1 /* -*- Mode: C; tab-width: 4; c-basic-offset: 4 -*- */
2 /* flow --- flow of strange bees */
5 static const char sccsid[] = "@(#)flow.c 5.00 2000/11/01 xlockmore";
9 * Copyright (c) 1996 by Tim Auckland <tda10.geo@yahoo.com>
10 * Incorporating some code from Stephen Davies Copyright (c) 2000
12 * Search code based on techniques described in "Strange Attractors:
13 * Creating Patterns in Chaos" by Julien C. Sprott
15 * Permission to use, copy, modify, and distribute this software and its
16 * documentation for any purpose and without fee is hereby granted,
17 * provided that the above copyright notice appear in all copies and that
18 * both that copyright notice and this permission notice appear in
19 * supporting documentation.
21 * This file is provided AS IS with no warranties of any kind. The author
22 * shall have no liability with respect to the infringement of copyrights,
23 * trade secrets or any patents by this file or any part thereof. In no
24 * event will the author be liable for any lost revenue or profits or
25 * other special, indirect and consequential damages.
27 * "flow" shows a variety of continuous phase-space flows around strange
28 * attractors. It includes the well-known Lorentz mask (the "Butterfly"
29 * of chaos fame), two forms of Rossler's "Folded Band" and Poincare'
30 * sections of the "Birkhoff Bagel" and Duffing's forced occilator. "flow"
31 * can now discover new attractors.
35 * 29-Oct-2004: [TDA] Discover Attractors unknown to science.
36 * Replace 2D rendering of Periodic Attractors with a 3D
37 * 'interrupted' rendering. Replace "-/+allow2d" with "-/+periodic"
38 * Replace all ODE formulae with completely generic forms.
39 * Add '-search' option to perform background high-speed discovery
40 * for completely new attractors without impacting rendering
42 * Use gaussian distribution for initial point positions and for
44 * Add "+dbuf" option to allow Double-Buffering to be turned off on
46 * Remove redundant '-zoom' option. Now automatically zooms if both
47 * rotation and riding are permitted.
48 * Replace dynamic bounding box with static one pre-calculated
49 * during discovery phase.
50 * Simplify and fix bounding box clipping code. Should now be safe
51 * to run without double buffer on all XFree86 servers if desired.
52 * 12-Oct-2004: [TDA] Merge Xscreensaver and Xlockmore branches
53 * Added Chalky's orbital camera, but made the zooming work by
54 * flying the camera rather than interpolating the view transforms.
55 * Added Chalky's Bounding Box, but time-averaged the boundaries to
56 * let the lost bees escape.
57 * Added Chalky's 'view-frustrum' clipping, but only applying it to
58 * the Bounding Box. Trails make clipping less useful.
59 * Added Chalky's "-slow" and "-freeze" options for compatibility,
60 * but haven't implemented the features, since the results are ugly
61 * and make no mathematical contribution.
62 * Added Double-Buffering as a work-around for a persistent XFree86
63 * bug that left debris on the screen.
64 * 21-Mar-2003: [TDA] Trails added (XLockmore branch)
65 * 01-Nov-2000: [TDA] Allocation checks (XLockmore branch)
66 * 21-Feb-2000: [Chalky] Major hackage (Stephen Davies, chalky@null.net)
67 * (Xscreensaver branch)
68 * Forced perspective mode, added 3d box around attractor which
69 * involved coding 3d-planar-clipping against the view-frustrum
70 * thingy. Also made view alternate between piggybacking on a 'bee'
71 * to zooming around outside the attractor. Most bees slow down and
72 * stop, to make the structure of the attractor more obvious.
73 * 28-Jan-1999: [TDA] Catch 'lost' bees in flow.c and disable them.
75 * I chose to disable them rather than reinitialise them because
76 * reinitialising can produce fake attractors.
77 * This has allowed me to relax some of the parameters and initial
78 * conditions slightly to catch some of the more extreme cases. As a
79 * result you may see some bees fly away at the start - these are the ones
80 * that 'missed' the attractor. If the bee with the camera should fly
81 * away the mode will restart :-)
82 * 31-Nov-1998: [TDA] Added Duffing (what a strange day that was :) DAB)
83 * Duffing's forced oscillator has been added to the formula list and
84 * the parameters section has been updated to display it in Poincare'
86 * 30-Nov-1998: [TDA] Added travelling perspective option
87 * A more exciting point-of-view has been added to all autonomous flows.
88 * This views the flow as seen by a particle moving with the flow. In the
89 * metaphor of the original code, I've attached a camera to one of the
91 * 30-Nov-1998: [TDA] Much code cleanup.
92 * 09-Apr-1997: [TDA] Ported to xlockmore-4
93 * 18-Jul-1996: Adapted from swarm.c Copyright (c) 1991 by Patrick J. Naughton.
94 * 31-Aug-1990: Adapted from xswarm by Jeff Butterworth. (butterwo@ncsc.org).
99 # define DEFAULTS "*delay: 10000 \n" \
102 "*cycles: 10000 \n" \
105 # define flow_handle_event 0
106 # include "xlockmore.h" /* in xscreensaver distribution */
107 #else /* STANDALONE */
108 # include "xlock.h" /* in xlockmore distribution */
109 #endif /* STANDALONE */
113 #define DEF_ROTATE "TRUE"
114 #define DEF_RIDE "TRUE"
115 #define DEF_BOX "TRUE"
116 #define DEF_PERIODIC "TRUE"
117 #define DEF_SEARCH "TRUE"
118 #define DEF_DBUF "TRUE"
123 static Bool periodicp;
127 static XrmOptionDescRec opts[] = {
128 {"-rotate", ".flow.rotate", XrmoptionNoArg, "on"},
129 {"+rotate", ".flow.rotate", XrmoptionNoArg, "off"},
130 {"-ride", ".flow.ride", XrmoptionNoArg, "on"},
131 {"+ride", ".flow.ride", XrmoptionNoArg, "off"},
132 {"-box", ".flow.box", XrmoptionNoArg, "on"},
133 {"+box", ".flow.box", XrmoptionNoArg, "off"},
134 {"-periodic", ".flow.periodic", XrmoptionNoArg, "on"},
135 {"+periodic", ".flow.periodic", XrmoptionNoArg, "off"},
136 {"-search", ".flow.search", XrmoptionNoArg, "on"},
137 {"+search", ".flow.search", XrmoptionNoArg, "off"},
138 {"-dbuf", ".flow.dbuf", XrmoptionNoArg, "on"},
139 {"+dbuf", ".flow.dbuf", XrmoptionNoArg, "off"},
142 static argtype vars[] = {
143 {&rotatep, "rotate", "Rotate", DEF_ROTATE, t_Bool},
144 {&ridep, "ride", "Ride", DEF_RIDE, t_Bool},
145 {&boxp, "box", "Box", DEF_BOX, t_Bool},
146 {&periodicp, "periodic", "Periodic", DEF_PERIODIC, t_Bool},
147 {&searchp, "search", "Search", DEF_SEARCH, t_Bool},
148 {&dbufp, "dbuf", "Dbuf", DEF_DBUF, t_Bool},
151 static OptionStruct desc[] = {
152 {"-/+rotate", "turn on/off rotating around attractor."},
153 {"-/+ride", "turn on/off ride in the flow."},
154 {"-/+box", "turn on/off bounding box."},
155 {"-/+periodic", "turn on/off periodic attractors."},
156 {"-/+search", "turn on/off search for new attractors."},
157 {"-/+dbuf", "turn on/off double buffering."},
160 ENTRYPOINT ModeSpecOpt flow_opts =
161 {sizeof opts / sizeof opts[0], opts,
162 sizeof vars / sizeof vars[0], vars, desc};
165 ModStruct flow_description = {
166 "flow", "init_flow", "draw_flow", "release_flow",
167 "refresh_flow", "init_flow", NULL, &flow_opts,
168 1000, 1024, 10000, -10, 200, 1.0, "",
169 "Shows dynamic strange attractors", 0, NULL
174 typedef struct { double x, y, z; } dvector;
176 #define N_PARS 20 /* Enough for Full Cubic or Periodic Cubic */
177 typedef dvector Par[N_PARS];
178 enum { /* Name the parameter indices to make it easier to write
181 X,XX,XXX,XXY,XXZ,XY,XYY,XYZ,XZ,XZZ,
184 SINY = XY /* OK to overlap in this case */
187 /* Camera target [TDA] */
194 #define IX(C) ((C) * segindex + sp->cnsegs[(C)])
195 #define B(t,b) (sp->p + (t) + (b) * sp->taillen)
196 #define X(t,b) (B((t),(b))->x)
197 #define Y(t,b) (B((t),(b))->y)
198 #define Z(t,b) (B((t),(b))->z)
199 #define balance_rand(v) ((LRAND()/MAXRAND*(v))-((v)/2)) /* random around 0 */
200 #define LOST_IN_SPACE 2000.0
201 #define INITIALSTEP 0.04
202 #define EYEHEIGHT 0.005
206 /* Points that make up the box (normalized coordinates) */
207 static const double box[][3] = {
242 /* Lines connecting the box dots */
243 static const double lines[][2] = {
244 {0,1}, {1,2}, {2,3}, {3,0}, /* box */
245 {4,5}, {5,6}, {6,7}, {7,4},
246 {0,4}, {1,5}, {2,6}, {3,7},
247 {4+4,5+4}, {5+4,6+4}, {6+4,7+4}, {7+4,4+4},
248 {4+8,5+8}, {5+8,6+8}, {6+8,7+8}, {7+8,4+8},
249 {4+12,5+12}, {5+12,6+12}, {6+12,7+12}, {7+12,4+12},
250 {4+16,5+16}, {5+16,6+16}, {6+16,7+16}, {7+16,4+16},
251 {4+20,5+20}, {5+20,6+20}, {6+20,7+20}, {7+20,4+20},
252 {4+24,5+24}, {5+24,6+24}, {6+24,7+24}, {7+24,4+24},
256 /* Variables used in rendering */
257 dvector cam[3]; /* camera flight path */
260 Pixmap buffer; /* Double Buffer */
261 dvector circle[2]; /* POV that circles around the scene */
262 dvector centre; /* centre */
263 int beecount; /* number of bees */
264 XSegment *csegs; /* bee lines */
266 XSegment *old_segs; /* old bee lines */
270 /* Variables common to iterators */
271 dvector (*ODE) (Par par, double x, double y, double z);
272 dvector range; /* Initial conditions */
273 double yperiod; /* ODE's where Y is periodic. */
275 /* Variables used in iterating main flow */
277 dvector *p; /* bee positions x[time][bee#] */
281 dvector mid; /* Effective bounding box */
284 /* second set of variables, used for parallel search */
295 static flowstruct *flows = (flowstruct *) NULL;
304 /* Generic 3D Cubic Polynomial. Includes all the Quadratics (Lorentz,
305 Rossler) and much more! */
307 /* I considered offering a seperate 'Quadratic' option, since Cubic is
308 clearly overkill for the standard examples, but the performance
309 difference is too small to measure. The compute time is entirely
310 dominated by the XDrawSegments calls anyway. [TDA] */
312 Cubic(Par a, double x, double y, double z)
315 d.x = a[C].x + a[X].x*x + a[XX].x*x*x + a[XXX].x*x*x*x + a[XXY].x*x*x*y +
316 a[XXZ].x*x*x*z + a[XY].x*x*y + a[XYY].x*x*y*y + a[XYZ].x*x*y*z +
317 a[XZ].x*x*z + a[XZZ].x*x*z*z + a[Y].x*y + a[YY].x*y*y +
318 a[YYY].x*y*y*y + a[YYZ].x*y*y*z + a[YZ].x*y*z + a[YZZ].x*y*z*z +
319 a[Z].x*z + a[ZZ].x*z*z + a[ZZZ].x*z*z*z;
321 d.y = a[C].y + a[X].y*x + a[XX].y*x*x + a[XXX].y*x*x*x + a[XXY].y*x*x*y +
322 a[XXZ].y*x*x*z + a[XY].y*x*y + a[XYY].y*x*y*y + a[XYZ].y*x*y*z +
323 a[XZ].y*x*z + a[XZZ].y*x*z*z + a[Y].y*y + a[YY].y*y*y +
324 a[YYY].y*y*y*y + a[YYZ].y*y*y*z + a[YZ].y*y*z + a[YZZ].y*y*z*z +
325 a[Z].y*z + a[ZZ].y*z*z + a[ZZZ].y*z*z*z;
327 d.z = a[C].z + a[X].z*x + a[XX].z*x*x + a[XXX].z*x*x*x + a[XXY].z*x*x*y +
328 a[XXZ].z*x*x*z + a[XY].z*x*y + a[XYY].z*x*y*y + a[XYZ].z*x*y*z +
329 a[XZ].z*x*z + a[XZZ].z*x*z*z + a[Y].z*y + a[YY].z*y*y +
330 a[YYY].z*y*y*y + a[YYZ].z*y*y*z + a[YZ].z*y*z + a[YZZ].z*y*z*z +
331 a[Z].z*z + a[ZZ].z*z*z + a[ZZZ].z*z*z*z;
336 /* 3D Cubic in (x,z) with periodic sinusoidal forcing term in x. y is
337 the independent periodic (time) axis. This includes Birkhoff's
338 Bagel and Duffing's Attractor */
340 Periodic(Par a, double x, double y, double z)
344 d.x = a[C].x + a[X].x*x + a[XX].x*x*x + a[XXX].x*x*x*x +
345 a[XXZ].x*x*x*z + a[XZ].x*x*z + a[XZZ].x*x*z*z + a[Z].x*z +
346 a[ZZ].x*z*z + a[ZZZ].x*z*z*z + a[SINY].x*sin(y);
350 d.z = a[C].z + a[X].z*x + a[XX].z*x*x + a[XXX].z*x*x*x +
351 a[XXZ].z*x*x*z + a[XZ].z*x*z + a[XZZ].z*x*z*z + a[Z].z*z +
352 a[ZZ].z*z*z + a[ZZZ].z*z*z*z;
357 /* Numerical integration of the ODE using 2nd order Runge Kutta.
358 Returns length^2 of the update, so that we can detect if the step
359 size needs reducing. */
361 Iterate(dvector *p, dvector(*ODE)(Par par, double x, double y, double z),
362 Par par, double step)
366 k1 = ODE(par, p->x, p->y, p->z);
370 k2 = ODE(par, p->x + k1.x, p->y + k1.y, p->z + k1.z);
374 k3.x = (k1.x + k2.x) / 2.0;
375 k3.y = (k1.y + k2.y) / 2.0;
376 k3.z = (k1.z + k2.z) / 2.0;
382 return k3.x*k3.x + k3.y*k3.y + k3.z*k3.z;
385 /* Memory functions */
387 #define deallocate(p,t) if (p!=NULL) {free(p); p=(t*)NULL; }
388 #define allocate(p,t,s) if ((p=(t*)malloc(sizeof(t)*s))==NULL)\
389 {free_flow(sp);return;}
392 free_flow(flowstruct *sp)
394 deallocate(sp->csegs, XSegment);
395 deallocate(sp->cnsegs, int);
396 deallocate(sp->old_segs, XSegment);
397 deallocate(sp->p, dvector);
400 /* Generate Gaussian random number: mean 0, "amplitude" A (actually
401 A is 3*standard deviation). */
403 /* Note this generates a pair of gaussian variables, so it saves one
404 to give out next time it's called */
409 static Bool ready = 0;
416 x = 2.0 * (double)LRAND() / MAXRAND - 1.0;
417 y = 2.0 * (double)LRAND() / MAXRAND - 1.0;
421 w = sqrt((-2 * log(w))/w);
428 /* Attempt to discover new atractors by sending a pair of bees on a
429 fast trip through the new flow and computing their Lyapunov
430 exponent. Returns False if the bees fly away.
432 If the bees stay bounded, the new bounds and the Lyapunov exponent
433 are stored in sp and the function returns True.
435 Repeat invocations continue the flow and improve the accuracy of
436 the bounds and the Lyapunov exponent. Set sp->count2 to zero to
439 Acts on alternate variable set, so that it can be run in parallel
440 with the main flow */
443 discover(ModeInfo * mi)
449 double dl2, df, rs, lsum = 0, s, maxv2 = 0, v2;
455 sp = &flows[MI_SCREEN(mi)];
457 if(sp->count2 == 0) {
458 /* initial conditions */
459 sp->p2[0].x = Gauss_Rand(sp->range.x);
460 sp->p2[0].y = (sp->yperiod > 0)?
461 balance_rand(sp->range.y) : Gauss_Rand(sp->range.y);
462 sp->p2[0].z = Gauss_Rand(sp->range.z);
464 /* 1000 steps to find an attractor */
465 /* Most cases explode out here */
466 for(N=0; N < 1000; N++){
467 Iterate(sp->p2, sp->ODE, sp->par2, sp->step2);
468 if(sp->yperiod > 0 && sp->p2[0].y > sp->yperiod)
469 sp->p2[0].y -= sp->yperiod;
470 if(fabs(sp->p2[0].x) > LOST_IN_SPACE ||
471 fabs(sp->p2[0].y) > LOST_IN_SPACE ||
472 fabs(sp->p2[0].z) > LOST_IN_SPACE) {
477 /* Small perturbation */
478 sp->p2[1].x = sp->p2[0].x + 0.000001;
479 sp->p2[1].y = sp->p2[0].y;
480 sp->p2[1].z = sp->p2[0].z;
483 /* Reset bounding box */
484 max.x = min.x = sp->p2[0].x;
485 max.y = min.y = sp->p2[0].y;
486 max.z = min.z = sp->p2[0].z;
488 /* Compute Lyapunov Exponent */
490 /* (Technically, we're only estimating the largest Lyapunov
491 Exponent, but that's all we need to know to determine if we
492 have a strange attractor.) [TDA] */
494 /* Fly two bees close together */
495 for(N=0; N < 5000; N++){
496 for(i=0; i< 2; i++) {
497 v2 = Iterate(sp->p2+i, sp->ODE, sp->par2, sp->step2);
498 if(sp->yperiod > 0 && sp->p2[i].y > sp->yperiod)
499 sp->p2[i].y -= sp->yperiod;
501 if(fabs(sp->p2[i].x) > LOST_IN_SPACE ||
502 fabs(sp->p2[i].y) > LOST_IN_SPACE ||
503 fabs(sp->p2[i].z) > LOST_IN_SPACE) {
506 if(v2 > maxv2) maxv2 = v2; /* Track max v^2 */
509 /* find bounding box */
510 if ( sp->p2[0].x < min.x ) min.x = sp->p2[0].x;
511 else if ( sp->p2[0].x > max.x ) max.x = sp->p2[0].x;
512 if ( sp->p2[0].y < min.y ) min.y = sp->p2[0].y;
513 else if ( sp->p2[0].y > max.y ) max.y = sp->p2[0].y;
514 if ( sp->p2[0].z < min.z ) min.z = sp->p2[0].z;
515 else if ( sp->p2[0].z > max.z ) max.z = sp->p2[0].z;
517 /* Measure how much we have to pull the two bees to prevent
519 dl.x = sp->p2[1].x - sp->p2[0].x;
520 dl.y = sp->p2[1].y - sp->p2[0].y;
521 dl.z = sp->p2[1].z - sp->p2[0].z;
523 dl2 = dl.x*dl.x + dl.y*dl.y + dl.z*dl.z;
527 sp->p2[1].x = sp->p2[0].x + rs * dl.x;
528 sp->p2[1].y = sp->p2[0].y + rs * dl.y;
529 sp->p2[1].z = sp->p2[0].z + rs * dl.z;
530 lsum = lsum + log(df);
532 l = M_LOG2E / 2 * lsum / nl / sp->step2;
536 /* Anything that didn't explode has a finite attractor */
537 /* If Lyapunov is negative then it probably hit a fixed point or a
538 * limit cycle. Positive Lyapunov indicates a strange attractor. */
542 sp->size2 = max.x - min.x;
544 if(s > sp->size2) sp->size2 = s;
546 if(s > sp->size2) sp->size2 = s;
548 sp->mid2.x = (max.x + min.x) / 2;
549 sp->mid2.y = (max.y + min.y) / 2;
550 sp->mid2.z = (max.z + min.z) / 2;
552 if(sqrt(maxv2) > sp->size2 * 0.2) {
553 /* Flowing too fast, reduce step size. This
554 helps to eliminate high-speed limit cycles,
555 which can show +ve Lyapunov due to integration
562 /* Sets up initial conditions for a flow without all the extra baggage
563 that goes with init_flow */
565 restart_flow(ModeInfo * mi)
572 sp = &flows[MI_SCREEN(mi)];
575 /* Re-Initialize point positions, velocities, etc. */
576 for (b = 0; b < sp->beecount; b++) {
577 X(0, b) = Gauss_Rand(sp->range.x);
578 Y(0, b) = (sp->yperiod > 0)?
579 balance_rand(sp->range.y) : Gauss_Rand(sp->range.y);
580 Z(0, b) = Gauss_Rand(sp->range.z);
584 /* Returns true if line was behind a clip plane, or it clips the line */
585 /* nx,ny,nz is the normal to the plane. d is the distance from the origin */
586 /* s and e are the end points of the line to be clipped */
588 clip(double nx, double ny, double nz, double d, dvector *s, dvector *e)
594 front1 = (nx*s->x + ny*s->y + nz*s->z >= -d);
595 front2 = (nx*e->x + ny*e->y + nz*e->z >= -d);
596 if (!front1 && !front2) return 1;
597 if (front1 && front2) return 0;
602 /* Find t in line equation */
603 t = ( -d - nx*s->x - ny*s->y - nz*s->z) / ( nx*w.x + ny*w.y + nz*w.z);
605 p.x = s->x + w.x * t;
606 p.y = s->y + w.y * t;
607 p.z = s->z + w.z * t;
609 /* Move clipped point to the intersection */
623 init_flow (ModeInfo * mi)
629 if ((flows = (flowstruct *) calloc(MI_NUM_SCREENS(mi),
630 sizeof (flowstruct))) == NULL)
633 sp = &flows[MI_SCREEN(mi)];
637 sp->taillen = MI_SIZE(mi);
638 if (sp->taillen < -MINTRAIL) {
639 /* Change by sqrt so it seems more variable */
640 sp->taillen = NRAND((int)sqrt((double) (-sp->taillen - MINTRAIL + 1)));
641 sp->taillen = sp->taillen * sp->taillen + MINTRAIL;
642 } else if (sp->taillen < MINTRAIL) {
643 sp->taillen = MINTRAIL;
646 if(!rotatep && !ridep) rotatep = True; /* We need at least one viewpoint */
648 /* Start camera at Orbit or Bee */
654 sp->chasetime = 1; /* Go directly to target */
658 sp->step2 = INITIALSTEP;
660 /* Zero parameter set */
661 memset(sp->par2, 0, N_PARS * sizeof(dvector));
663 /* Set up standard examples */
664 switch (NRAND((periodicp) ? 5 : 3)) {
672 sp->par2[Y].x = 10 + balance_rand(5*0); /* a */
673 sp->par2[X].x = - sp->par2[Y].x; /* -a */
674 sp->par2[X].y = 28 + balance_rand(5*0); /* b */
678 sp->par2[Z].z = - 2 + balance_rand(1*0); /* -c */
688 sp->par2[Z].x = -2 + balance_rand(1); /* a */
690 sp->par2[Y].y = 0.2 + balance_rand(0.1); /* b */
691 sp->par2[C].z = 0.2 + balance_rand(0.1); /* c */
693 sp->par2[Z].z = -5.7;
699 z' = 0.2 + z(x - 5.7)
701 name = "RosslerCone";
703 sp->par2[Z].x = -2; /* a */
705 sp->par2[Y].y = 0.2; /* b */
706 sp->par2[ZZ].y = -0.331 + balance_rand(0.01); /* c */
709 sp->par2[Z].z = -5.7;
715 z' = 0.7x + az(0.1 - x^2)
719 sp->par2[SINY].x = 0.35 + balance_rand(0.25); /* b */
720 sp->par2[C].y = 1.57; /* c */
722 sp->par2[Z].z = 1 + balance_rand(0.5); /* a/10 */
723 sp->par2[XXZ].z = -10 * sp->par2[Z].z; /* -a */
724 sp->yperiod = 2 * M_PI;
728 x' = -ax - z/2 - z^3/8 + b sin(y)
733 sp->par2[X].x = -0.2 + balance_rand(0.1); /* a */
734 sp->par2[Z].x = -0.5;
735 sp->par2[ZZZ].x = -0.125;
736 sp->par2[SINY].x = 27.0 + balance_rand(3.0); /* b */
737 sp->par2[C].y = 1.33; /* c */
739 sp->yperiod = 2 * M_PI;
747 if(sp->yperiod > 0) {
749 /* periodic flows show either uniform distribution or a
750 snapshot on the 'time' axis */
751 sp->range.y = NRAND(2)? sp->yperiod : 0;
757 /* Run discoverer to set up bounding box, etc. Lyapunov will
758 probably be innaccurate, since we're only running it once, but
759 we're using known strange attractors so it should be ok. */
761 if(MI_IS_VERBOSE(mi))
763 "flow: Lyapunov exponent: %g, step: %g, size: %g (%s)\n",
764 sp->lyap2, sp->step2, sp->size2, name);
765 /* Install new params */
766 sp->lyap = sp->lyap2;
767 sp->size = sp->size2;
769 sp->step = sp->step2;
770 memcpy(sp->par, sp->par2, sizeof(sp->par2));
772 sp->count2 = 0; /* Reset search */
775 sp->beecount = MI_COUNT(mi);
776 if (sp->beecount < 0) { /* random variations */
777 sp->beecount = NRAND(-sp->beecount) + 1; /* Minimum 1 */
780 # ifdef HAVE_COCOA /* Don't second-guess Quartz's double-buffering */
784 if(dbufp) { /* Set up double buffer */
785 if (sp->buffer != None)
786 XFreePixmap(MI_DISPLAY(mi), sp->buffer);
787 sp->buffer = XCreatePixmap(MI_DISPLAY(mi), MI_WINDOW(mi),
788 MI_WIDTH(mi), MI_HEIGHT(mi), MI_DEPTH(mi));
790 sp->buffer = MI_WINDOW(mi);
792 /* no "NoExpose" events from XCopyArea wanted */
793 XSetGraphicsExposures(MI_DISPLAY(mi), MI_GC(mi), False);
795 /* Make sure we're using 'thin' lines */
796 XSetLineAttributes(MI_DISPLAY(mi), MI_GC(mi), 0, LineSolid, CapNotLast,
799 /* Clear the background (may be slow depending on user prefs). */
802 /* Allocate memory. */
803 if (sp->csegs == NULL) {
804 allocate(sp->csegs, XSegment,
805 (sp->beecount + BOX_L) * MI_NPIXELS(mi) * sp->taillen);
806 allocate(sp->cnsegs, int, MI_NPIXELS(mi));
807 allocate(sp->old_segs, XSegment, sp->beecount * sp->taillen);
808 allocate(sp->p, dvector, sp->beecount * sp->taillen);
811 /* Initialize point positions, velocities, etc. */
814 /* Set up camera tail */
815 X(1, 0) = sp->cam[1].x = 0;
816 Y(1, 0) = sp->cam[1].y = 0;
817 Z(1, 0) = sp->cam[1].z = 0;
821 draw_flow (ModeInfo * mi)
825 double M[3][3]; /* transformation matrix */
826 flowstruct *sp = NULL;
832 sp = &flows[MI_SCREEN(mi)];
833 if (sp->csegs == NULL)
836 #ifdef HAVE_COCOA /* Don't second-guess Quartz's double-buffering */
837 XClearWindow (MI_DISPLAY(mi), MI_WINDOW(mi));
840 /* multiplier for indexing segment arrays. Used in IX macro, etc. */
841 segindex = (sp->beecount + BOX_L) * sp->taillen;
844 if(sp->count2 == 0) { /* start new search */
845 sp->step2 = INITIALSTEP;
846 /* Pick random parameters. Actual range is irrelevant
847 since parameter scale determines flow speed but not
849 for(i=0; i< N_PARS; i++) {
850 sp->par2[i].x = Gauss_Rand(1.0);
851 sp->par2[i].y = Gauss_Rand(1.0);
852 sp->par2[i].z = Gauss_Rand(1.0);
855 if(!discover(mi)) { /* Flow exploded, reset. */
859 sp->count2 = 0; /* Attractor found, but it's not strange */
860 }else if(sp->count2 > 1000000) { /* This one will do */
861 sp->count2 = 0; /* Reset search */
862 if(MI_IS_VERBOSE(mi))
864 "flow: Lyapunov exponent: %g, step: %g, size: %g (unnamed)\n",
865 sp->lyap2, sp->step2, sp->size2);
866 /* Install new params */
867 sp->lyap = sp->lyap2;
868 sp->size = sp->size2;
870 sp->step = sp->step2;
871 memcpy(sp->par, sp->par2, sizeof(sp->par2));
873 /* If we're allowed to zoom out, do so now, so that we
874 get a look at the new attractor. */
875 if(sp->chaseto == BEE && rotatep) {
879 /* Reset initial conditions, so we don't get
880 misleading artifacts in the particle density. */
886 /* Reset segment buffers */
887 for (col = 0; col < MI_NPIXELS(mi); col++)
890 MI_IS_DRAWN(mi) = True;
892 /* Calculate circling POV [Chalky]*/
893 sp->circle[1] = sp->circle[0];
894 sp->circle[0].x = sp->size * 2 * sin(sp->count / 100.0) *
895 (-0.6 + 0.4 *cos(sp->count / 500.0)) + sp->mid.x;
896 sp->circle[0].y = sp->size * 2 * cos(sp->count / 100.0) *
897 (0.6 + 0.4 *cos(sp->count / 500.0)) + sp->mid.y;
898 sp->circle[0].z = sp->size * 2 * sin(sp->count / 421.0) + sp->mid.z;
900 /* Timed chase instead of Chalkie's Bistable oscillator [TDA] */
901 if(rotatep && ridep) {
902 if(sp->chaseto == BEE && NRAND(1000) == 0){
905 }else if(NRAND(4000) == 0){
911 /* Set up orientation matrix */
913 double x[3], p[3], x2=0, xp=0;
916 /* Chasetime is here to guarantee the camera makes it all the
917 way to the target in a finite number of steps. */
918 if(sp->chasetime > 1)
921 if(sp->chaseto == BEE){
922 /* Camera Head targets bee 0 */
923 sp->cam[0].x += (X(0, 0) - sp->cam[0].x)/sp->chasetime;
924 sp->cam[0].y += (Y(0, 0) - sp->cam[0].y)/sp->chasetime;
925 sp->cam[0].z += (Z(0, 0) - sp->cam[0].z)/sp->chasetime;
927 /* Camera Tail targets previous position of bee 0 */
928 sp->cam[1].x += (X(1, 0) - sp->cam[1].x)/sp->chasetime;
929 sp->cam[1].y += (Y(1, 0) - sp->cam[1].y)/sp->chasetime;
930 sp->cam[1].z += (Z(1, 0) - sp->cam[1].z)/sp->chasetime;
932 /* Camera Wing targets bee 1 */
933 sp->cam[2].x += (X(0, 1) - sp->cam[2].x)/sp->chasetime;
934 sp->cam[2].y += (Y(0, 1) - sp->cam[2].y)/sp->chasetime;
935 sp->cam[2].z += (Z(0, 1) - sp->cam[2].z)/sp->chasetime;
937 /* Camera Head targets Orbiter */
938 sp->cam[0].x += (sp->circle[0].x - sp->cam[0].x)/sp->chasetime;
939 sp->cam[0].y += (sp->circle[0].y - sp->cam[0].y)/sp->chasetime;
940 sp->cam[0].z += (sp->circle[0].z - sp->cam[0].z)/sp->chasetime;
942 /* Camera Tail targets diametrically opposite the middle
943 of the bounding box from the Orbiter */
945 (2*sp->circle[0].x - sp->mid.x - sp->cam[1].x)/sp->chasetime;
947 (2*sp->circle[0].y - sp->mid.y - sp->cam[1].y)/sp->chasetime;
949 (2*sp->circle[0].z - sp->mid.z - sp->cam[1].z)/sp->chasetime;
950 /* Camera Wing targets previous position of Orbiter */
951 sp->cam[2].x += (sp->circle[1].x - sp->cam[2].x)/sp->chasetime;
952 sp->cam[2].y += (sp->circle[1].y - sp->cam[2].y)/sp->chasetime;
953 sp->cam[2].z += (sp->circle[1].z - sp->cam[2].z)/sp->chasetime;
956 /* Viewpoint from Tail of camera */
957 sp->centre.x=sp->cam[1].x;
958 sp->centre.y=sp->cam[1].y;
959 sp->centre.z=sp->cam[1].z;
962 x[0] = sp->cam[0].x - sp->cam[1].x;
963 x[1] = sp->cam[0].y - sp->cam[1].y;
964 x[2] = sp->cam[0].z - sp->cam[1].z;
967 p[0] = sp->cam[2].x - sp->cam[1].x;
968 p[1] = sp->cam[2].y - sp->cam[1].y;
969 p[2] = sp->cam[2].z - sp->cam[1].z;
972 /* So long as X and P don't collide, these can be used to form
973 three mutually othogonal axes: X, (X x P) x X and X x P.
974 After being normalised to unit length, these form the
975 Orientation Matrix. */
978 x2+= x[i]*x[i]; /* X . X */
979 xp+= x[i]*p[i]; /* X . P */
980 M[0][i] = x[i]; /* X */
983 for(i=0; i<3; i++) /* (X x P) x X */
984 M[1][i] = x2*p[i] - xp*x[i]; /* == (X . X) P - (X . P) X */
986 M[2][0] = x[1]*p[2] - x[2]*p[1]; /* X x P */
987 M[2][1] = -x[0]*p[2] + x[2]*p[0];
988 M[2][2] = x[0]*p[1] - x[1]*p[0];
993 for(i=0; i<3; i++) A+=M[j][i]*M[j][i]; /* sum squares */
996 for(i=0; i<3; i++) M[j][i]/=A;
999 if(sp->chaseto == BEE) {
1000 X(0, 1)=X(0, 0)+M[1][0]*sp->step; /* adjust neighbour */
1001 Y(0, 1)=Y(0, 0)+M[1][1]*sp->step;
1002 Z(0, 1)=Z(0, 0)+M[1][2]*sp->step;
1006 /* <=- Bounding Box -=> */
1008 for (b = 0; b < BOX_L; b++) {
1010 /* Chalky's clipping code, Only used for the box */
1011 /* clipping trails is slow and of little benefit. [TDA] */
1012 int p1 = lines[b][0];
1013 int p2 = lines[b][1];
1015 double x1=box[p1][0]* sp->size/2 + sp->mid.x - sp->centre.x;
1016 double y1=box[p1][1]* sp->size/2 + sp->mid.y - sp->centre.y;
1017 double z1=box[p1][2]* sp->size/2 + sp->mid.z - sp->centre.z;
1018 double x2=box[p2][0]* sp->size/2 + sp->mid.x - sp->centre.x;
1019 double y2=box[p2][1]* sp->size/2 + sp->mid.y - sp->centre.y;
1020 double z2=box[p2][2]* sp->size/2 + sp->mid.z - sp->centre.z;
1022 A1.x=M[0][0]*x1 + M[0][1]*y1 + M[0][2]*z1;
1023 A1.y=M[1][0]*x1 + M[1][1]*y1 + M[1][2]*z1;
1024 A1.z=M[2][0]*x1 + M[2][1]*y1 + M[2][2]*z1 + EYEHEIGHT * sp->size;
1025 A2.x=M[0][0]*x2 + M[0][1]*y2 + M[0][2]*z2;
1026 A2.y=M[1][0]*x2 + M[1][1]*y2 + M[1][2]*z2;
1027 A2.z=M[2][0]*x2 + M[2][1]*y2 + M[2][2]*z2 + EYEHEIGHT * sp->size;
1029 /* Clip in 3D before projecting down to 2D. A 2D clip
1030 after projection wouldn't be able to handle lines that
1032 if (clip(1, 0, 0,-1, &A1, &A2) || /* Screen */
1033 clip(1, 2, 0, 0, &A1, &A2) || /* Left */
1034 clip(1,-2, 0, 0, &A1, &A2) || /* Right */
1035 clip(1,0, 2.0*MI_WIDTH(mi)/MI_HEIGHT(mi), 0, &A1, &A2)||/*UP*/
1036 clip(1,0,-2.0*MI_WIDTH(mi)/MI_HEIGHT(mi), 0, &A1, &A2))/*Down*/
1039 /* Colour according to bee */
1040 col = b % (MI_NPIXELS(mi) - 1);
1042 sp->csegs[IX(col)].x1 = MI_WIDTH(mi)/2 + MI_WIDTH(mi) * A1.y/A1.x;
1043 sp->csegs[IX(col)].y1 = MI_HEIGHT(mi)/2 + MI_WIDTH(mi) * A1.z/A1.x;
1044 sp->csegs[IX(col)].x2 = MI_WIDTH(mi)/2 + MI_WIDTH(mi) * A2.y/A2.x;
1045 sp->csegs[IX(col)].y2 = MI_HEIGHT(mi)/2 + MI_WIDTH(mi) * A2.z/A2.x;
1051 for (b = 0; b < sp->beecount; b++) {
1052 if(fabs(X(0, b)) > LOST_IN_SPACE ||
1053 fabs(Y(0, b)) > LOST_IN_SPACE ||
1054 fabs(Z(0, b)) > LOST_IN_SPACE){
1055 if(sp->chaseto == BEE && b == 0){
1056 /* Lost camera bee. Need to replace it since
1057 rerunning init_flow could lose us a hard-won new
1058 attractor. Try moving it very close to a random
1059 other bee. This way we have a good chance of being
1060 close to the attractor and not forming a false
1061 artifact. If we've lost many bees this may need to
1063 /* Don't worry about camera wingbee. It stays close
1064 to the main camera bee no matter what happens. */
1065 int newb = 1 + NRAND(sp->beecount - 1);
1066 X(0, 0) = X(0, newb) + 0.001;
1067 Y(0, 0) = Y(0, newb);
1068 Z(0, 0) = Z(0, newb);
1069 if(MI_IS_VERBOSE(mi))
1071 "flow: resetting lost camera near bee %d\n",
1077 /* Age the tail. It's critical this be fast since
1078 beecount*taillen can be large. */
1079 memmove(B(1, b), B(0, b), (sp->taillen - 1) * sizeof(dvector));
1081 Iterate(B(0,b), sp->ODE, sp->par, sp->step);
1083 /* Don't show wingbee since he's not quite in the flow. */
1084 if(sp->chaseto == BEE && b == 1) continue;
1086 /* Colour according to bee */
1087 col = b % (MI_NPIXELS(mi) - 1);
1089 /* Fill the segment lists. */
1091 begin = 0; /* begin new trail */
1092 end = MIN(sp->taillen, sp->count); /* short trails at first */
1093 for(i=0; i < end; i++){
1094 double x = X(i,b)-sp->centre.x;
1095 double y = Y(i,b)*(sp->yperiod < 0? (sp->size/sp->yperiod) :1)
1097 double z = Z(i,b)-sp->centre.z;
1098 double XM=M[0][0]*x + M[0][1]*y + M[0][2]*z;
1099 double YM=M[1][0]*x + M[1][1]*y + M[1][2]*z;
1100 double ZM=M[2][0]*x + M[2][1]*y + M[2][2]*z + EYEHEIGHT * sp->size;
1103 swarm++; /* count the remaining bees */
1104 if(sp->yperiod > 0 && Y(i,b) > sp->yperiod){
1106 Y(i,b) -= sp->yperiod;
1107 /* hide tail to prevent streaks in Y. Streaks in X,Z
1108 are ok, they help to outline the Poincare'
1110 for(j = i; j < end; j++) Y(j,b) = Y(i,b);
1115 if(XM <= 0){ /* off screen - new trail */
1119 absx = MI_WIDTH(mi)/2 + MI_WIDTH(mi) * YM/XM;
1120 absy = MI_HEIGHT(mi)/2 + MI_WIDTH(mi) * ZM/XM;
1121 /* Performance bottleneck */
1122 if(absx <= 0 || absx >= MI_WIDTH(mi) ||
1123 absy <= 0 || absy >= MI_HEIGHT(mi)) {
1124 /* off screen - new trail */
1128 if(i > begin) { /* complete previous segment */
1129 sp->csegs[IX(col)].x2 = absx;
1130 sp->csegs[IX(col)].y2 = absy;
1134 if(i < end -1){ /* start new segment */
1135 sp->csegs[IX(col)].x1 = absx;
1136 sp->csegs[IX(col)].y1 = absy;
1142 XSetForeground(MI_DISPLAY(mi), MI_GC(mi), MI_BLACK_PIXEL(mi));
1143 if (dbufp) { /* In Double Buffer case, prepare off-screen copy */
1144 /* For slow systems, this can be the single biggest bottleneck
1145 in the program. These systems may be better of not using
1146 the double buffer. */
1147 XFillRectangle(MI_DISPLAY(mi), sp->buffer, MI_GC(mi), 0, 0,
1148 MI_WIDTH(mi), MI_HEIGHT(mi));
1149 } else { /* Otherwise, erase previous segment list directly */
1150 XDrawSegments(MI_DISPLAY(mi), sp->buffer, MI_GC(mi),
1151 sp->old_segs, sp->nold_segs);
1155 if (MI_NPIXELS(mi) > 2){ /* colour */
1157 for (col = 0; col < MI_NPIXELS(mi) - 1; col++)
1158 if (sp->cnsegs[col] > 0) {
1159 if(sp->cnsegs[col] > mn) mn = sp->cnsegs[col];
1160 XSetForeground(MI_DISPLAY(mi), MI_GC(mi), MI_PIXEL(mi, col+1));
1161 /* This is usually the biggest bottleneck on most
1162 systems. The maths load is insignificant compared
1164 XDrawSegments(MI_DISPLAY(mi), sp->buffer, MI_GC(mi),
1165 sp->csegs + col * segindex, sp->cnsegs[col]);
1167 } else { /* mono handled seperately since xlockmore uses '1' for
1169 XSetForeground(MI_DISPLAY(mi), MI_GC(mi), MI_WHITE_PIXEL(mi));
1170 XDrawSegments(MI_DISPLAY(mi), sp->buffer, MI_GC(mi),
1171 sp->csegs, sp->cnsegs[0]);
1173 if (dbufp) { /* In Double Buffer case, this updates the screen */
1174 XCopyArea(MI_DISPLAY(mi), sp->buffer, MI_WINDOW(mi), MI_GC(mi), 0, 0,
1175 MI_WIDTH(mi), MI_HEIGHT(mi), 0, 0);
1176 } else { /* Otherwise, screen is already updated. Copy segments
1177 to erase-list to be erased directly next time. */
1179 for (col = 0; col < MI_NPIXELS(mi) - 1; col++) {
1180 memcpy(sp->old_segs + c, sp->csegs + col * segindex,
1181 sp->cnsegs[col] * sizeof(XSegment));
1182 c += sp->cnsegs[col];
1187 if(sp->count > 1 && swarm == 0) { /* all gone */
1188 if(MI_IS_VERBOSE(mi))
1189 fprintf(stdout, "flow: all gone at %d\n", sp->count);
1193 if(sp->count++ > MI_CYCLES(mi)){ /* Time's up. If we haven't
1194 found anything new by now we
1195 should pick a new standard
1202 reshape_flow(ModeInfo * mi, int width, int height)
1209 release_flow (ModeInfo * mi)
1211 if (flows != NULL) {
1214 for (screen = 0; screen < MI_NUM_SCREENS(mi); screen++)
1215 free_flow(&flows[screen]);
1217 flows = (flowstruct *) NULL;
1222 refresh_flow (ModeInfo * mi)
1224 if(!dbufp) MI_CLEARWINDOW(mi);
1227 XSCREENSAVER_MODULE ("Flow", flow)
1229 #endif /* MODE_flow */