2 * We try to find an optimal triangle grid
10 #define INITIAL_F "initial"
12 static double edgewise_vertex_displacement_cost(const Vertices vertices);
13 static double noncircular_rim_cost(Vertices vertices);
15 static void compute_vertex_areas(const Vertices vertices, double areas[N]);
16 static double best_energy= DBL_MAX;
17 static void flushoutput(void);
19 static void cost(double *energy, double tweight, double tcost);
20 #define COST(weight, compute) cost(&energy, (weight), (compute))
22 /*---------- main energy computation and subroutines ----------*/
24 static double compute_energy(Vertices vertices) {
25 double vertex_areas[N], energy;
27 compute_vertex_areas(vertices,vertex_areas);
29 printf("cost > energy |");
31 COST(1000.0, edgewise_vertex_displacement_cost(vertices));
32 COST(1.0, graph_layout_cost(vertices,vertex_areas));
33 COST(1e3, noncircular_rim_cost(vertices));
35 printf("| total %# e |", energy);
36 if (energy < best_energy) {
40 best_f= fopen(BEST_F ".new","wb"); if (!best_f) diee("fopen new best");
41 r= fwrite(vertices,sizeof(vertices),1,best_f); if (r!=1) diee("fwrite");
42 if (fclose(best_f)) diee("fclose new best");
43 if (rename(BEST_F ".new", BEST_F)) diee("rename install new best");
51 static void cost(double *energy, double tweight, double tcost) {
52 double tenergy= tweight * tcost;
53 printf(" %# e > %# e |", tcost, tenergy);
57 static void flushoutput(void) {
58 if (fflush(stdout) || ferror(stdout)) { perror("stdout"); exit(-1); }
61 static void compute_vertex_areas(const Vertices vertices, double areas[N]) {
71 double e1v[D3], e2v[D3], av[D3];
73 e1v[k]= vertices[v1][k] - vertices[v0][k];
74 e2v[k]= vertices[v2][k] - vertices[v0][k];
80 areas[v0]= total / count;
84 /*---------- use of GSL ----------*/
86 /* We want to do multidimensional minimisation.
88 * We don't think there are any local minima. Or at least, if there
89 * are, the local minimum which will be found from the starting
90 * state is the one we want.
92 * We don't want to try to provide a derivative of the cost
93 * function. That's too tedious (and anyway the polynomial
94 * approximation to our our cost function sometimes has high degree
95 * in the inputs which means the quadratic model implied by most of
96 * the gradient descent minimisers is not ideal).
98 * This eliminates most of the algorithms. Nelder and Mead's
99 * simplex algorithm is still available and we will try that.
101 * In our application we are searching for the optimal locations of
102 * N actualvertices in D3 (3) dimensions - ie, we are searching for
103 * the optimal metapoint in an N*D3-dimensional space.
105 * So eg with X=Y=100, the simplex will contain 300 metavertices
106 * each of which is an array of 300 doubles for the actualvertex
107 * coordinates. Hopefully this won't be too slow ...
110 static void gsldie(const char *what, int status) {
111 fprintf(stderr,"gsl function failed: %s: %s\n", what, gsl_strerror(status));
115 static gsl_multimin_fminimizer *minimiser;
117 static const stop_epsilon= 1e-4;
121 static double minfunc_f(const gsl_vector *x, void *params) {
122 assert(x->size == DIM);
123 assert(x->stride == 1);
124 return compute_energy((Vertices)x->data);
127 int main(int argc, const char *const *argv) {
128 struct gsl_multimin_function multimin_function;
130 Vertices initial, step_size;
132 gsl_vector initial_gsl, step_size_gsl;
135 if (argc>1) { fputs("takes no arguments\n",stderr); exit(8); }
137 minimiser= gsl_multimin_fminimizer_alloc
138 (gsl_multimin_fminimizer_nmsimplex, DIM);
139 if (!minimiser) { perror("alloc minimiser"); exit(-1); }
141 multimin_function.f= minfunc_f;
142 multimin_function.n= DIM;
143 multimin_function.params= 0;
145 initial_f= fopen(INITIAL_F,"rb"); if (!initial_f) diee("fopen initial");
146 errno= 0; r= fread(initial,sizeof(initial),1,initial_f);
147 if (r!=1) diee("fread");
150 initial_gsl.size= DIM;
151 initial_gsl.stride= 1;
152 initial_gsl.block= 0;
153 initial_gsl.owner= 0;
154 step_size_gsl= initial_gsl;
156 initial_gsl.data= initial;
157 step_size_gsl.data= step_size;
160 K step_size[v][k]= 1e-3;
161 FOR_RIM_VERTEX(vx,vy,v)
162 step_size[v][3] *= 0.1;
164 for (vy=0; vy<Y; vy+=Y-1)
166 for (i=0; i<DIM; i++) step_size[i]= step_size;
169 step_size= gsl_vector_alloc(DIM); if (!step_size) gsldie("alloc step");
170 gsl_vector_set_all(step_size, 1e-3);
175 r= gsl_multimin_fminimizer_set(minimiser, &multimin_function,
176 &initial_gsl, &step_size);
177 if (r) { gsldie("fminimizer_set",r); }
180 r= gsl_multimin_fminimizer_iterate(minimiser);
181 if (r) { gsldie("fminimizer_iterate",r); }
183 size= gsl_multimin_fminimizer_size(minimiser);
184 r= gsl_multimin_test_size(size, stop_epsilon);
186 printf("size %# e, r=%d\n", size, r);
189 if (r==GSL_SUCCESS) break;
190 assert(r==GSL_CONTINUE);
194 /*---------- Edgewise vertex displacement ----------*/
214 * Find R', the `expected' location of R, by
215 * reflecting S in M (the midpoint of QP).
221 * Giving energy contribution:
229 * (The dimensions of this are those of F_vd.)
231 * By symmetry, this calculation gives the same answer with R and S
232 * exchanged. Looking at the projection in the RMS plane:
238 * R' ,' 2d" = |SS'| = |RR'| = 2d
240 * `-._ ,' By congruent triangles,
241 * ` M with M' = midpoint of RS,
242 * ,' `-._ |MM'| = |RR'|/2 = d
245 * ,' M' _ , - ' d = |MM'|
249 * We choose this value for l (rather than |RM|+|MS|, say, or |RM|)
250 * because we want this symmetry and because we're happy to punish
251 * bending more than uneveness in the metric.
253 * In practice to avoid division by zero we'll add epsilon to l^3
254 * and the huge energy ought then to be sufficient for the model to
255 * avoid being close to R=S.
258 static double edgewise_vertex_displacement_cost(const Vertices vertices) {
259 static const l3_epsison= 1e-6;
261 int pi,e,qi,ri,si, k;
262 double m[D3], mprime[D3], b, d2, l, sigma_bd2_l3;
265 ri= EDGE_END2(pi,(e+1)%V6); if (r<0) continue;
266 si= EDGE_END2(pi,(e+5)%V6); if (s<0) continue;
267 assert(ri == EDGE_END2(qi,(e+2)%V6));
268 assert(si == EDGE_END2(qi,(e+4)%V6));
270 K m[k]= (vertices[pi][k] + vertices[qi][k]) * 0.5;
271 K mprime[k]= (vertices[ri][k] + vertices[si][k]) * 0.5;
272 b= hypotD(vertices[pi], vertices[qi]);
273 d2= hypotD2(m, mprime);
274 l= hypotD(vertices[ri][k] - vertices[si][k]);
275 l3 = l*l*l + l3_epsilon;
277 sigma_bd2_l3 += b * d2 / l3;
282 /*---------- noncircular rim cost ----------*/
284 static double noncircular_rim_cost(Vertices vertices) {
288 FOR_RIM_VERTEX(vy,vx,v) {
290 /* By symmetry, nearest point on circle is the one with
291 * the same angle subtended at the z axis. */
292 oncircle[0]= vertices[v][0];
293 oncircle[1]= vertices[v][1];
295 double mult= 1.0/ magnD(oncircle);
298 double d2= hypotD2(vertices[v], oncircle);