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 addcost(double *energy, double tweight, double tcost);
22 #define COST(weight, compute) addcost(&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(1e4, edgewise_vertex_displacement_cost(vertices));
34 COST(1e2, graph_layout_cost(vertices,vertex_areas));
35 COST(1e4, 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");
57 static void addcost(double *energy, double tweight, double tcost) {
58 double tenergy= tweight * tcost;
59 printf(" %# e > %# e |", tcost, tenergy);
63 static void compute_vertex_areas(const Vertices vertices, double areas[N]) {
64 int v0,v1,v2, e1,e2, k;
75 double e1v[D3], e2v[D3], av[D3];
77 e1v[k]= vertices[v1][k] - vertices[v0][k];
78 e2v[k]= vertices[v2][k] - vertices[v0][k];
84 areas[v0]= total / count;
88 /*---------- use of GSL ----------*/
90 /* We want to do multidimensional minimisation.
92 * We don't think there are any local minima. Or at least, if there
93 * are, the local minimum which will be found from the starting
94 * state is the one we want.
96 * We don't want to try to provide a derivative of the cost
97 * function. That's too tedious (and anyway the polynomial
98 * approximation to our our cost function sometimes has high degree
99 * in the inputs which means the quadratic model implied by most of
100 * the gradient descent minimisers is not ideal).
102 * This eliminates most of the algorithms. Nelder and Mead's
103 * simplex algorithm is still available and we will try that.
105 * In our application we are searching for the optimal locations of
106 * N actualvertices in D3 (3) dimensions - ie, we are searching for
107 * the optimal metapoint in an N*D3-dimensional space.
109 * So eg with X=Y=100, the simplex will contain 300 metavertices
110 * each of which is an array of 300 doubles for the actualvertex
111 * coordinates. Hopefully this won't be too slow ...
114 static gsl_multimin_fminimizer *minimiser;
116 static const double stop_epsilon= 1e-4;
118 static double minfunc_f(const gsl_vector *x, void *params) {
119 assert(x->size == DIM);
120 assert(x->stride == 1);
121 return compute_energy((const double(*)[D3])x->data);
124 int main(int argc, const char *const *argv) {
125 gsl_multimin_function multimin_function;
127 Vertices initial, step_size;
129 gsl_vector initial_gsl, step_size_gsl;
132 if (argc>1) { fputs("takes no arguments\n",stderr); exit(8); }
134 minimiser= gsl_multimin_fminimizer_alloc
135 (gsl_multimin_fminimizer_nmsimplex, DIM);
136 if (!minimiser) { perror("alloc minimiser"); exit(-1); }
138 multimin_function.f= minfunc_f;
139 multimin_function.n= DIM;
140 multimin_function.params= 0;
142 initial_f= fopen(INITIAL_F,"rb"); if (!initial_f) diee("fopen initial");
143 errno= 0; r= fread(initial,sizeof(initial),1,initial_f);
144 if (r!=1) diee("fread");
147 initial_gsl.size= DIM;
148 initial_gsl.stride= 1;
149 initial_gsl.block= 0;
150 initial_gsl.owner= 0;
151 step_size_gsl= initial_gsl;
153 initial_gsl.data= &initial[0][0];
154 step_size_gsl.data= &step_size[0][0];
157 K step_size[v][k]= 0.01;
159 // FOR_RIM_VERTEX(vx,vy,v)
160 // step_size[v][3] *= 0.1;
162 GA( gsl_multimin_fminimizer_set(minimiser, &multimin_function,
163 &initial_gsl, &step_size_gsl) );
166 GA( gsl_multimin_fminimizer_iterate(minimiser) );
168 size= gsl_multimin_fminimizer_size(minimiser);
169 r= gsl_multimin_test_size(size, stop_epsilon);
171 printf("%*s size %# e, r=%d\n", 135,"", size, r);
174 if (r==GSL_SUCCESS) break;
175 assert(r==GSL_CONTINUE);
180 /*---------- Edgewise vertex displacement ----------*/
200 * Find R', the `expected' location of R, by
201 * reflecting S in M (the midpoint of QP).
207 * Giving energy contribution:
215 * (The dimensions of this are those of F_vd.)
217 * By symmetry, this calculation gives the same answer with R and S
218 * exchanged. Looking at the projection in the RMS plane:
224 * R' ,' 2d" = |SS'| = |RR'| = 2d
226 * `-._ ,' By congruent triangles,
227 * ` M with M' = midpoint of RS,
228 * ,' `-._ |MM'| = |RR'|/2 = d
231 * ,' M' _ , - ' d = |MM'|
235 * We choose this value for l (rather than |RM|+|MS|, say, or |RM|)
236 * because we want this symmetry and because we're happy to punish
237 * bending more than uneveness in the metric.
239 * In practice to avoid division by zero we'll add epsilon to l^3
240 * and the huge energy ought then to be sufficient for the model to
241 * avoid being close to R=S.
244 static double edgewise_vertex_displacement_cost(const Vertices vertices) {
245 static const double l3_epsilon= 1e-6;
247 int pi,e,qi,ri,si, k;
248 double m[D3], mprime[D3], b, d2, l, sigma_bd2_l3=0;
251 ri= EDGE_END2(pi,(e+1)%V6); if (ri<0) continue;
252 si= EDGE_END2(pi,(e+5)%V6); if (si<0) continue;
254 K m[k]= (vertices[pi][k] + vertices[qi][k]) * 0.5;
255 K mprime[k]= (vertices[ri][k] + vertices[si][k]) * 0.5;
256 b= hypotD(vertices[pi], vertices[qi]);
257 d2= hypotD2(m, mprime);
258 l= hypotD(vertices[ri], vertices[si]);
259 double l3 = l*l*l + l3_epsilon;
261 sigma_bd2_l3 += b * d2 / l3;
266 /*---------- noncircular rim cost ----------*/
268 static double noncircular_rim_cost(const Vertices vertices) {
272 FOR_RIM_VERTEX(vy,vx,v) {
274 /* By symmetry, nearest point on circle is the one with
275 * the same angle subtended at the z axis. */
276 oncircle[0]= vertices[v][0];
277 oncircle[1]= vertices[v][1];
279 double mult= 1.0/ magnD(oncircle);
282 double d2= hypotD2(vertices[v], oncircle);