2 * We try to find an optimal triangle grid
9 #include <gsl/gsl_errno.h>
10 #include <gsl/gsl_multimin.h>
13 #define INITIAL_F "initial"
15 static double edgewise_vertex_displacement_cost(const Vertices vertices);
16 static double noncircular_rim_cost(const Vertices vertices);
18 static void compute_vertex_areas(const Vertices vertices, double areas[N]);
19 static double best_energy= DBL_MAX;
21 static void cost(double *energy, double tweight, double tcost);
22 #define COST(weight, compute) cost(&energy, (weight), (compute))
24 /*---------- main energy computation and subroutines ----------*/
26 static double compute_energy(const Vertices vertices) {
27 double vertex_areas[N], energy;
29 compute_vertex_areas(vertices,vertex_areas);
31 printf("cost > energy |");
33 COST(1000.0, edgewise_vertex_displacement_cost(vertices));
34 COST(1.0, graph_layout_cost(vertices,vertex_areas));
35 COST(1e3, noncircular_rim_cost(vertices));
37 printf("| total %# e |", energy);
38 if (energy < best_energy) {
44 best_f= fopen(BEST_F ".new","wb"); if (!best_f) diee("fopen new best");
45 r= fwrite(vertices,sizeof(vertices),1,best_f); if (r!=1) diee("fwrite");
46 if (fclose(best_f)) diee("fclose new best");
47 if (rename(BEST_F ".new", BEST_F)) diee("rename install new best");
55 static void cost(double *energy, double tweight, double tcost) {
56 double tenergy= tweight * tcost;
57 printf(" %# e > %# e |", tcost, tenergy);
61 static void compute_vertex_areas(const Vertices vertices, double areas[N]) {
62 int v0,v1,v2, e1,e2, k;
73 double e1v[D3], e2v[D3], av[D3];
75 e1v[k]= vertices[v1][k] - vertices[v0][k];
76 e2v[k]= vertices[v2][k] - vertices[v0][k];
82 areas[v0]= total / count;
86 /*---------- use of GSL ----------*/
88 /* We want to do multidimensional minimisation.
90 * We don't think there are any local minima. Or at least, if there
91 * are, the local minimum which will be found from the starting
92 * state is the one we want.
94 * We don't want to try to provide a derivative of the cost
95 * function. That's too tedious (and anyway the polynomial
96 * approximation to our our cost function sometimes has high degree
97 * in the inputs which means the quadratic model implied by most of
98 * the gradient descent minimisers is not ideal).
100 * This eliminates most of the algorithms. Nelder and Mead's
101 * simplex algorithm is still available and we will try that.
103 * In our application we are searching for the optimal locations of
104 * N actualvertices in D3 (3) dimensions - ie, we are searching for
105 * the optimal metapoint in an N*D3-dimensional space.
107 * So eg with X=Y=100, the simplex will contain 300 metavertices
108 * each of which is an array of 300 doubles for the actualvertex
109 * coordinates. Hopefully this won't be too slow ...
112 static gsl_multimin_fminimizer *minimiser;
114 static const double stop_epsilon= 1e-4;
116 static double minfunc_f(const gsl_vector *x, void *params) {
117 assert(x->size == DIM);
118 assert(x->stride == 1);
119 return compute_energy((const double(*)[D3])x->data);
122 int main(int argc, const char *const *argv) {
123 gsl_multimin_function multimin_function;
125 Vertices initial, step_size;
127 gsl_vector initial_gsl, step_size_gsl;
130 if (argc>1) { fputs("takes no arguments\n",stderr); exit(8); }
132 minimiser= gsl_multimin_fminimizer_alloc
133 (gsl_multimin_fminimizer_nmsimplex, DIM);
134 if (!minimiser) { perror("alloc minimiser"); exit(-1); }
136 multimin_function.f= minfunc_f;
137 multimin_function.n= DIM;
138 multimin_function.params= 0;
140 initial_f= fopen(INITIAL_F,"rb"); if (!initial_f) diee("fopen initial");
141 errno= 0; r= fread(initial,sizeof(initial),1,initial_f);
142 if (r!=1) diee("fread");
145 initial_gsl.size= DIM;
146 initial_gsl.stride= 1;
147 initial_gsl.block= 0;
148 initial_gsl.owner= 0;
149 step_size_gsl= initial_gsl;
151 initial_gsl.data= (double*)initial;
152 step_size_gsl.data= (double*)step_size;
155 K step_size[v][k]= 1e-3;
156 FOR_RIM_VERTEX(vx,vy,v)
157 step_size[v][3] *= 0.1;
159 GA( gsl_multimin_fminimizer_set(minimiser, &multimin_function,
160 &initial_gsl, &step_size_gsl) );
163 GA( gsl_multimin_fminimizer_iterate(minimiser) );
165 size= gsl_multimin_fminimizer_size(minimiser);
166 r= gsl_multimin_test_size(size, stop_epsilon);
168 printf("size %# e, r=%d\n", size, r);
171 if (r==GSL_SUCCESS) break;
172 assert(r==GSL_CONTINUE);
177 /*---------- Edgewise vertex displacement ----------*/
197 * Find R', the `expected' location of R, by
198 * reflecting S in M (the midpoint of QP).
204 * Giving energy contribution:
212 * (The dimensions of this are those of F_vd.)
214 * By symmetry, this calculation gives the same answer with R and S
215 * exchanged. Looking at the projection in the RMS plane:
221 * R' ,' 2d" = |SS'| = |RR'| = 2d
223 * `-._ ,' By congruent triangles,
224 * ` M with M' = midpoint of RS,
225 * ,' `-._ |MM'| = |RR'|/2 = d
228 * ,' M' _ , - ' d = |MM'|
232 * We choose this value for l (rather than |RM|+|MS|, say, or |RM|)
233 * because we want this symmetry and because we're happy to punish
234 * bending more than uneveness in the metric.
236 * In practice to avoid division by zero we'll add epsilon to l^3
237 * and the huge energy ought then to be sufficient for the model to
238 * avoid being close to R=S.
241 static double edgewise_vertex_displacement_cost(const Vertices vertices) {
242 static const double l3_epsilon= 1e-6;
244 int pi,e,qi,ri,si, k;
245 double m[D3], mprime[D3], b, d2, l, sigma_bd2_l3=0;
248 ri= EDGE_END2(pi,(e+1)%V6); if (ri<0) continue;
249 si= EDGE_END2(pi,(e+5)%V6); if (si<0) continue;
250 assert(ri == EDGE_END2(qi,(e+2)%V6));
251 assert(si == EDGE_END2(qi,(e+4)%V6));
253 K m[k]= (vertices[pi][k] + vertices[qi][k]) * 0.5;
254 K mprime[k]= (vertices[ri][k] + vertices[si][k]) * 0.5;
255 b= hypotD(vertices[pi], vertices[qi]);
256 d2= hypotD2(m, mprime);
257 l= hypotD(vertices[ri], vertices[si]);
258 double l3 = l*l*l + l3_epsilon;
260 sigma_bd2_l3 += b * d2 / l3;
265 /*---------- noncircular rim cost ----------*/
267 static double noncircular_rim_cost(const Vertices vertices) {
271 FOR_RIM_VERTEX(vy,vx,v) {
273 /* By symmetry, nearest point on circle is the one with
274 * the same angle subtended at the z axis. */
275 oncircle[0]= vertices[v][0];
276 oncircle[1]= vertices[v][1];
278 double mult= 1.0/ magnD(oncircle);
281 double d2= hypotD2(vertices[v], oncircle);