---

Measure.C

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/***************************************************************************
 *cr
 *cr            (C) Copyright 1995-2011 The Board of Trustees of the
 *cr                        University of Illinois
 *cr                         All Rights Reserved
 *cr
 ***************************************************************************/

/***************************************************************************
 * RCS INFORMATION:
 *
 *      $RCSfile: Measure.C,v $
 *      $Author: johns $        $Locker:  $             $State: Exp $
 *      $Revision: 1.134 $       $Date: 2011/10/20 22:46:19 $
 *
 ***************************************************************************
 * DESCRIPTION:
 *   Code to measure atom distances, angles, dihedrals, etc.
 ***************************************************************************/

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include "Measure.h"
#include "AtomSel.h"
#include "Matrix4.h"
#include "utilities.h"
#include "fitrms.h"
#include "ResizeArray.h"
#include "SpatialSearch.h"  // for find_within()
#include "MoleculeList.h"
#include "Inform.h"
#include "Timestep.h"
#include "VMDApp.h"
#include "WKFThreads.h"
#include "WKFUtils.h"

// Standard functions available to everyone
static const char *measure_error_messages[] = {
  "no atom selection",                                  // -1
  "no atoms in selection",                              // -2
  "incorrect number of weights for selection",          // -3
  "internal error: NULL pointer given",                 // -4
  "bad weight sum, would cause divide by zero",         // -5
  "molecule was deleted(?)",                            // -6
  "cannot understand weight parameter",                 // -7
  "non-number given as parameter",                      // -8
  "two selections don't have the same number of atoms", // -9
  "internal error: out of range",                       // -10
  "no coordinates in selection",                        // -11
  "couldn't compute eigenvalue/vectors",                // -12
  "unknown Tcl problem",                                // -13
  "no atom radii",                                      // -14
  "order parameter contains out-of-range atom index",   // -15
  "order parameter not supported in old RMS fit mode",  // -16
  "specified frames are out of range",                  // -17
  "both selections must reference the same molecule",   // -18
  "one atom appears twice in list",                     // -19
  "molecule contains no frames",                        // -20
  "invalid atom id",                                    // -21
  "cutoff must be smaller than cell dimension",         // -22
  "Zero volmap gridsize"                                // -23
};
  
const char *measure_error(int errnum) {
  if (errnum >= 0 || errnum < -23) 
    return "bad error number";
  return measure_error_messages[-errnum - 1];
}

int measure_move(const AtomSel *sel, float *framepos, const Matrix4 &mat) {
  if (!sel)       return MEASURE_ERR_NOSEL;
  if (!framepos)  return MEASURE_ERR_NOFRAMEPOS;

  // and apply it to the coordinates
  int i;
  float *pos = framepos;
  if (mat.mat[3]==0 && mat.mat[7]==0 && mat.mat[11]==0 && mat.mat[15]==1) {
    // then this is just a rotation followed by a translation.  Do it
    // the fast way.
    const float dx = mat.mat[12];
    const float dy = mat.mat[13];
    const float dz = mat.mat[14];
    const float *m = mat.mat;
    if (sel->selected == sel->num_atoms) {
      for (i=0; i<sel->num_atoms; i++) {
        float x = pos[0]*m[0] + pos[1]*m[4] + pos[2]*m[8] + dx;
        float y = pos[0]*m[1] + pos[1]*m[5] + pos[2]*m[9] + dy;
        float z = pos[0]*m[2] + pos[1]*m[6] + pos[2]*m[10] + dz;
        pos[0] = x;
        pos[1] = y;
        pos[2] = z;
        pos += 3;
      }
    } else {
      pos += sel->firstsel*3;
      for (i=sel->firstsel; i<=sel->lastsel; i++) {
        if (sel->on[i]) {
          float x = pos[0]*m[0] + pos[1]*m[4] + pos[2]*m[8] + dx;
          float y = pos[0]*m[1] + pos[1]*m[5] + pos[2]*m[9] + dy;
          float z = pos[0]*m[2] + pos[1]*m[6] + pos[2]*m[10] + dz;
          pos[0] = x;
          pos[1] = y;
          pos[2] = z;
        }
        pos += 3;
      }
    }
  } else {
    pos += sel->firstsel*3;
    for (i=sel->firstsel; i<=sel->lastsel; i++) {
      if (sel->on[i]) {
        mat.multpoint3d(pos, pos);
      }
      pos += 3;
    }
  }
  return MEASURE_NOERR;
}

// compute the sum of a set of weights which correspond either to
// all atoms, or to selected atoms
int measure_sumweights(const AtomSel *sel, int numweights, 
                       const float *weights, float *weightsum) {
  if (!sel)       return MEASURE_ERR_NOSEL;
  if (!weights)   return MEASURE_ERR_NOWEIGHT;
  if (!weightsum) return MEASURE_ERR_GENERAL;

  double sum = 0;

  if (numweights == sel->num_atoms) {
    int i;
    for (i=sel->firstsel; i<=sel->lastsel; i++) {
      if (sel->on[i]) {
        sum += weights[i];
      }
    }
  } else if (numweights == sel->selected) {
    int i, j;
    for (j=0,i=sel->firstsel; i<=sel->lastsel; i++) {
      if (sel->on[i]) {
        sum += weights[j];
        j++;
      }
    }
  } else {
    return MEASURE_ERR_BADWEIGHTPARM;  
  }

  *weightsum = (float) sum;
  return MEASURE_NOERR;
}


// compute the center of mass
// return -5 if total weight == 0, otherwise 0 for success.
int measure_center(const AtomSel *sel, const float *framepos,
                   const float *weight, float *com) {
  if (!sel)      return MEASURE_ERR_NOSEL;
  if (!framepos) return MEASURE_ERR_NOFRAMEPOS;
  if (!weight)   return MEASURE_ERR_NOWEIGHT;
  if (!com)      return MEASURE_ERR_NOCOM;   

  int i, j;
  float x, y, z, w;
  j=0;
  w=x=y=z=0;
  for (i=sel->firstsel; i<=sel->lastsel; i++) {
    if (sel->on[i]) {
      float tw = weight[j];
      w += tw;
      x += tw * framepos[3*i  ];
      y += tw * framepos[3*i+1];
      z += tw * framepos[3*i+2];
      j++;
    }
  }

  if (w == 0) {
    return MEASURE_ERR_BADWEIGHTSUM;
  }

  com[0] = x / w;
  com[1] = y / w;
  com[2] = z / w;

  return MEASURE_NOERR;
}


// compute the axis aligned aligned bounding box for the selected atoms
int measure_minmax(int num, const int *on, const float *framepos, 
                   const float *radii, float *min_coord, float *max_coord) {
  int i;
  float minx, miny, minz;
  float maxx, maxy, maxz;

  if (!on)                      return MEASURE_ERR_NOSEL;
  if (num == 0)                 return MEASURE_ERR_NOATOMS;
  if (!min_coord || !max_coord) return MEASURE_ERR_NOMINMAXCOORDS;

  vec_zero(min_coord);
  vec_zero(max_coord);

  // find first selected atom
  for (i=0; i<num; i++) if (on[i]) break;

  // if no atoms are selected return { 0 0 0 } for min and max
  if (i==num) return MEASURE_NOERR;

  // initialize minmax coords to the first selected atom
  if (radii == NULL) {
    // calculate bounding box of atom centers
    minx = maxx = framepos[i*3  ];
    miny = maxy = framepos[i*3+1];
    minz = maxz = framepos[i*3+2];
  } else {
    // calculate bounding box for atoms /w given radii 
    minx = framepos[i*3  ] - radii[i];
    maxx = framepos[i*3  ] + radii[i];
    miny = framepos[i*3+1] - radii[i];
    maxy = framepos[i*3+1] + radii[i];
    minz = framepos[i*3+2] - radii[i];
    maxz = framepos[i*3+2] + radii[i];
  }

  // continue looping from there until we finish
  if (radii == NULL) {
    // calculate bounding box of atom centers
    for (i++; i<num; i++) {
      if (on[i]) {
        int ind = i * 3;
        float tmpx = framepos[ind  ];
        if (tmpx < minx) minx = tmpx; 
        if (tmpx > maxx) maxx = tmpx;
  
        float tmpy = framepos[ind+1];
        if (tmpy < miny) miny = tmpy; 
        if (tmpy > maxy) maxy = tmpy;
  
        float tmpz = framepos[ind+2];
        if (tmpz < minz) minz = tmpz; 
        if (tmpz > maxz) maxz = tmpz;
      }
    }
  } else {
    // calculate bounding box for atoms /w given radii 
    for (i++; i<num; i++) {
      if (on[i]) {
        int ind = i * 3;
        float mintmpx = framepos[ind  ] - radii[i];
        float maxtmpx = framepos[ind  ] + radii[i];
        if (mintmpx < minx) minx = mintmpx;
        if (maxtmpx > maxx) maxx = maxtmpx;
  
        float mintmpy = framepos[ind+1] - radii[i];
        float maxtmpy = framepos[ind+1] + radii[i];
        if (mintmpy < miny) miny = mintmpy; 
        if (maxtmpy > maxy) maxy = maxtmpy;
  
        float mintmpz = framepos[ind+2] - radii[i];
        float maxtmpz = framepos[ind+2] + radii[i];
        if (mintmpz < minz) minz = mintmpz; 
        if (maxtmpz > maxz) maxz = maxtmpz;
      }
    }
  }

  // set the final min/max output values
  min_coord[0]=minx;
  min_coord[1]=miny;
  min_coord[2]=minz;
  max_coord[0]=maxx;
  max_coord[1]=maxy;
  max_coord[2]=maxz;

  return MEASURE_NOERR;
}


// Calculate average position of selected atoms over selected frames
extern int measure_avpos(const AtomSel *sel, MoleculeList *mlist, 
                         int start, int end, int step, float *avpos) {
  if (!sel)                     return MEASURE_ERR_NOSEL;
  if (sel->num_atoms == 0)      return MEASURE_ERR_NOATOMS;

  Molecule *mymol = mlist->mol_from_id(sel->molid());
  int maxframes = mymol->numframes();
  
  // accept value of -1 meaning "all" frames
  if (end == -1)
    end = maxframes-1;

  if (maxframes == 0 || start < 0 || start > end || 
      end >= maxframes || step <= 0)
    return MEASURE_ERR_BADFRAMERANGE;

  int i;
  for (i=0; i<(3*sel->selected); i++)
    avpos[i] = 0.0f;

  int frame, avcount, j;
  for (avcount=0,frame=start; frame<=end; avcount++,frame+=step) {
    const float *framepos = (mymol->get_frame(frame))->pos;
    for (j=0,i=sel->firstsel; i<=sel->lastsel; i++) {
      if (sel->on[i]) {
        avpos[j*3    ] += framepos[i*3    ];
        avpos[j*3 + 1] += framepos[i*3 + 1];
        avpos[j*3 + 2] += framepos[i*3 + 2];
        j++;
      }
    } 
  }

  float avinv = 1.0f / (float) avcount;
  for (j=0; j<(3*sel->selected); j++) {
    avpos[j] *= avinv;
  } 

  return MEASURE_NOERR;
}


// Calculate dipole moment for selected atoms
extern int measure_dipole(const AtomSel *sel, MoleculeList *mlist, 
                          float *dipole, int unitsdebye, int usecenter) {
  if (!sel)                     return MEASURE_ERR_NOSEL;
  if (sel->num_atoms == 0)      return MEASURE_ERR_NOATOMS;

  Molecule *mymol = mlist->mol_from_id(sel->molid());
  double  rvec[3] = {0, 0, 0};
  double qrvec[3] = {0, 0, 0};
  double mrvec[3] = {0, 0, 0};
  double totalq = 0.0;
  double totalm = 0.0;
  int i;

  // get atom coordinates
  const float *framepos = sel->coordinates(mlist);

  // get atom charges
  const float *q = mymol->charge();
  const float *m = mymol->mass();

  for (i=sel->firstsel; i<=sel->lastsel; i++) {
    if (sel->on[i]) {
      int ind = i * 3;
      rvec[0] += framepos[ind    ];
      rvec[1] += framepos[ind + 1];
      rvec[2] += framepos[ind + 2];

      qrvec[0] += framepos[ind    ] * q[i];
      qrvec[1] += framepos[ind + 1] * q[i];
      qrvec[2] += framepos[ind + 2] * q[i];

      mrvec[0] += framepos[ind    ] * m[i];
      mrvec[1] += framepos[ind + 1] * m[i];
      mrvec[2] += framepos[ind + 2] * m[i];

      totalq += q[i];
      totalm += m[i];
    }
  }

  // fall back to geometrical center when bad or no masses
  if (totalm < 0.0001)
    usecenter=1;
  
  switch (usecenter) {
    case 1:
    {
        double rscale = totalq / sel->selected; 
        dipole[0] = (float) (qrvec[0] - (rvec[0] * rscale)); 
        dipole[1] = (float) (qrvec[1] - (rvec[1] * rscale)); 
        dipole[2] = (float) (qrvec[2] - (rvec[2] * rscale)); 
        break;
    }

    case -1: 
    {
        double mscale = totalq / totalm; 
        dipole[0] = (float) (qrvec[0] - (mrvec[0] * mscale)); 
        dipole[1] = (float) (qrvec[1] - (mrvec[1] * mscale)); 
        dipole[2] = (float) (qrvec[2] - (mrvec[2] * mscale)); 
        break;
    }
    
    case 0: // fallthrough
    default: 
    {
        dipole[0] = (float) qrvec[0];
        dipole[1] = (float) qrvec[1];
        dipole[2] = (float) qrvec[2];
        break;
    }
  }

  // According to the values in
  // http://www.physics.nist.gov/cuu/Constants/index.html
  // 1 e*A = 4.80320440079 D
  // 1 D = 1E-18 Fr*cm = 0.208194346224 e*A
  // 1 e*A = 1.60217653 E-29 C*m
  // 1 C*m = 6.24150947961 E+28 e*A
  // 1 e*A = 1.88972613458 e*a0
  // 1 e*a0 = 0.529177208115 e*A

  if (unitsdebye) {
    // 1 e*A = 4.80320440079 D 
    // latest CODATA (2006) gives:
    //         4.80320425132073913031
    dipole[0] *= 4.80320425132f;
    dipole[1] *= 4.80320425132f;
    dipole[2] *= 4.80320425132f;
  }
 
  return MEASURE_NOERR;
}


// Calculate RMS fluctuation of selected atoms over selected frames
extern int measure_rmsf(const AtomSel *sel, MoleculeList *mlist, 
                        int start, int end, int step, float *rmsf) {
  if (!sel)                     return MEASURE_ERR_NOSEL;
  if (sel->num_atoms == 0)      return MEASURE_ERR_NOATOMS;

  Molecule *mymol = mlist->mol_from_id(sel->molid());
  int maxframes = mymol->numframes();

  // accept value of -1 meaning "all" frames
  if (end == -1)
    end = maxframes-1;

  if (maxframes == 0 || start < 0 || start > end ||
      end >= maxframes || step <= 0)
    return MEASURE_ERR_BADFRAMERANGE;

  int i;
  for (i=0; i<sel->selected; i++)
    rmsf[i] = 0.0f;

  int rc; 
  float *avpos = new float[3*sel->selected];
  rc = measure_avpos(sel, mlist, start, end, step, avpos);

  if (rc != MEASURE_NOERR) {
    delete [] avpos;
    return rc;
  }

  // calculate per-atom variance here
  int frame, avcount, j;
  for (avcount=0,frame=start; frame<=end; avcount++,frame+=step) {
    const float *framepos = (mymol->get_frame(frame))->pos;
    for (j=0,i=sel->firstsel; i<=sel->lastsel; i++) {
      if (sel->on[i]) {
        rmsf[j] += distance2(&avpos[3*j], &framepos[3*i]);
        j++;
      }
    }
  }

  float avinv = 1.0f / (float) avcount;
  for (j=0; j<sel->selected; j++) {
    rmsf[j] = sqrtf(rmsf[j] * avinv);
  }

  delete [] avpos;
  return MEASURE_NOERR;
}


//  rgyr := sqrt(sum (mass(n) ( r(n) - r(com) )^2)/sum(mass(n)))
//  The return value, a float, is put in 'float *rgyr'
//  The function return value is 0 if ok, <0 if not
int measure_rgyr(const AtomSel *sel, MoleculeList *mlist, const float *weight, 
                 float *rgyr) {
  if (!rgyr) return MEASURE_ERR_NORGYR;
  if (!sel) return MEASURE_ERR_NOSEL;
  if (!mlist) return MEASURE_ERR_GENERAL;

  const float *framepos = sel->coordinates(mlist);

  // compute the center of mass with the current weights
  float com[3];
  int ret_val = measure_center(sel, framepos, weight, com);
  if (ret_val < 0) 
    return ret_val;
  
  // measure center of gyration
  int i, j;
  float total_w, sum;

  total_w=sum=0;
  for (j=0,i=sel->firstsel; i<=sel->lastsel; i++) {
    if (sel->on[i]) {
      float w = weight[j];
      total_w += w;
      sum += w * distance2(framepos + 3*i, com);
      j++;
    } 
  }

  if (total_w == 0) {
    return MEASURE_ERR_BADWEIGHTSUM;
  }

  // and finalize the computation
  *rgyr = sqrtf(sum/total_w);

  return MEASURE_NOERR;
}


//  1) if num == sel.selected ; assumes there is one weight per 
//           selected atom
//  2) if num == sel.num_atoms; assumes weight[i] is for atom[i]
//  returns 0 and value in rmsd if good
//   return < 0 if invalid
//  Function is::=  rmsd = 
//    sqrt(sum(weight(n) * sqr(r1(i(n))-r2(i(n))))/sum(weight(n)) / N
int measure_rmsd(const AtomSel *sel1, const AtomSel *sel2,
                 int num, const float *framepos1, const float *framepos2,
                 float *weight, float *rmsd) {
  if (!sel1 || !sel2)   return MEASURE_ERR_NOSEL;
  if (sel1->selected < 1 || sel2->selected < 1) return MEASURE_ERR_NOSEL;
  if (!weight || !rmsd) return MEASURE_ERR_NOWEIGHT;

  // the number of selected atoms must be the same
  if (sel1->selected != sel2->selected) return MEASURE_ERR_MISMATCHEDCNT;

  // need to know how to traverse the list of weights
  // there could be 1 weight per atom (sel_flg == 1) or 
  // 1 weight per selected atom (sel_flg == 0)
  int sel_flg;
  if (num == sel1->num_atoms) {
    sel_flg = 1; // using all elements
  } else {
    sel_flg = 0; // using elements from selection
  }

  // temporary variables
  float tmp_w;
  int w_index = 0;              // the term in the weight field to use
  int sel1ind = sel1->firstsel; // start from the first selected atom
  int sel2ind = sel2->firstsel; // start from the first selected atom
  float wsum = 0;
  float rmsdsum = 0;

  *rmsd = 10000000; // if we bail out, return a huge value

  // compute the rmsd
  int count = sel1->selected;
  while (count-- > 0) {
    // find next 'on' atom in sel1 and sel2
    // loop is safe since we already stop the on count > 0 above
    while (!sel1->on[sel1ind])
      sel1ind++;
    while (!sel2->on[sel2ind])
      sel2ind++;

    // the weight offset to use depends on how many terms there are
    if (sel_flg == 0) {
      tmp_w = weight[w_index++];
    } else {
      tmp_w = weight[sel1ind]; // use the first selection for the weights
    }

    // sum the calculated rmsd and weight values
    rmsdsum += tmp_w * distance2(framepos1 + 3*sel1ind, framepos2 + 3*sel2ind);
    wsum += tmp_w;

    // and advance to the next atom pair
    sel1ind++;
    sel2ind++;
  }

  // check weight sum
  if (wsum == 0) {
    return MEASURE_ERR_BADWEIGHTSUM;
  }

  // finish the rmsd calcs
  *rmsd = sqrtf(rmsdsum / wsum);

  return MEASURE_NOERR; // and say rmsd is OK
}

/* jacobi.C, taken from Numerical Recipes and specialized to 3x3 case */

#define ROTATE(a,i,j,k,l) g=a[i][j];h=a[k][l];a[i][j]=g-s*(h+g*tau);\
        a[k][l]=h+s*(g-h*tau);

static int jacobi(float a[4][4], float d[3], float v[3][3])
{
  int n=3;
  int j,iq,ip,i;
  float tresh,theta,tau,t,sm,s,h,g,c,*b,*z;
  
  b=new float[n];
  z=new float[n];
  for (ip=0;ip<n;ip++) {
    for (iq=0;iq<n;iq++) v[ip][iq]=0.0;
    v[ip][ip]=1.0;
  }
  for (ip=0;ip<n;ip++) {
    b[ip]=d[ip]=a[ip][ip];
    z[ip]=0.0;
  }
  for (i=1;i<=50;i++) {
    sm=0.0;
    for (ip=0;ip<n-1;ip++) {
      for (iq=ip+1;iq<n;iq++)
        sm += (float) fabs(a[ip][iq]);
    }
    if (sm == 0.0) {
      delete [] z;
      delete [] b;
      return 0; // Exit normally
    }
    if (i < 4)
      tresh=0.2f*sm/(n*n);
    else
      tresh=0.0f;
    for (ip=0;ip<n-1;ip++) {
      for (iq=ip+1;iq<n;iq++) {
        g=100.0f * fabsf(a[ip][iq]);
        if (i > 4 && (float)(fabs(d[ip])+g) == (float)fabs(d[ip])
            && (float)(fabs(d[iq])+g) == (float)fabs(d[iq]))
          a[ip][iq]=0.0;
        else if (fabs(a[ip][iq]) > tresh) {
          h=d[iq]-d[ip];
          if ((float)(fabs(h)+g) == (float)fabs(h))
            t=(a[ip][iq])/h;
          else {
            theta=0.5f*h/(a[ip][iq]);
            t=1.0f/(fabsf(theta)+sqrtf(1.0f+theta*theta));
            if (theta < 0.0f) t = -t;
          }
          c=1.0f/sqrtf(1+t*t);
          s=t*c;
          tau=s/(1.0f+c);
          h=t*a[ip][iq];
          z[ip] -= h;
          z[iq] += h;
          d[ip] -= h;
          d[iq] += h;
          a[ip][iq]=0.0;
          for (j=0;j<=ip-1;j++) {
            ROTATE(a,j,ip,j,iq)
              }
          for (j=ip+1;j<=iq-1;j++) {
            ROTATE(a,ip,j,j,iq)
              }
          for (j=iq+1;j<n;j++) {
            ROTATE(a,ip,j,iq,j)
              }
          for (j=0;j<n;j++) {
            ROTATE(v,j,ip,j,iq)
              }
        }
      }
    }
    for (ip=0;ip<n;ip++) {
      b[ip] += z[ip];
      d[ip]=b[ip];
      z[ip]=0.0;
    }
  }
  delete [] b;
  delete [] z;
  return 1; // Failed to converge
}
#undef ROTATE

static int transvecinv(const double *v, Matrix4 &res) {
  double x, y, z;
  x=v[0];
  y=v[1];
  z=v[2];
  if (x == 0.0 && y == 0.0) {
    if (z == 0.0) {
      return -1;
    }
    if (z > 0)
      res.rot(90, 'y');
    else
      res.rot(-90, 'y');
    return 0;
  }
  double theta = atan2(x,y);
  double length = sqrt(x*x + y*y);
  double phi = atan2(z,length);
  res.rot((float) RADTODEG(theta), 'z');
  res.rot((float) RADTODEG(-phi),  'y');
  return 0;
}
 
static int transvec(const double *v, Matrix4 &res) {
  double x, y, z;
  x=v[0];
  y=v[1];
  z=v[2];
  if (x == 0.0 && y == 0.0) {
    if (z == 0.0) {
      return -1;
    }
    if (z > 0)
      res.rot(-90, 'y');
    else
      res.rot(90, 'y');
    return 0;
  }
  double theta = atan2(x,y);
  double length = sqrt(x*x + y*y);
  double phi = atan2(z,length);
  res.rot((float) RADTODEG(phi),  'y');
  res.rot((float) RADTODEG(-theta), 'z');
  return 0;
}

static Matrix4 myfit2(const float *x, const float *y, 
               const float *comx, const float *comy) {

  Matrix4 res;
  double dx[3], dy[3];
  dx[0] = x[0] - comx[0];
  dx[1] = x[1] - comx[1];
  dx[2] = x[2] - comx[2];
  dy[0] = y[0] - comy[0];
  dy[1] = y[1] - comy[1];
  dy[2] = y[2] - comy[2];

  res.translate(-comx[0], -comx[1], -comx[2]);
  transvecinv(dx, res); 
  transvec(dy, res);
  res.translate(comy[0], comy[1], comy[2]); 
  return res;
}

static Matrix4 myfit3(const float *x1, const float *x2, 
                      const float *y1, const float *y2,
                      const float *comx, const float *comy) {
   
  Matrix4 mx, my, rx1, ry1;
  double dx1[3], dy1[3], angle;
  float dx2[3], dy2[3], x2t[3], y2t[3];

  for (int i=0; i<3; i++) {
    dx1[i] = x1[i] - comx[i];
    dx2[i] = x2[i] - comx[i];
    dy1[i] = y1[i] - comy[i];
    dy2[i] = y2[i] - comy[i];
  }

  transvecinv(dx1, rx1);
  rx1.multpoint3d(dx2, x2t);
  angle = atan2(x2t[2], x2t[1]);
  mx.translate(-comx[0], -comx[1], -comy[2]);
  mx.multmatrix(rx1);
  mx.rot((float) RADTODEG(angle), 'x'); 

  transvecinv(dy1, ry1);
  ry1.multpoint3d(dy2, y2t);
  angle = atan2(y2t[2], y2t[1]);
  my.translate(-comy[0], -comy[1], -comy[2]);
  my.multmatrix(ry1);
  my.rot((float) RADTODEG(angle), 'x');
  
  my.inverse();
  mx.multmatrix(my);
  return mx;
}

// find the best fit alignment to take the first structure into the second
// Put the result in the matrix 'mat'
//  This algorithm comes from Kabsch, Acta Cryst. (1978) A34, 827-828.
// Need the 2nd weight for the com calculation
int measure_fit(const AtomSel *sel1, const AtomSel *sel2, const float *x, 
                const float *y, const float *weight, 
                const int *order, Matrix4 *mat) {
  float comx[3];
  float comy[3];
  int num = sel1->selected;
  int ret_val;
  ret_val = measure_center(sel1, x, weight, comx);
  if (ret_val < 0) {
    return ret_val;
  }
  ret_val = measure_center(sel2, y, weight, comy);
  if (ret_val < 0) {
    return ret_val;
  }

  // the Kabsch method won't work of the number of atoms is less than 4
  // (and won't work in some cases of n > 4; I think it works so long as
  // three or more planes are needed to intersect all the data points
  switch (sel1->selected) {
  case 1: { // simple center of mass alignment
    Matrix4 tmp;
    tmp.translate(-comx[0], -comx[1], -comx[2]);
    tmp.translate(comy[0], comy[1], comy[2]);
    memcpy(mat->mat, tmp.mat, 16*sizeof(float));
    return MEASURE_NOERR;
  }
  case 3:
  case 2: { 
    // find the first (n-1) points (from each molecule)
    int pts[6], count = 0;
    int n;
    for (n=sel1->firstsel; n<=sel1->lastsel; n++) {
      if (sel1->on[n]) {
        pts[count++] = n;
        if (sel1->selected == 2) {
          count++;                   // will put y data in pts[3]
          break;
        }
        if (count == 2) break;
      }
    }
    for (n=sel2->firstsel; n<=sel2->lastsel; n++) {
      if (sel2->on[n]) {
        pts[count++] = n;
        if (sel1->selected == 2) {
          count++;
          break;
        }
        if (count == 4) break;
      }
    }
    if (count != 4) {
      return MEASURE_ERR_MISMATCHEDCNT;
    }
    
    // reorder the sel2 atoms according to the order parameter
    if (order != NULL) {
      int i; 
      int tmp[6];
      memcpy(tmp, pts, sizeof(pts));
      for (i=0; i<num; i++) {
        pts[i + num] = tmp[num + order[i]]; // order indices are 0-based
      } 
    }

    if (sel1->selected == 2) {
      *mat = myfit2(x+3*pts[0],y+3*pts[2], comx, comy);
      ret_val = 0;
    } else {
      *mat = myfit3(x+3*pts[0],x+3*pts[1],y+3*pts[2],y+3*pts[3],comx,comy);
      ret_val = 0;
    }  
    if (ret_val != 0) {
      return MEASURE_ERR_GENERAL;
    }

    return 0;
  }
  default:
    break;
  }
  // at this point I know all the data values are good
 

  // use the new RMS fit implementation by default unless told otherwise 
  char *opt = getenv("VMDRMSFITMETHOD");
  if (!opt || strcmp(opt, "oldvmd")) {
    int i, k;
    float *v1, *v2;
    v1 = new float[3*num];
    v2 = new float[3*num];
    for (k=0, i=sel1->firstsel; i<=sel1->lastsel; i++) {
      if (sel1->on[i]) {
        int ind = 3 * i;
        v1[k++] = x[ind    ];
        v1[k++] = x[ind + 1];
        v1[k++] = x[ind + 2];
      }
    }
    for (k=0, i=sel2->firstsel; i<=sel2->lastsel; i++) {
      if (sel2->on[i]) {
        int ind = 3 * i;
        v2[k++] = y[ind    ];
        v2[k++] = y[ind + 1];
        v2[k++] = y[ind + 2];
      }
    }

    // reorder the sel2 atoms according to the order parameter
    if (order != NULL) {
      int i; 
      float *tmp = new float[3*num];
      memcpy(tmp, v2, 3*num*sizeof(float));
      for (i=0; i<num; i++) {
        int ind = 3 * i;
        int idx = 3 * order[i]; // order indices are 0-based
        v2[ind    ] = tmp[idx    ];
        v2[ind + 1] = tmp[idx + 1];
        v2[ind + 2] = tmp[idx + 2];
      } 
      delete[] tmp;
    }

    float tmp[16];
    // MatrixFitRMS returns RMS distance of fitted molecule.  Would be nice
    // to return this information to the user since it would make computing
    // the fitted RMSD much faster (no need to get the matrix, apply the
    // transformation, recompute RMS).
    MatrixFitRMS(num, v1, v2, weight, tmp);

    delete [] v1;
    delete [] v2;
    //fprintf(stderr, "got err %f\n", err);
    // output from FitRMS is a 3x3 matrix, plus a pre-translation stored in
    // row 3, and a post-translation stored in column 3.
    float pre[3], post[3];
    for (i=0; i<3; i++) {
      post[i] = tmp[4*i+3];
      tmp[4*i+3] = 0;
    }
    for (i=0; i<3; i++) {
      pre[i] = tmp[12+i];
      tmp[12+i] = 0;
    }
    Matrix4 result;
    result.translate(pre);
    result.multmatrix(Matrix4(tmp));
    result.translate(post);
    memcpy(mat->mat, result.mat, 16*sizeof(float));
    return 0;
  }

  // the old RMS fit code doesn't support reordering of sel2 currently
  if (order != NULL) {
    return MEASURE_ERR_NOTSUP;
  }

  // a) compute R = r(i,j) = sum( w(n) * (y(n,i)-comy(i)) * (x(n,j)-comx(j)))
  Matrix4 R;
  int i,j;
  float scale = (float) num * num;
  for (i=0; i<3; i++) {
    for (j=0; j<3; j++) {
      float tmp = 0;
      int nx = 0, ny = 0, k = 0; 
      while (nx < sel1->num_atoms && ny < sel2->num_atoms) { 
        if (!sel1->on[nx]) {
          nx++;
          continue;
        }
        if (!sel2->on[ny]) {
          ny++;
          continue;
        }

        // found both, so get data
        
        tmp += weight[k] * (y[3*ny+i] - comy[i]) * (x[3*nx+j] - comx[j]) /
          scale;
        nx++;
        ny++;
        k++;
      }
      R.mat[4*i+j] = tmp;
    }
  }

  // b) 1) form R~R
  Matrix4 Rt;
  for (i=0; i<3; i++) {
    for (j=0; j<3; j++) {
      Rt.mat[4*i+j] = R.mat[4*j+i];
    }
  }
  Matrix4 RtR(R);
  RtR.multmatrix(Rt);

  // b) 2) find the eigenvalues and eigenvectors
  float evalue[3];
  float evector[3][3];
  float tmpmat[4][4];
  for (i=0; i<4; i++)
    for (j=0; j<4; j++)
      tmpmat[i][j]=RtR.mat[4*i+j];

  if(jacobi(tmpmat,evalue,evector) != 0) return MEASURE_ERR_NONZEROJACOBI;
  // transposition the evector matrix to put the vectors in rows
  float vectmp;
  vectmp=evector[0][1]; evector[0][1]=evector[1][0]; evector[1][0]=vectmp;
  vectmp=evector[0][2]; evector[0][2]=evector[2][0]; evector[2][0]=vectmp;
  vectmp=evector[2][1]; evector[2][1]=evector[1][2]; evector[1][2]=vectmp;


  // b) 4) sort so that the eigenvalues are from largest to smallest
  //      (or rather so a[0] is eigenvector with largest eigenvalue, ...)
  float *a[3];
  a[0] = evector[0];
  a[1] = evector[1];
  a[2] = evector[2];
#define SWAP(qq,ww) {                                           \
    float v; float *v1;                                         \
    v = evalue[qq]; evalue[qq] = evalue[ww]; evalue[ww] = v;    \
    v1 = a[qq]; a[qq] = a[ww]; a[ww] = v1;                      \
}
  if (evalue[0] < evalue[1]) {
    SWAP(0, 1);
  }
  if (evalue[0] < evalue[2]) {
    SWAP(0, 2);
  }
  if (evalue[1] < evalue[2]) {
    SWAP(1, 2);
  }

  
  // c) determine b(i) = R*a(i)
  float b[3][3];

  Rt.multpoint3d(a[0], b[0]);
  vec_normalize(b[0]);

  Rt.multpoint3d(a[1], b[1]);
  vec_normalize(b[1]);

  Rt.multpoint3d(a[2], b[2]);
  vec_normalize(b[2]);

  // d) compute U = u(i,j) = sum(b(k,i) * a(k,j))
  Matrix4 U;
  for (i=0; i<3; i++) {
    for (j=0; j<3; j++) {
      float *tmp = &(U.mat[4*j+i]);
      *tmp = 0;
      for (int k=0; k<3; k++) {
        *tmp += b[k][i] * a[k][j];
      }
    }
  }

  // Check the determinant of U.  If it's negative, we need to
  // flip the sign of the last row.
  float *um = U.mat;
  float det = 
    um[0]*(um[4+1]*um[8+2] - um[4+2]*um[8+1]) -
    um[1]*(um[4+0]*um[8+2] - um[4+2]*um[8+0]) +
    um[2]*(um[4+0]*um[8+1] - um[4+1]*um[8+0]);
  if (det < 0) {
    for (int q=0; q<3; q++) um[8+q] = -um[8+q];
  }

  // e) apply the offset for com
  Matrix4 tx;
  tx.translate(-comx[0], -comx[1], -comx[2]);
  Matrix4 ty;
  ty.translate(comy[0], comy[1], comy[2]);
  //  U.multmatrix(com);
  ty.multmatrix(U);
  ty.multmatrix(tx);
  memcpy(mat->mat, ty.mat, 16*sizeof(float));
  return MEASURE_NOERR;
}

// For different values of the random seed, the computed SASA's of brH.pdb 
// converge to within 1% of each other when the number of points is about
// 500.  We therefore use 500 as the default number.
#define NPTS 500 
extern int measure_sasa(const AtomSel *sel, const float *framepos,
    const float *radius, float srad, float *sasa, 
    ResizeArray<float> *sasapts, const AtomSel *restrictsel,
    const int *nsamples) {

  // check arguments
  if (!sel) return MEASURE_ERR_NOSEL;
  if (!sel->selected) {
    *sasa = 0;
    return MEASURE_NOERR;
  }
  if (!framepos) return MEASURE_ERR_NOFRAMEPOS;
  if (!radius)   return MEASURE_ERR_NORADII;
  if (restrictsel && restrictsel->num_atoms != sel->num_atoms)
    return MEASURE_ERR_MISMATCHEDCNT;

  int i;
  int npts = nsamples ? *nsamples : NPTS;
  float maxrad = -1;


  // find biggest atom radius 
  for (i=0; i<sel->num_atoms; i++) {
    float rad = radius[i];
    if (maxrad < rad) maxrad = rad;
  }

  // Find atoms within maxrad of each other.  
  // build a list of pairs for each atom
  ResizeArray<int> *pairlist = new ResizeArray<int>[sel->num_atoms];
  {
    GridSearchPair *pairs;
    pairs = vmd_gridsearch1(framepos, sel->num_atoms, sel->on, 
                            2.0f * (maxrad + srad), 0, sel->num_atoms * 1000);

    GridSearchPair *p, *tmp; 
    for (p = pairs; p != NULL; p = tmp) {
      int ind1=p->ind1;
      int ind2=p->ind2;
      pairlist[ind1].append(ind2);
      pairlist[ind2].append(ind1);
      tmp = p->next;
      free(p);
    }
  }

  static const float RAND_MAX_INV = 1.0f/VMD_RAND_MAX;
  // Seed the random number generator before each calculation.  This gives
  // reproducible results and still allows a more accurate answer to be
  // obtained by increasing the samples size.  I don't know if this is a
  // "good" seed value or not, I just picked something random-looking.
  vmd_srandom(38572111);

  // All the spheres use the same random points.  
  float *spherepts = new float[3*npts];
  for (i=0; i<npts; i++) {
    float u1 = (float) vmd_random();
    float u2 = (float) vmd_random();
    float z = 2.0f*u1*RAND_MAX_INV -1.0f;
    float phi = (float) (2.0f*VMD_PI*u2*RAND_MAX_INV);
    float R = sqrtf(1.0f-z*z);
    spherepts[3*i  ] = R*cosf(phi);
    spherepts[3*i+1] = R*sinf(phi);
    spherepts[3*i+2] = z;
  }

  const float prefac = (float) (4 * VMD_PI / npts);
  float totarea = 0.0f;
  // compute area for each atom based on its pairlist
  for (i=sel->firstsel; i<=sel->lastsel; i++) {
    if (sel->on[i]) {
      // only atoms in restrictsel contribute
      if (restrictsel && !restrictsel->on[i]) continue;
      const float *loc = framepos+3*i;
      float rad = radius[i]+srad;
      float surfpos[3];
      int surfpts = npts;
      const ResizeArray<int> &nbrs = pairlist[i];
      for (int j=0; j<npts; j++) {
        surfpos[0] = loc[0] + rad*spherepts[3*j  ];
        surfpos[1] = loc[1] + rad*spherepts[3*j+1];
        surfpos[2] = loc[2] + rad*spherepts[3*j+2];
        int on = 1;
        for (int k=0; k<nbrs.num(); k++) {
          int ind = nbrs[k];
          const float *nbrloc = framepos+3*ind;
          float radsq = radius[ind]+srad; radsq *= radsq;
          float dx = surfpos[0]-nbrloc[0];
          float dy = surfpos[1]-nbrloc[1];
          float dz = surfpos[2]-nbrloc[2];
          if (dx*dx + dy*dy + dz*dz <= radsq) {
            on = 0;
            break;
          }
        }
        if (on) {
          if (sasapts) {
            sasapts->append(surfpos[0]);
            sasapts->append(surfpos[1]);
            sasapts->append(surfpos[2]);
          }
        } else {
          surfpts--;
        }
      }
      float atomarea = prefac * rad * rad * surfpts;
      totarea += atomarea;
    }
  }

  delete [] pairlist;
  delete [] spherepts;
  *sasa = totarea;
  return 0;
}


//
// Calculate g(r)
//

// find the minimum image distance for one coordinate component 
// and square the result (orthogonal cells only).
static float fix_pbc_n_sqr(float delta, const float boxby2) {
  while (delta >= boxby2) { delta -= 2.0f * boxby2; }
  while (delta < -boxby2) { delta += 2.0f * boxby2; }
  return delta * delta;
}

// calculate the minimum distance between two positions with respect 
// to the periodic cell (orthogonal cells only).
static float min_dist_with_pbc(const float *a, const float *b, 
                                const float *boxby2) {
  float distsqr;
  distsqr  = fix_pbc_n_sqr(a[0] - b[0], boxby2[0]);
  distsqr += fix_pbc_n_sqr(a[1] - b[1], boxby2[1]);
  distsqr += fix_pbc_n_sqr(a[2] - b[2], boxby2[2]);
  return sqrtf(distsqr);
}

static inline double spherical_cap(const double &radius, const double &boxby2) {
  return (VMD_PI / 3.0 * (radius - boxby2) * (radius - boxby2)
          * ( 2.0 * radius + boxby2));
}


typedef struct {
  int threadid;
  int threadcount;
  int count_o_start;
  int count_o_end;
  const float *olist;
  int count_i;
  const float *ilist;
  int count_h;
  int *hlist;
  float delta;
  const float *boxby2;
  wkfmsgtimer *msgtp;
  int curframe;
  int maxframe;
} gofrparms_t;
    
// calculate the non-normalized pair-distribution function
// for two lists of atom coordinates and store the resulting
// histogram in the hlist array. orthogonal cell version.
//
// NOTE: this is just the workhorse function. special issues related
// to atoms present in both lists have to be dealt with in the uplevel
// functions, that then also can do various post-processing steps.
extern "C" void * measure_gofr_orth(void *voidparms) {
  // handle per-thread parameters
  gofrparms_t *parms = (gofrparms_t *) voidparms;
  const int count_o_start = parms->count_o_start;
  const int count_o_end   = parms->count_o_end;
  const int count_i       = parms->count_i;
  const int count_h       = parms->count_h;
  const float *olist      = parms->olist;
  const float *ilist      = parms->ilist;
  const float *boxby2     = parms->boxby2;
  wkfmsgtimer *msgtp         = parms->msgtp;
  const int curframe      = parms->curframe;
  const int maxframe      = parms->maxframe;
  int *hlist              = parms->hlist;
  // other local variables
  int i, j, idx;
  float dist;
  const float deltascale = 1.0f / parms->delta;
  int msgcnt=0;

  // loop over the chunk of pairs that was associated to this thread.
  for (i=count_o_start; i<count_o_end; ++i) {

    // print progress messages only for thread(s) that have
    // a timer defined (usually only tid==0).
    if (msgtp && wkf_msg_timer_timeout(msgtp)) {
      char tmpbuf[1024];
      if (msgcnt==0) 
        sprintf(tmpbuf, "timestep %d of %d", curframe, maxframe);
      else
        sprintf(tmpbuf, "timestep %d of %d: (%6.2f %% complete)", curframe, maxframe, 
                (100.0f * i) / (float) (count_o_end - count_o_start + 1));
      msgInfo << "vmd_measure_gofr_orth: " << tmpbuf << sendmsg;
      msgcnt++;
      // XXX we should update the display here...
    }

    for (j=0; j<count_i; ++j) {
      // calculate distance and add to histogram
      dist = min_dist_with_pbc(&olist[i*3], &ilist[j*3], boxby2);
      idx = (int) (dist * deltascale);
      if ((idx >= 0) && (idx < count_h)) 
        ++hlist[idx];
    }
  }

  return MEASURE_NOERR;
}

// main entry point for 'measure gofr'.
// tasks:
// - sanity check on arguments.
// - select proper algorithm for PBC treatment.
int measure_gofr(AtomSel *sel1, AtomSel *sel2, MoleculeList *mlist,
                 const int count_h, double *gofr, double *numint, double *histog,
                 const float delta, int first, int last, int step, int *framecntr,
                 int usepbc, int selupdate) {
  int i, j, frame;
  float a, b, c, alpha, beta, gamma;
  int isortho=0;    // orthogonal unit cell not assumed by default.
  int duplicates=0; // counter for duplicate atoms in both selections.

  // initialize a/b/c/alpha/beta/gamma to arbitrary defaults to please the compiler.
  a=b=c=9999999.0;
  alpha=beta=gamma=90.0;

  // reset counter for total, skipped, and _orth processed frames.
  framecntr[0]=framecntr[1]=framecntr[2]=0;

  // First round of sanity checks.
  // neither list can be undefined
  if (!sel1 || !sel2 ) {
    return MEASURE_ERR_NOSEL;
  }

  // make sure that both selections are from the same molecule
  // so that we know that PBC unit cell info is the same for both
  if (sel2->molid() != sel1->molid()) {
    return MEASURE_ERR_MISMATCHEDMOLS;
  }

  Molecule *mymol = mlist->mol_from_id(sel1->molid());
  int maxframe = mymol->numframes() - 1;
  int nframes = 0;

  if (last == -1)
    last = maxframe;

  if ((last < first) || (last < 0) || (step <=0) || (first < -1)
      || (maxframe < 0) || (last > maxframe)) {
      msgErr << "measure gofr: bad frame range given." 
             << " max. allowed frame#: " << maxframe << sendmsg;
    return MEASURE_ERR_BADFRAMERANGE;
  }

  // test for non-orthogonal PBC cells, zero volume cells, etc.
  if (usepbc) { 
    for (isortho=1, nframes=0, frame=first; frame <=last; ++nframes, frame += step) {
      const Timestep *ts;

      if (first == -1) {
        // use current frame only. don't loop.
        ts = sel1->timestep(mlist);
        frame=last;
      } else {
        ts = mymol->get_frame(frame);
      }
      // get periodic cell information for current frame
      a = ts->a_length;
      b = ts->b_length;
      c = ts->c_length;
      alpha = ts->alpha;
      beta = ts->beta;
      gamma = ts->gamma;

      // check validity of PBC cell side lengths
      if (fabsf(a*b*c) < 0.0001) {
        msgErr << "measure gofr: unit cell volume is zero." << sendmsg;
        return MEASURE_ERR_GENERAL;
      }

      // check PBC unit cell shape to select proper low level algorithm.
      if ((alpha != 90.0) || (beta != 90.0) || (gamma != 90.0)) {
        isortho=0;
      }
    }
  } else {
    // initialize a/b/c/alpha/beta/gamma to arbitrary defaults
    isortho=1;
    a=b=c=9999999.0;
    alpha=beta=gamma=90.0;
  }

  // until we can handle non-orthogonal periodic cells, this is fatal
  if (!isortho) {
    msgErr << "measure gofr: only orthorhombic cells are supported (for now)." << sendmsg;
    return MEASURE_ERR_GENERAL;
  }

  // clear the result arrays
  for (i=0; i<count_h; ++i) {
    gofr[i] = numint[i] = histog[i] = 0.0;
  }

  // pre-allocate coordinate buffers of the max size we'll
  // ever need, so we don't have to reallocate if/when atom
  // selections are updated on-the-fly
  float *sel1coords = new float[3*sel1->num_atoms];
  float *sel2coords = new float[3*sel2->num_atoms];

  // setup status message timer
  wkfmsgtimer *msgt = wkf_msg_timer_create(5);

  // threading setup.
  wkf_thread_t *threads;
  gofrparms_t  *parms;
#if defined(VMDTHREADS)
  int numprocs = wkf_thread_numprocessors();
#else
  int numprocs = 1;
#endif

  threads = new wkf_thread_t[numprocs];
  memset(threads, 0, numprocs * sizeof(wkf_thread_t));

  // allocate and (partially) initialize array of per-thread parameters
  parms = new gofrparms_t[numprocs];
  for (i=0; i<numprocs; ++i) {
    parms[i].threadid = i;
    parms[i].threadcount = numprocs;
    parms[i].delta = (float) delta;
    parms[i].msgtp = NULL;
    parms[i].count_h = count_h;
    parms[i].hlist = new int[count_h];
  }

  msgInfo << "measure gofr: using multi-threaded implementation with " 
          << numprocs << ((numprocs > 1) ? " CPUs" : " CPU") << sendmsg;

  for (nframes=0,frame=first; frame <=last; ++nframes, frame += step) {
    const Timestep *ts1, *ts2;

    if (frame  == -1) {
      // use current frame only. don't loop.
      ts1 = sel1->timestep(mlist);
      ts2 = sel2->timestep(mlist);
      frame=last;
    } else {
      sel1->which_frame = frame;
      sel2->which_frame = frame;
      ts1 = ts2 = mymol->get_frame(frame); // requires sels from same mol
    }

    if (usepbc) {
      // get periodic cell information for current frame
      a     = ts1->a_length;
      b     = ts1->b_length;
      c     = ts1->c_length;
      alpha = ts1->alpha;
      beta  = ts1->beta;
      gamma = ts1->gamma;
    }

    // compute half periodic cell size
    float boxby2[3];
    boxby2[0] = 0.5f * a;
    boxby2[1] = 0.5f * b;
    boxby2[2] = 0.5f * c;

    // update the selections if the user desires it
    if (selupdate) {
      if (sel1->change(NULL, mymol) != AtomSel::PARSE_SUCCESS)
        msgErr << "measure gofr: failed to evaluate atom selection update";
      if (sel2->change(NULL, mymol) != AtomSel::PARSE_SUCCESS)
        msgErr << "measure gofr: failed to evaluate atom selection update";
    }

    // check for duplicate atoms in the two lists, as these will have
    // to be subtracted back out of the first histogram slot
    if (sel2->molid() == sel1->molid()) {
      int i;
      for (i=0, duplicates=0; i<sel1->num_atoms; ++i) {
        if (sel1->on[i] && sel2->on[i])
          ++duplicates;
      }
    }

    // copy selected atoms to the two coordinate lists
    // requires that selections come from the same molecule
    const float *framepos = ts1->pos;
    for (i=0, j=0; i<sel1->num_atoms; ++i) {
      if (sel1->on[i]) {
        int a = i*3;
        sel1coords[j    ] = framepos[a    ];
        sel1coords[j + 1] = framepos[a + 1];
        sel1coords[j + 2] = framepos[a + 2];
        j+=3;
      }
    }
    framepos = ts2->pos;
    for (i=0, j=0; i<sel2->num_atoms; ++i) {
      if (sel2->on[i]) {
        int a = i*3;
        sel2coords[j    ] = framepos[a    ];
        sel2coords[j + 1] = framepos[a + 1];
        sel2coords[j + 2] = framepos[a + 2];
        j+=3;
      }
    }

    // clear the histogram for this frame
    // and set up argument structure for threaded execution.
    int maxframe = (int) ((last - first + 1) / ((float) step));
    for (i=0; i<numprocs; ++i) {
      memset(parms[i].hlist, 0, count_h * sizeof(int));
      parms[i].boxby2 = boxby2;
      parms[i].curframe = frame;
      parms[i].maxframe = maxframe;
    }
    parms[0].msgtp = msgt;
    
    if (isortho && sel1->selected && sel2->selected) {
      int count_o = sel1->selected;
      int count_i = sel2->selected;
      const float *olist = sel1coords;
      const float *ilist = sel2coords;
      // make sure the outer loop is the longer one to have 
      // better threading efficiency and cache utilization.
      if (count_o < count_i) {
        count_o = sel2->selected;
        count_i = sel1->selected;
        olist = sel2coords;
        ilist = sel1coords;
      }

      // distribute outer loop across threads in fixed size chunks.
      // this should work very well for small numbers of threads.
      // thrdelta is the chunk size and we need it to be at least 1 
      // _and_ numprocs*thrdelta >= count_o.
      int thrdelta = (count_o + (numprocs-1)) / numprocs;
      int o_min = 0;
      int o_max = thrdelta;
      for (i=0; i<numprocs; ++i, o_min += thrdelta, o_max += thrdelta) {
        if (o_max >  count_o)  o_max = count_o; // curb loop to max
        if (o_min >= count_o)  o_max = - 1;     // no work for this thread. too little data.
        parms[i].count_o_start = o_min;
        parms[i].count_o_end   = o_max;
        parms[i].count_i       = count_i;
        parms[i].olist         = olist;
        parms[i].ilist         = ilist;
      }
      
      // do the gofr calculation for orthogonal boxes.
      // XXX. non-orthogonal box not supported yet. detected and handled above.
#if defined(VMDTHREADS)
      for (i=0; i<numprocs; ++i) {
        wkf_thread_create(&threads[i], measure_gofr_orth, &parms[i]);
      }
      for (i=0; i<numprocs; ++i) {
        wkf_thread_join(threads[i], NULL);
      } 
#else
      measure_gofr_orth((void *) &parms[0]);
#endif
      ++framecntr[2]; // frame processed with _orth algorithm
    } else {
      ++framecntr[1]; // frame skipped
    }
    ++framecntr[0];   // total frames.

    // correct the first histogram slot for the number of atoms that are 
    // present in both lists. they'll end up in the first histogram bin. 
    // we subtract only from the first thread histogram which is always defined.
    parms[0].hlist[0] -= duplicates;

    // in case of going 'into the edges', we should cut 
    // off the part that is not properly normalized to 
    // not confuse people that don't know about this.
    int h_max=count_h;
    float smallside=a;
    if (isortho && usepbc) {
      if(b < smallside) {
        smallside=b;
      }
      if(c < smallside) {
        smallside=c;
      }
      h_max=(int) (sqrtf(0.5f)*smallside/delta) +1;
      if (h_max > count_h) {
        h_max=count_h;
      }
    }
    // compute normalization function.
    double all=0.0;
    double pair_dens = 0.0;
    if (sel1->selected && sel2->selected) {
      if (usepbc) {
        pair_dens = a * b * c / ((double)sel1->selected * (double)sel2->selected - (double)duplicates);
      } else { // assume a particle volume of 30 \AA^3 (~ 1 water).
        pair_dens = 30.0 * (double)sel1->selected / 
          ((double)sel1->selected * (double)sel2->selected - (double)duplicates);
      }
    }
    
    // XXX for orthogonal boxes, we can reduce this to rmax < sqrt(0.5)*smallest side
    for (i=0; i<h_max; ++i) {
      // radius of inner and outer sphere that form the spherical slice
      double r_in  = delta * (double)i;
      double r_out = delta * (double)(i+1);
      double slice_vol = 4.0 / 3.0 * VMD_PI
        * ((r_out * r_out * r_out) - (r_in * r_in * r_in));

      if (isortho && usepbc) {
        // add correction for 0.5*box < r <= sqrt(0.5)*box
        if (r_out > boxby2[0]) {
          slice_vol -= 2.0 * spherical_cap(r_out, boxby2[0]);
        }
        if (r_out > boxby2[1]) {
          slice_vol -= 2.0 * spherical_cap(r_out, boxby2[1]);
        }
        if (r_out > boxby2[2]) {
          slice_vol -= 2.0 * spherical_cap(r_out, boxby2[2]);
        }
        if (r_in > boxby2[0]) {
          slice_vol += 2.0 * spherical_cap(r_in, boxby2[0]);
        }
        if (r_in > boxby2[1]) {
          slice_vol += 2.0 * spherical_cap(r_in, boxby2[1]);
        }
        if (r_in > boxby2[2]) {
          slice_vol += 2.0 * spherical_cap(r_in, boxby2[2]);
        }
      }

      double normf = pair_dens / slice_vol;
      double histv = 0.0;
      for (j=0; j<numprocs; ++j) {
        histv += (double) parms[j].hlist[i];
      }
      gofr[i] += normf * histv;
      all     += histv;
      if (sel1->selected) {
        numint[i] += all / (double)(sel1->selected);
      }
      histog[i] += histv;
    }
  }
  delete [] sel1coords;
  delete [] sel2coords;

  double norm = 1.0 / (double) nframes;
  for (i=0; i<count_h; ++i) {
    gofr[i]   *= norm;
    numint[i] *= norm;
    histog[i] *= norm;
  }
  wkf_msg_timer_destroy(msgt);

  // release thread-related storage
  for (i=0; i<numprocs; ++i) {
    delete [] parms[i].hlist;
  }
  delete [] threads;
  delete [] parms;

  return MEASURE_NOERR;
}


int measure_geom(MoleculeList *mlist, int *molid, int *atmid, ResizeArray<float> *gValues,
                 int frame, int first, int last, int defmolid, int geomtype) {
  int i, ret_val;
  for(i=0; i < geomtype; i++) {
    // make sure an atom is not repeated in this list
    if(i > 0 && molid[i-1]==molid[i] && atmid[i-1]==atmid[i]) {
      printf("measure_geom: %i/%i %i/%i\n", molid[i-1],atmid[i-1],molid[i],atmid[i]);
      return MEASURE_ERR_REPEATEDATOM;
    }
  }

  float value;
  int max_ts, orig_ts;

  // use the default molecule to determine which frames to cycle through
  Molecule *mol = mlist->mol_from_id(defmolid);
  if( !mol )
    return MEASURE_ERR_NOMOLECULE;
  
  // get current frame number and make sure there are frames
  if((orig_ts = mol->frame()) < 0)
    return MEASURE_ERR_NOFRAMES;
  
  // get the max frame number and determine frame range
  max_ts = mol->numframes()-1;
  if (frame<0) {
    if (first<0 && last<0) first = last = orig_ts; 
    if (last<0 || last>max_ts) last = max_ts;
    if (first<0) first = 0;
  } else {
    if (frame>max_ts) frame = max_ts;
    first = last = frame; 
  }
  
  // go through all the frames, calculating values
  for(i=first; i <= last; i++) {
    mol->override_current_frame(i);
    switch (geomtype) {
    case MEASURE_BOND:
      if ((ret_val=calculate_bond(mlist, molid, atmid, &value))<0)
        return ret_val;
      gValues->append(value);
      break;
    case MEASURE_ANGLE:
      if ((ret_val=calculate_angle(mlist, molid, atmid, &value))<0)
        return ret_val;
      gValues->append(value);
      break;
    case MEASURE_DIHED:
      if ((ret_val=calculate_dihed(mlist, molid, atmid, &value))<0)
        return ret_val;
      gValues->append(value);
      break;
    }
  }
  
  // reset the current frame
  mol->override_current_frame(orig_ts);
  
  return MEASURE_NOERR;
}
  
  
// calculate the value of this geometry, and return it
int calculate_bond(MoleculeList *mlist, int *molid, int *atmid, float *value) {

  // get coords to calculate distance 
  int ret_val;
  float pos1[3], pos2[3];
  if ((ret_val=normal_atom_coord(mlist->mol_from_id(molid[0]), atmid[0], pos1))<0)
    return ret_val;
  if ((ret_val=normal_atom_coord(mlist->mol_from_id(molid[1]), atmid[1], pos2))<0)
    return ret_val;
  
  vec_sub(pos2, pos2, pos1);
  *value = norm(pos2);

  return MEASURE_NOERR;
}

// calculate the value of this geometry, and return it
int calculate_angle(MoleculeList *mlist, int *molid, int *atmid, float *value) {

  // get coords to calculate distance 
  int ret_val;
  float pos1[3], pos2[3], pos3[3], r1[3], r2[3];
  if((ret_val=normal_atom_coord(mlist->mol_from_id(molid[0]), atmid[0], pos1))<0)
    return ret_val;
  if((ret_val=normal_atom_coord(mlist->mol_from_id(molid[1]), atmid[1], pos2))<0)
    return ret_val;
  if((ret_val=normal_atom_coord(mlist->mol_from_id(molid[2]), atmid[2], pos3))<0)
    return ret_val;

  vec_sub(r1, pos1, pos2);
  vec_sub(r2, pos3, pos2);
  *value = angle(r1, r2);

  return MEASURE_NOERR;
}

// calculate the value of this geometry, and return it
int calculate_dihed(MoleculeList *mlist, int *molid, int *atmid, float *value) {

  // get coords to calculate distance 
  int ret_val;
  float pos1[3], pos2[3], pos3[3], pos4[3]; 
  if((ret_val=normal_atom_coord(mlist->mol_from_id(molid[0]), atmid[0], pos1))<0)
    return ret_val;
  if((ret_val=normal_atom_coord(mlist->mol_from_id(molid[1]), atmid[1], pos2))<0)
    return ret_val;
  if((ret_val=normal_atom_coord(mlist->mol_from_id(molid[2]), atmid[2], pos3))<0)
    return ret_val;
  if((ret_val=normal_atom_coord(mlist->mol_from_id(molid[3]), atmid[3], pos4))<0)
    return ret_val;

  *value = dihedral(pos1, pos2, pos3, pos4);

  return MEASURE_NOERR;
}


// for the given Molecule, find the UNTRANSFORMED coords for the given atom
// return Molecule pointer if successful, NULL otherwise.
int normal_atom_coord(Molecule *mol, int a, float *pos) {
  Timestep *now;

  int cell[3];
  memset(cell, 0, 3*sizeof(int));

  // get the molecule pointer, and get the coords for the current timestep
  int ret_val = check_mol(mol, a);
  if (ret_val<0) 
    return ret_val;

  if ((now = mol->current())) {
    memcpy((void *)pos, (void *)(now->pos + 3*a), 3*sizeof(float));
    
    // Apply periodic image transformation before returning
    Matrix4 mat;
    now->get_transform_from_cell(cell, mat);
    mat.multpoint3d(pos, pos);
    
    return MEASURE_NOERR;
  }
  
  // if here, error (i.e. molecule contains no frames)
  return MEASURE_ERR_NOFRAMES;
}


// check whether the given molecule & atom index is OK
// if OK, return Molecule pointer; otherwise, return NULL
int check_mol(Molecule *mol, int a) {

  if (!mol)
    return MEASURE_ERR_NOMOLECULE;
  if (a < 0 || a >= mol->nAtoms)
    return MEASURE_ERR_BADATOMID;
  
  return MEASURE_NOERR;
}


int measure_energy(MoleculeList *mlist, int *molid, int *atmid, int natoms, ResizeArray<float> *gValues,
                 int frame, int first, int last, int defmolid, double *params, int geomtype) {
  int i, ret_val;
  for(i=0; i < natoms; i++) {
    // make sure an atom is not repeated in this list
    if(i > 0 && molid[i-1]==molid[i] && atmid[i-1]==atmid[i]) {
      printf("measure_energy: %i/%i %i/%i\n", molid[i-1],atmid[i-1],molid[i],atmid[i]);
      return MEASURE_ERR_REPEATEDATOM;
    }
  }

  float value;
  int max_ts, orig_ts;

  // use the default molecule to determine which frames to cycle through
  Molecule *mol = mlist->mol_from_id(defmolid);
  if( !mol )
    return MEASURE_ERR_NOMOLECULE;
  
  // get current frame number and make sure there are frames
  if((orig_ts = mol->frame()) < 0)
    return MEASURE_ERR_NOFRAMES;
  
  // get the max frame number and determine frame range
  max_ts = mol->numframes()-1;
  if (frame==-1) {
    if (first<0 && last<0) first = last = orig_ts; 
    if (last<0 || last>max_ts) last = max_ts;
    if (first<0) first = 0;
  } else {
    if (frame>max_ts || frame==-2) frame = max_ts;
    first = last = frame; 
  }
  
  // go through all the frames, calculating values
  for(i=first; i <= last; i++) {
    mol->override_current_frame(i);
    switch (geomtype) {
    case MEASURE_BOND:
      if ((ret_val=compute_bond_energy(mlist, molid, atmid, &value, (float) params[0], (float) params[1]))<0)
        return ret_val;
      gValues->append(value);
      break;
    case MEASURE_ANGLE:
      if ((ret_val=compute_angle_energy(mlist, molid, atmid, &value, (float) params[0], (float) params[1], (float) params[2], (float) params[3]))<0)
        return ret_val;
      gValues->append(value);
      break;
    case MEASURE_DIHED:
      if ((ret_val=compute_dihed_energy(mlist, molid, atmid, &value, (float) params[0], int(params[1]), (float) params[2]))<0)
        return ret_val;
      gValues->append(value);
      break;
    case MEASURE_IMPRP:
      if ((ret_val=compute_imprp_energy(mlist, molid, atmid, &value, (float) params[0], (float) params[1]))<0)
        return ret_val;
      gValues->append(value);
      break;
    case MEASURE_VDW:
      if ((ret_val=compute_vdw_energy(mlist, molid, atmid, &value, (float) params[0], (float) params[1], (float) params[2], (float) params[3], (float) params[4], (float) params[5]))<0)
        return ret_val;
      gValues->append(value);
      break;
    case MEASURE_ELECT:
      if ((ret_val=compute_elect_energy(mlist, molid, atmid, &value, (float) params[0], (float) params[1], (bool) params[2], (bool) params[3], (float) params[4]))<0)
        return ret_val;
      gValues->append(value);
      break;
    }
  }
  
  // reset the current frame
  mol->override_current_frame(orig_ts);
  
  return MEASURE_NOERR;
}
  
// calculate the energy of this geometry
int compute_bond_energy(MoleculeList *mlist, int *molid, int *atmid, float *energy, float k, float x0) {
  int ret_val;
  float dist;

  // Get the coordinates
  if ((ret_val=calculate_bond(mlist, molid, atmid, &dist))<0)
        return ret_val;
  float x = dist-x0;
  *energy = k*x*x;

  return MEASURE_NOERR;
}

// calculate the energy of this geometry
int compute_angle_energy(MoleculeList *mlist, int *molid, int *atmid, float *energy,
                         float k, float x0, float kub, float s0) {
  int ret_val;
  float value;

  // Get the coordinates
  if ((ret_val=calculate_angle(mlist, molid, atmid, &value))<0)
        return ret_val;
  float x = (float) DEGTORAD((value-x0));
  float s = 0.0f;

  if (kub>0.0f) {
    int twoatoms[2];
    twoatoms[0] = atmid[0];
    twoatoms[1] = atmid[2];
    if ((ret_val=calculate_bond(mlist, molid, twoatoms, &value))<0)
      return ret_val;
    s = value-s0;
  }

  *energy = k*x*x + kub*s*s;

  return MEASURE_NOERR;
}

// calculate the energy of this geometry
int compute_dihed_energy(MoleculeList *mlist, int *molid, int *atmid, float *energy,
                         float k, int n, float delta) {
  int ret_val;
  float value;

  // Get the coordinates
  if ((ret_val=calculate_dihed(mlist, molid, atmid, &value))<0)
        return ret_val;
  *energy = k*(1+cosf((float) (DEGTORAD((n*value-delta)))));

  return MEASURE_NOERR;
}

// calculate the energy of this geometry
int compute_imprp_energy(MoleculeList *mlist, int *molid, int *atmid, float *energy,
                         float k, float x0) {
  int ret_val;
  float value;

  // Get the coordinates
  if ((ret_val=calculate_dihed(mlist, molid, atmid, &value))<0)
        return ret_val;
  float x = (float) (DEGTORAD((value-x0)));
  *energy = k*x*x;

  return MEASURE_NOERR;
}

// Calculate the VDW energy for specified pair of atoms
// VDW energy:                                               
// Evdw = eps * ((Rmin/dist)**12 - 2*(Rmin/dist)**6)         
// eps = sqrt(eps1*eps2),  Rmin = Rmin1+Rmin2                
int compute_vdw_energy(MoleculeList *mlist, int *molid, int *atmid, float *energy, float eps1, float rmin1,
                         float eps2, float rmin2, float cutoff, float switchdist) {
  int ret_val;
  float dist;

  // Get the coordinates
  if ((ret_val=calculate_bond(mlist, molid, atmid, &dist))<0)
    return ret_val;

  float sw=1.0;
  if (switchdist>0.0 && cutoff>0.0) {
    if (dist>=cutoff) {
      sw = 0.0;
    } else if (dist>=switchdist) {
      // This is the CHARMM switching function
      float dist2 = dist*dist;
      float cut2 = cutoff*cutoff;
      float switch2 = switchdist*switchdist;
      float s = cut2-dist2;
      float range = cut2-switch2;
      sw = s*s*(cut2+2*dist2-3*switch2)/(range*range*range);
    }
  }

  float term6 = (float) powf((rmin1+rmin2)/dist,6);
  *energy = sqrtf(eps1*eps2)*(term6*term6 - 2.0f*term6)*sw;

  return MEASURE_NOERR;
}

int compute_elect_energy(MoleculeList *mlist, int *molid, int *atmid, float *energy, float q1, float q2,
                         bool flag1, bool flag2, float cutoff) {
  int ret_val;
  float dist;

  // Get the coordinates
  if ((ret_val=calculate_bond(mlist, molid, atmid, &dist))<0)
    return ret_val;

  // Get atom charges
  if (!flag1) q1 = mlist->mol_from_id(molid[0])->charge()[atmid[0]];
  if (!flag2) q2 = mlist->mol_from_id(molid[0])->charge()[atmid[1]];

  if (cutoff>0.0) {
    if (dist<cutoff) {
      float efac = 1.0f-dist*dist/(cutoff*cutoff);
      *energy = 332.0636f*q1*q2/dist*efac*efac;
    } else {
      *energy = 0.0f;
    }
  } else {
    *energy = 332.0636f*q1*q2/dist;
  }

  return MEASURE_NOERR;
}
 

// Compute the center of mass for a given selection.
// The result is put in rcom which has to have a size of at least 3.
static void center_of_mass(AtomSel *sel, MoleculeList *mlist, float *rcom) {
  int i;
  float m = 0, mtot = 0;
  Molecule *mol = mlist->mol_from_id(sel->molid());

  // get atom masses
  const float *mass = mol->mass();

  // get atom coordinates
  const float *pos = sel->coordinates(mlist);

  memset(rcom, 0, 3*sizeof(float));

  // center of mass
  for (i=sel->firstsel; i<=sel->lastsel; i++) {
    if (sel->on[i]) {
      int ind = i * 3;

      m = mass[i];

      rcom[0] += m*pos[ind    ];
      rcom[1] += m*pos[ind + 1];
      rcom[2] += m*pos[ind + 2];

      // total mass
      mtot += m;
    }
  }

  rcom[0] /= mtot;
  rcom[1] /= mtot;
  rcom[2] /= mtot;
}


// Calculate principle axes and moments of inertia for selected atoms.
// The corresponding eigenvalues are also returned, they can be used
// to see if two axes are equivalent. The center of mass will be put
// in parameter rcom.
// The user can provide his own set of coordinates in coor. If this
// parameter is NULL then the coordinates from the selection are used.
extern int measure_inertia(AtomSel *sel, MoleculeList *mlist, const float *coor, float rcom[3],
                           float priaxes[3][3], float itensor[4][4], float evalue[3]) {
  if (!sel)                     return MEASURE_ERR_NOSEL;
  if (sel->num_atoms == 0)      return MEASURE_ERR_NOATOMS;

  Molecule *mol = mlist->mol_from_id(sel->molid());

  float x, y, z, m;
  float Ixx=0, Iyy=0, Izz=0, Ixy=0, Ixz=0, Iyz=0;
  int i,j=0;

  // need to put 3x3 inertia tensor into 4x4 matrix for jacobi eigensolver
  // itensor = {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 1}};
  memset(itensor, 0, 16*sizeof(float));
  itensor[3][3] = 1.0;

  // compute center of mass
  center_of_mass(sel, mlist, rcom);

  // get atom coordinates
  const float *pos = sel->coordinates(mlist);

  // get atom masses
  const float *mass = mol->mass();


  // moments of inertia tensor
  for (i=sel->firstsel; i<=sel->lastsel; i++) {
    if (sel->on[i]) {
      // position relative to COM
      if (coor) {
        // use user provided coordinates
        x = coor[j*3    ] - rcom[0];
        y = coor[j*3 + 1] - rcom[1];
        z = coor[j*3 + 2] - rcom[2];
        j++;
      } else {
        // use coordinates from selection
        x = pos[i*3    ] - rcom[0];
        y = pos[i*3 + 1] - rcom[1];
        z = pos[i*3 + 2] - rcom[2];
      }

      m = mass[i];

      Ixx += m*(y*y+z*z);
      Iyy += m*(x*x+z*z);
      Izz += m*(x*x+y*y);
      Ixy -= m*x*y;
      Ixz -= m*x*z;
      Iyz -= m*y*z;
    }
  }

  itensor[0][0] = Ixx;
  itensor[1][1] = Iyy;
  itensor[2][2] = Izz;
  itensor[0][1] = Ixy;
  itensor[1][0] = Ixy;
  itensor[0][2] = Ixz;
  itensor[2][0] = Ixz;
  itensor[1][2] = Iyz;
  itensor[2][1] = Iyz;

  // Find the eigenvalues and eigenvectors of moments of inertia tensor.
  // The eigenvectors correspond to the principle axes of inertia.
  float evector[3][3];
  if (jacobi(itensor,evalue,evector) != 0) return MEASURE_ERR_NONZEROJACOBI;

  // transpose the evector matrix to put the vectors in rows
  float vectmp;
  vectmp=evector[0][1]; evector[0][1]=evector[1][0]; evector[1][0]=vectmp;
  vectmp=evector[0][2]; evector[0][2]=evector[2][0]; evector[2][0]=vectmp;
  vectmp=evector[2][1]; evector[2][1]=evector[1][2]; evector[1][2]=vectmp;


  // sort so that the eigenvalues are from largest to smallest
  // (or rather so a[0] is eigenvector with largest eigenvalue, ...)
  float *a[3];
  a[0] = evector[0];
  a[1] = evector[1];
  a[2] = evector[2];
  // The code for SWAP is copied from measure_fit().
  // It swaps rows in the eigenvector matrix.
#define SWAP(qq,ww) {                                           \
    float v; float *v1;                                         \
    v = evalue[qq]; evalue[qq] = evalue[ww]; evalue[ww] = v;    \
    v1 = a[qq]; a[qq] = a[ww]; a[ww] = v1;                      \
}
  if (evalue[0] < evalue[1]) {
    SWAP(0, 1);
  }
  if (evalue[0] < evalue[2]) {
    SWAP(0, 2);
  }
  if (evalue[1] < evalue[2]) {
    SWAP(1, 2);
  }

#if 0
  // If the 2nd and 3rd eigenvalues are identical and not close to zero
  // then the corresponding axes are not unique. 
  if (evalue[1]/evalue[0]>0.1 && fabs(evalue[1]-evalue[2])/evalue[0]<0.05
      && fabs(evalue[0]-evalue[1])/evalue[0]>0.05) {
    msgInfo << "Principal axes of inertia 2 and 3 are not unique!" << sendmsg;
  }
#endif

  for (i=0; i<3; i++) {
    for (j=0; j<3; j++) 
      priaxes[i][j] = a[i][j];
  }

  return MEASURE_NOERR;
}

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