_index = 0 ;
293 advance_in_time ( time_step, red_position[array_index], red_position[new_index] ) ;
294 advance_in_time ( time_step, grn_position[array_index], grn_position[new_index] ) ;
295 array_index = new_index ;
297 deltax = red_position[array_index][0] - grn_position[array_index][0] ;
298 deltay = red_position[array_index][1] - grn_position[array_index][1] ;
299 deltaz = red_position[array_index][2] - grn_position[array_index][2] ;
300 distance = sqrt ( deltax * deltax + deltay * deltay + deltaz * deltaz ) ;
302 if ( animate == 2 ) animate = 0 ;
305 glutPostRedisplay () ;
310 /* The Main Program */
312 int main ( int argc, char *argv[] )
316 /* Initialize the random number generator */
319 /* Set up the OpenGL parameters */
320 glEnable ( GL_DEPTH_TEST ) ;
321 glClearColor ( 0.0, 0.0, 0.0, 0.0 ) ;
322 glClearDepth ( 1.0 ) ;
324 /* Initialize GLUT */
325 glutInitWindowSize ( 600, 600 ) ;
326 glutInit ( &pargc, argv ) ;
327 glutInitDisplayMode ( GLUT_RGB | GLUT_DOUBLE | GLUT_DEPTH ) ;
329 /* Create the window */
330 glutCreateWindow ( "Lorenz Attractor" ) ;
331 glutKeyboardFunc ( key_cb ) ;
332 glutMouseFunc ( mouse_cb ) ;
333 glutSpecialFunc ( special_cb ) ;
334 glutDisplayFunc ( display_cb ) ;
335 glutReshapeFunc ( reshape_cb ) ;
336 glutTimerFunc ( 30, timer_cb, 0 ) ;
338 /* Initialize the attractor: The easiest way is to call the keyboard callback with an
339 * argument of 'r' for Reset.
341 key_cb ( 'r', 0, 0 ) ;
343 /* Enter the GLUT main loop */