X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=moebius2.git;a=blobdiff_plain;f=energy.c;h=308aebfa3f890aea945bafc0aa38fc1feb18f9c8;hp=139e7e1304d11937fe5d5261e6dba81d9ad55c3e;hb=6df67fc7061e6fae41fec8c05b9e49e4e393c733;hpb=2c2c1cc2745430048709dc3b11aef7071d802c86

diff --git a/energy.c b/energy.c
index 139e7e1..308aebf 100644
--- a/energy.c
+++ b/energy.c
@@ -3,299 +3,461 @@
  */
 
 #include "common.h"
-#include "bgl.h"
+#include "minimise.h"
 #include "mgraph.h"
+#include "parallel.h"
 
-#define BEST_F "best"
-#define INITIAL_F "initial"
+double vertex_areas[N], vertex_mean_edge_lengths[N], edge_lengths[N][V6];
 
-static double edgewise_vertex_displacement_cost(const Vertices vertices);
-static double noncircular_rim_cost(Vertices vertices);
-
-static void compute_vertex_areas(const Vertices vertices, double areas[N]);
 static double best_energy= DBL_MAX;
-static void flushoutput(void);
 
-static void cost(double *energy, double tweight, double tcost);
-#define COST(weight, compute) cost(&energy, (weight), (compute))
+static void addcost(double *energy, double tweight, double tcost, int pr);
+
+/*---------- main energy computation, weights, etc. ----------*/
+
+typedef double CostComputation(const Vertices vertices, int section);
+typedef void PreComputation(const Vertices vertices, int section);
+
+typedef struct {
+  double weight;
+  CostComputation *fn;
+} CostContribution;
+
+#define NPRECOMPS ((sizeof(precomps)/sizeof(precomps[0])))
+#define NCOSTS ((sizeof(costs)/sizeof(costs[0])))
+#define COST(weight, compute) { (weight),(compute) },
+
+static PreComputation *const precomps[]= {
+  compute_edge_lengths,
+  compute_vertex_areas
+};
+
+static const CostContribution costs[]= {
+
+#if XBITS==3
+#define STOP_EPSILON 1e-6
+    COST(  3e3,   line_bending_cost)
+    COST(  3e3,   edge_length_variation_cost)
+    COST( 0.4e3,  rim_proximity_cost)
+    COST(  1e6,   edge_angle_cost)
+                  #define EDGE_ANGLE_COST_CIRCCIRCRAT (0.5/1.7)
+//    COST(  1e1,   small_triangles_cost)
+    COST(  1e12,   noncircular_rim_cost)
+#endif
+
+#if XBITS==4
+#define STOP_EPSILON 1.2e-4
+    COST(  3e5,   line_bending_cost)
+    COST( 10e3,   edge_length_variation_cost)
+    COST( 9.0e3,  rim_proximity_cost) // 5e1 is too much
+				      // 2.5e1 is too little
+    // 0.2e1 grows compared to previous ?
+    // 0.6e0 shrinks compared to previous ?
+
+//    COST(  1e12,   edge_angle_cost)
+                  #define EDGE_ANGLE_COST_CIRCCIRCRAT (0.5/1.3)
+    COST(  1e18,   noncircular_rim_cost)
+#endif
+
+#if XBITS>=5 /* nonsense follows but never mind */
+#define STOP_EPSILON 1e-6
+    COST(  3e5,   line_bending_cost)
+    COST( 10e2,   edge_length_variation_cost)
+    COST( 9.0e1,  rim_proximity_cost) // 5e1 is too much
+				      // 2.5e1 is too little
+    // 0.2e1 grows compared to previous ?
+    // 0.6e0 shrinks compared to previous ?
+
+    COST(  1e12,   edge_angle_cost)
+                  #define EDGE_ANGLE_COST_CIRCCIRCRAT (0.5/1.3)
+    COST(  1e18,   noncircular_rim_cost)
+#endif
+
+};
+
+const double edge_angle_cost_circcircrat= EDGE_ANGLE_COST_CIRCCIRCRAT;
+
+void energy_init(void) {
+  stop_epsilon= STOP_EPSILON;
+}
+
+/*---------- energy computation machinery ----------*/
+
+void compute_energy_separately(const struct Vertices *vs,
+			 int section, void *energies_v, void *totals_v) {
+  double *energies= energies_v;
+  int ci;
+  
+  for (ci=0; ci<NPRECOMPS; ci++) {
+    precomps[ci](vs->a, section);
+    inparallel_barrier();
+  }
+  for (ci=0; ci<NCOSTS; ci++)
+    energies[ci]= costs[ci].fn(vs->a, section);
+}
+
+void compute_energy_combine(const struct Vertices *vertices,
+			 int section, void *energies_v, void *totals_v) {
+  int ci;
+  double *energies= energies_v;
+  double *totals= totals_v;
+
+  for (ci=0; ci<NCOSTS; ci++)
+    totals[ci] += energies[ci];
+}
+
+double compute_energy(const struct Vertices *vs) {
+  static int bests_unprinted;
+
+  double totals[NCOSTS], energy;
+  int ci, printing;
+
+  printing= printing_check(pr_cost,0);
 
-/*---------- main energy computation and subroutines ----------*/
+  if (printing) printf("%15lld c>e |", evaluations);
 
-static double compute_energy(Vertices vertices) {
-  double vertex_areas[N], energy;
+  for (ci=0; ci<NCOSTS; ci++)
+    totals[ci]= 0;
+
+  inparallel(vs,
+	     compute_energy_separately,
+	     compute_energy_combine,
+	     sizeof(totals) /* really, size of energies */,
+	     totals);
 
-  compute_vertex_areas(vertices,vertex_areas);
   energy= 0;
-  printf("cost > energy |");
+  for (ci=0; ci<NCOSTS; ci++)
+    addcost(&energy, costs[ci].weight, totals[ci], printing);
+
+  if (printing) printf("| total %# e |", energy);
 
-  COST(1000.0, edgewise_vertex_displacement_cost(vertices));
-  COST(1.0,    graph_layout_cost(vertices,vertex_areas));
-  COST(1e3,    noncircular_rim_cost(vertices));
-  
-  printf("| total %# e |", energy);
   if (energy < best_energy) {
-    FILE *best;
-    printf(" BEST");
-    
-    best_f= fopen(BEST_F ".new","wb");  if (!best_f) diee("fopen new best");
-    r= fwrite(vertices,sizeof(vertices),1,best_f); if (r!=1) diee("fwrite");
+    FILE *best_f;
+    int r;
+
+    if (printing) {
+      printf(" BEST");
+      if (bests_unprinted) printf(" [%4d]",bests_unprinted);
+      bests_unprinted= 0;
+    } else {
+      bests_unprinted++;
+    }
+
+    best_f= fopen(best_file_tmp,"wb");  if (!best_f) diee("fopen new out");
+    r= fwrite(vs->a,sizeof(vs->a),1,best_f); if (r!=1) diee("fwrite");
     if (fclose(best_f)) diee("fclose new best");
-    if (rename(BEST_F ".new", BEST_F)) diee("rename install new best");
+    if (rename(best_file_tmp,best_file)) diee("rename install new best");
+
+    best_energy= energy;
+  }
+  if (printing) {
+    putchar('\n');
+    flushoutput();
   }
-  putchar('\n');
-  flushoutput();
 
+  evaluations++;
   return energy;
-}    
+}
 
-static void cost(double *energy, double tweight, double tcost) {
+static void addcost(double *energy, double tweight, double tcost, int pr) {
   double tenergy= tweight * tcost;
-  printf(" %# e > %# e |", tcost, tenergy);
+  if (pr) printf(" %# e x %g > %# e* |", tcost, tweight, tenergy);
   *energy += tenergy;
 }
 
-static void flushoutput(void) {
-  if (fflush(stdout) || ferror(stdout)) { perror("stdout"); exit(-1); }
+/*---------- Precomputations ----------*/
+
+void compute_edge_lengths(const Vertices vertices, int section) {
+  int v1,e,v2;
+
+  FOR_EDGE(v1,e,v2, OUTER)
+    edge_lengths[v1][e]= hypotD(vertices[v1],vertices[v2]);
 }
 
-static void compute_vertex_areas(const Vertices vertices, double areas[N]) {
-  FOR_VERTEX(v0) {
-    double total= 0.0;
+void compute_vertex_areas(const Vertices vertices, int section) {
+  int v0,v1,v2, e1,e2;
+//  int k;
+
+  FOR_VERTEX(v0, OUTER) {
+    double total= 0.0, edges_total=0;
     int count= 0;
-    
+
     FOR_VEDGE(v0,e1,v1) {
       e2= (e1+1) % V6;
       v2= EDGE_END2(v0,e2);
       if (v2<0) continue;
-      
-      double e1v[D3], e2v[D3], av[D3];
-      K {
-	e1v[k]= vertices[v1][k] - vertices[v0][k];
-	e2v[k]= vertices[v2][k] - vertices[v0][k];
-      }
-      xprod(av, e1v, e2v);
-      total += hypotD1(av);
+
+      edges_total += edge_lengths[v0][e1];
+
+//      double e1v[D3], e2v[D3], av[D3];
+//      K {
+//	e1v[k]= vertices[v1][k] - vertices[v0][k];
+//	e2v[k]= vertices[v2][k] - vertices[v0][k];
+//      }
+//      xprod(av, e1v, e2v);
+//      total += magnD(av);
+
       count++;
     }
-    areas[v0]= total / count;
+    vertex_areas[v0]= total / count;
+    vertex_mean_edge_lengths[v0]= edges_total / count;
   }
 }
 
-/*---------- use of GSL ----------*/
+/*---------- Edgewise vertex displacement ----------*/
 
-  /* We want to do multidimensional minimisation.
+  /*
+   * Definition:
+   *
+   *    At each vertex Q, in each direction e:
+   *
+   *                   	              e
+   *                           Q ----->----- R
+   *		          _,-'\__/
+   *   	       	      _,-'     	 delta
+   * 		   P '
    *
-   * We don't think there are any local minima.  Or at least, if there
-   * are, the local minimum which will be found from the starting
-   * state is the one we want.
+   *                      r
+   *       cost    = delta          (we use r=3)
+   *           Q,e
    *
-   * We don't want to try to provide a derivative of the cost
-   * function.  That's too tedious (and anyway the polynomial
-   * approximation to our our cost function sometimes has high degree
-   * in the inputs which means the quadratic model implied by most of
-   * the gradient descent minimisers is not ideal).
    *
-   * This eliminates most of the algorithms.  Nelder and Mead's
-   * simplex algorithm is still available and we will try that.
+   * Calculation:
    *
-   * In our application we are searching for the optimal locations of
-   * N actualvertices in D3 (3) dimensions - ie, we are searching for
-   * the optimal metapoint in an N*D3-dimensional space.
-   * 
-   * So eg with X=Y=100, the simplex will contain 300 metavertices
-   * each of which is an array of 300 doubles for the actualvertex
-   * coordinates.  Hopefully this won't be too slow ...
+   *      Let vector A = PQ
+   *                 B = QR
+   *
+   *   		       -1   A . B
+   *      delta =  tan     -------
+   *     		  | A x B |
+   *
+   *      which is always in the range 0..pi because the denominator
+   *      is nonnegative.  We add epsilon to |AxB| to avoid division
+   *      by zero.
+   *
+   *                     r
+   *      cost    = delta
+   *          Q,e
    */
 
-static void gsldie(const char *what, int status) {
-  fprintf(stderr,"gsl function failed: %s: %s\n", what, gsl_strerror(status));
-  exit(-1);
+double line_bending_cost(const Vertices vertices, int section) {
+  static const double axb_epsilon= 1e-6;
+  static const double exponent_r= 4;
+
+  int pi,e,qi,ri, k;
+  double  a[D3], b[D3], axb[D3];
+  double total_cost= 0;
+
+  FOR_EDGE(qi,e,ri, OUTER) {
+    pi= EDGE_END2(qi,(e+3)%V6); if (pi<0) continue;
+
+//if (!(qi&XMASK)) fprintf(stderr,"%02x-%02x-%02x (%d)\n",pi,qi,ri,e);
+
+    K a[k]= -vertices[pi][k] + vertices[qi][k];
+    K b[k]= -vertices[qi][k] + vertices[ri][k];
+
+    xprod(axb,a,b);
+
+    double delta= atan2(magnD(axb) + axb_epsilon, dotprod(a,b));
+    double cost= pow(delta,exponent_r);
+
+    total_cost += cost;
+  }
+  return total_cost;
 }
 
-static gsl_multimin_fminimizer *minimiser;
+/*---------- edge length variation ----------*/
 
-static const stop_epsilon= 1e-4;
+  /*
+   * Definition:
+   *
+   *    See the diagram above.
+   *                                r
+   *       cost    = ( |PQ| - |QR| )
+   *           Q,e
+   */
 
-#define DIM (N*D3)
+double edge_length_variation_cost(const Vertices vertices, int section) {
+  double diff, cost= 0, exponent_r= 2;
+  int q, e,r, eback;
 
-static double minfunc_f(const gsl_vector *x, void *params) {
-  assert(x->size == DIM);
-  assert(x->stride == 1);
-  return compute_energy((Vertices)x->data);
+  FOR_EDGE(q,e,r, OUTER) {
+    eback= edge_reverse(q,e);
+    diff= edge_lengths[q][e] - edge_lengths[q][eback];
+    cost += pow(diff,exponent_r);
+  }
+  return cost;
 }
 
-int main(int argc, const char *const *argv) {
-  struct gsl_multimin_function multimin_function;
-  double size;
-  Vertices initial, step_size;
-  FILE *initial;
-  gsl_vector initial_gsl, step_size_gsl;
-  int r;
-  
-  if (argc>1) { fputs("takes no arguments\n",stderr); exit(8); }
-
-  minimiser= gsl_multimin_fminimizer_alloc
-    (gsl_multimin_fminimizer_nmsimplex, DIM);
-  if (!minimiser) { perror("alloc minimiser"); exit(-1); }
-
-  multimin_function.f= minfunc_f;
-  multimin_function.n= DIM;
-  multimin_function.params= 0;
-
-  initial_f= fopen(INITIAL_F,"rb");  if (!initial_f) diee("fopen initial");
-  errno= 0; r= fread(initial,sizeof(initial),1,initial_f);
-  if (r!=1) diee("fread");
-  fclose(initial_f);
-
-  initial_gsl.size= DIM;
-  initial_gsl.stride= 1;
-  initial_gsl.block= 0;
-  initial_gsl.owner= 0;
-  step_size_gsl= initial_gsl;
-
-  initial_gsl.data= initial;
-  step_size_gsl.data= step_size;
-
-  FOR_VERTEX(v)
-    K step_size[v][k]= 1e-3;
-  FOR_RIM_VERTEX(vx,vy,v)
-    step_size[v][3] *= 0.1;
-  
-  for (vy=0; vy<Y; vy+=Y-1)
-  for (vx=0; vx<x
-  for (i=0; i<DIM; i++) step_size[i]= step_size;
-  
+/*---------- rim proximity cost ----------*/
+
+static void find_nearest_oncircle(double oncircle[D3], const double p[D3]) {
+  /* By symmetry, nearest point on circle is the one with
+   * the same angle subtended at the z axis. */
+  oncircle[0]= p[0];
+  oncircle[1]= p[1];
+  oncircle[2]= 0;
+  double mult= 1.0/ magnD(oncircle);
+  oncircle[0] *= mult;
+  oncircle[1] *= mult;
+}
 
-  step_size= gsl_vector_alloc(DIM);  if (!step_size) gsldie("alloc step");
-  gsl_vector_set_all(step_size, 1e-3);
+double rim_proximity_cost(const Vertices vertices, int section) {
+  double oncircle[3], cost=0;
+  int v;
 
-  assert(step_
-  step_
+  FOR_VERTEX(v, OUTER) {
+    int y= v >> YSHIFT;
+    int nominal_edge_distance= y <= Y/2 ? y : Y-1-y;
+    if (nominal_edge_distance==0) continue;
 
-  r= gsl_multimin_fminimizer_set(minimiser, &multimin_function,
-				 &initial_gsl, &step_size);
-  if (r) { gsldie("fminimizer_set",r); }
-  
-  for (;;) {
-    r= gsl_multimin_fminimizer_iterate(minimiser);
-    if (r) { gsldie("fminimizer_iterate",r); }
+    find_nearest_oncircle(oncircle, vertices[v]);
 
-    size= gsl_multimin_fminimizer_size(minimiser);
-    r= gsl_multimin_test_size(size, stop_epsilon);
+    cost +=
+      vertex_mean_edge_lengths[v] *
+      (nominal_edge_distance*nominal_edge_distance) /
+      (hypotD2(vertices[v], oncircle) + 1e-6);
+  }
+  return cost;
+}
 
-    printf("size %# e, r=%d\n", size, r);
-    flushoutput();
+/*---------- noncircular rim cost ----------*/
 
-    if (r==GSL_SUCCESS) break;
-    assert(r==GSL_CONTINUE);
+double noncircular_rim_cost(const Vertices vertices, int section) {
+  int vy,vx,v;
+  double cost= 0.0;
+  double oncircle[3];
+
+  FOR_RIM_VERTEX(vy,vx,v, OUTER) {
+    find_nearest_oncircle(oncircle, vertices[v]);
+
+    double d2= hypotD2(vertices[v], oncircle);
+    cost += d2*d2;
   }
+  return cost;
 }
 
-/*---------- Edgewise vertex displacement ----------*/
+/*---------- overly sharp edge cost ----------*/
 
   /*
-   *  
-   *
    *
    *   	            Q `-_
-   *              / |    `-_
-   *   R' - _ _ _/_ |       `-.
-   *    .       /   M - - - - - S
-   *    .      /    |      _,-'
-   *    .     /     |  _,-'
-   *    .    /    , P '
-   *    .   /  ,-'
-   *    .  /,-'
-   *    . /'
+   *              / |    `-_		               P'Q' ------ S'
+   *             /  |       `-.		             _,' `.       .
+   *            /   |           S	          _,'     :      .
+   *           /    |      _,-'                _,'        :r    .r
+   *          /     |  _,-'		   R' '           `.   .
+   *         /    , P '			       ` .   r     :  .
+   *        /  ,-'			            `  .   : 
+   *       /,-'				                 ` C'
+   *      /'
    *   	 R
    *
    *
    *
-   *  Find R', the `expected' location of R, by
-   *  reflecting S in M (the midpoint of QP).
-   *
-   *  Let 2d = |RR'|
-   *       b = |PQ|
-   *       l = |RS|
+   *  Let delta =  angle between two triangles' normals
    *
    *  Giving energy contribution:
    *
-   *                               2
-   *                            b d
-   *    E             =  F   .  ----
-   *     vd, edge PQ      vd      3
-   *                             l
-   *
-   *  (The dimensions of this are those of F_vd.)
-   *
-   *  By symmetry, this calculation gives the same answer with R and S
-   *  exchanged.  Looking at the projection in the RMS plane:
+   *                                   2
+   *    E             =  F   .  delta
+   *     vd, edge PQ      vd
+   */
+
+double edge_angle_cost(const Vertices vertices, int section) {
+  double pq1[D3], rp[D3], ps[D3], rp_2d[D3], ps_2d[D3], rs_2d[D3];
+  double a,b,c,s,r;
+  const double minradius_base= 0.2;
+
+  int pi,e,qi,ri,si, k;
+//  double our_epsilon=1e-6;
+  double total_cost= 0;
+
+  FOR_EDGE(pi,e,qi, OUTER) {
+//    if (!(RIM_VERTEX_P(pi) || RIM_VERTEX_P(qi))) continue;
+
+    si= EDGE_END2(pi,(e+V6-1)%V6);  if (si<0) continue;
+    ri= EDGE_END2(pi,(e   +1)%V6);  if (ri<0) continue;
+
+    K {
+      pq1[k]= -vertices[pi][k] + vertices[qi][k];
+      rp[k]=  -vertices[ri][k] + vertices[pi][k];
+      ps[k]=  -vertices[pi][k] + vertices[si][k];
+    }
+
+    normalise(pq1,1,1e-6);
+    xprod(rp_2d, rp,pq1); /* projects RP into plane normal to PQ */
+    xprod(ps_2d, ps,pq1); /* likewise PS */
+    K rs_2d[k]= rp_2d[k] + ps_2d[k];
+    /* radius of circumcircle of R'P'S' from Wikipedia
+     * `Circumscribed circle' */
+    a= magnD(rp_2d);
+    b= magnD(ps_2d);
+    c= magnD(rs_2d);
+    s= 0.5*(a+b+c);
+    r= a*b*c / sqrt((a+b+c)*(a-b+c)*(b-c+a)*(c-a+b) + 1e-6);
+
+    double minradius= minradius_base + edge_angle_cost_circcircrat*(a+b);
+    double deficit= minradius - r;
+    if (deficit < 0) continue;
+    double cost= deficit*deficit;
+
+    total_cost += cost;
+  }
+
+  return total_cost;
+}
+
+/*---------- small triangles cost ----------*/
+
+  /*
    *
+   *   	            Q `-_
+   *              / |    `-_
+   *             /  |       `-.
+   *            /   |           S
+   *           /    |      _,-'
+   *          /     |  _,-'
+   *         /    , P '
+   *        /  ,-'
+   *       /,-'
+   *      /'
+   *   	 R
    *
-   *	       	       	       S'
-   *	   	    	     ,'
-   *	       	       	   ,'
-   *   	R'           	 ,'    	       	2d" = |SS'| = |RR'| = 2d
-   *	  `-._         ,'
-   *   	      `-._   ,'	      		By congruent triangles,
-   *              ` M	      		with M' = midpoint of RS,
-   *	       	  ,'  `-._     	       	|MM'| = |RR'|/2 = d
-   *	       	,'     	  `-._
-   *	      ,'       	      ` S       So use
-   *	    ,'	     M' _ , - '            d = |MM'|
-   *	  ,'   _ , - '
-   *	 R - '
+   *  Let delta =  angle between two triangles' normals
    *
-   *  We choose this value for l (rather than |RM|+|MS|, say, or |RM|)
-   *  because we want this symmetry and because we're happy to punish
-   *  bending more than uneveness in the metric.
+   *  Giving energy contribution:
    *
-   *  In practice to avoid division by zero we'll add epsilon to l^3
-   *  and the huge energy ought then to be sufficient for the model to
-   *  avoid being close to R=S.
+   *                                   2
+   *    E             =  F   .  delta
+   *     vd, edge PQ      vd
    */
 
-static double edgewise_vertex_displacement_cost(const Vertices vertices) {
-  static const l3_epsison= 1e-6;
+double small_triangles_cost(const Vertices vertices, int section) {
+  double pq[D3], ps[D3];
+  double x[D3];
+  int pi,e,qi,si, k;
+//  double our_epsilon=1e-6;
+  double total_cost= 0;
 
-  int pi,e,qi,ri,si, k;
-  double m[D3], mprime[D3], b, d2, l, sigma_bd2_l3;
-
-  FOR_EDGE(pi,e,qi) {
-    ri= EDGE_END2(pi,(e+1)%V6); if (r<0) continue;
-    si= EDGE_END2(pi,(e+5)%V6); if (s<0) continue;
-    assert(ri == EDGE_END2(qi,(e+2)%V6));
-    assert(si == EDGE_END2(qi,(e+4)%V6));
-    
-    K m[k]= (vertices[pi][k] + vertices[qi][k]) * 0.5;
-    K mprime[k]= (vertices[ri][k] + vertices[si][k]) * 0.5;
-    b= hypotD(vertices[pi], vertices[qi]);
-    d2= hypotD2(m, mprime);
-    l= hypotD(vertices[ri][k] - vertices[si][k]);
-    l3 = l*l*l + l3_epsilon;
-
-    sigma_bd2_l3 += b * d2 / l3;
-  }
-  return sigma_bd2_l3;
-}
+  FOR_EDGE(pi,e,qi, OUTER) {
+//    if (!(RIM_VERTEX_P(pi) || RIM_VERTEX_P(qi))) continue;
 
-/*---------- noncircular rim cost ----------*/
+    si= EDGE_END2(pi,(e+V6-1)%V6);  if (si<0) continue;
 
-static double noncircular_rim_cost(Vertices vertices) {
-  int vy,vx,v;
-  double cost= 0.0;
-  
-  FOR_RIM_VERTEX(vy,vx,v) {
-    double oncircle[3];
-    /* By symmetry, nearest point on circle is the one with
-     * the same angle subtended at the z axis. */
-    oncircle[0]= vertices[v][0];
-    oncircle[1]= vertices[v][1];
-    oncircle[2]= 0;
-    double mult= 1.0/ magnD(oncircle);
-    oncircle[0] *= mult;
-    oncircle[1] *= mult;
-    double d2= hypotD2(vertices[v], oncircle);
-    cost += d2*d2;
+    K {
+      pq[k]= vertices[qi][k] - vertices[pi][k];
+      ps[k]= vertices[si][k] - vertices[pi][k];
+    }
+    xprod(x, pq,ps);
+
+    double cost= 1/(magnD2(x) + 0.01);
+
+//double cost= pow(magnD(spqxpqr), 3);
+//assert(dot>=-1 && dot <=1);
+//double cost= 1-dot;
+    total_cost += cost;
   }
+
+  return total_cost;
 }