EdgeMesher.cpp

Go to the documentation of this file.

#include "meshkit/EdgeMesher.hpp"
#include "meshkit/Matrix.hpp"
#include "meshkit/MKCore.hpp"
#include "meshkit/ModelEnt.hpp"
#include "meshkit/SizingFunction.hpp"
#include "meshkit/RegisterMeshOp.hpp"
#include "moab/ReadUtilIface.hpp"
#include <vector>
#include <math.h>

namespace MeshKit
{

//---------------------------------------------------------------------------//
//Entity Type initilization for edge meshing
moab::EntityType EdgeMesher_tps[] = {moab::MBVERTEX, moab::MBEDGE, moab::MBMAXTYPE};
const moab::EntityType* EdgeMesher::output_types()
  { return EdgeMesher_tps; }

//---------------------------------------------------------------------------//
// Construction Function for Edge Mesher
EdgeMesher::EdgeMesher(MKCore *mk_core, const MEntVector &me_vec) 
        : MeshScheme(mk_core, me_vec), schemeType(EQUAL), ratio(1.2)
{

}
#if 0
//---------------------------------------------IBERRCHK(g_err, "Trouble get the adjacent geometric nodes on a surface.");------------------------------//
// measure function: compute the distance between the parametric coordinate 
// ustart and the parametric coordinate uend
double EdgeMesher::measure(iGeom::EntityHandle ent, double ustart, double uend) const
{
        double umin, umax;

        //get the minimal and maximal parametrical coordinates of the edge
        iGeom::Error err = mk_core()->igeom_instance()->getEntURange(ent, umin, umax);
        IBERRCHK(err, "Trouble getting parameter range for edge.");

        if (umin == umax) throw Error(MK_BAD_GEOMETRIC_EVALUATION, "Edge evaluated to same parameter umax and umin.");

        //compute the distance for edges
        double measure;
        err = mk_core()->igeom_instance()->measure(&ent, 1, &measure);

        IBERRCHK(err, "Trouble getting edge measure.");

        return measure * (uend - ustart) / (umax - umin);
}
#endif
//---------------------------------------------------------------------------//
// setup function: set up the number of intervals for edge meshing through the 
// sizing function
void EdgeMesher::setup_this()
{
    //compute the number of intervals for the associated ModelEnts, from the size set on them
    //the sizing function they point to, or a default sizing function
  for (MEntSelection::iterator mit = mentSelection.begin(); mit != mentSelection.end(); mit++)
  {
    ModelEnt *me = mit->first;

      //first check to see whether entity is meshed
    if (me->get_meshed_state() >= COMPLETE_MESH || me->mesh_intervals() > 0)
      continue;
    
    SizingFunction *sf = mk_core()->sizing_function(me->sizing_function_index());
    if (!sf && me -> mesh_intervals() < 0 && me -> interval_firmness() == DEFAULT &&
        mk_core()->sizing_function(0))
      sf = mk_core()->sizing_function(0);
    
    if (!sf && me -> mesh_intervals() < 0 && me -> interval_firmness() == DEFAULT)
    {
        //no sizing set, just assume default #intervals as 4
      me->mesh_intervals(4);
      me->interval_firmness(DEFAULT);
    }
    else
    {
        //check # intervals first, then size, and just choose for now
      if (sf->intervals() > 0)
      {
        if (me->constrain_even() && sf->intervals()%2)
          me -> mesh_intervals(sf->intervals()+1);
        else
          me -> mesh_intervals(sf->intervals());
        me -> interval_firmness(HARD);
      }
      else if (sf->size()>0)
      {
        int intervals = me->measure()/sf->size();
        if (!intervals) intervals++;
        if (me->constrain_even() && intervals%2) intervals++;
        me->mesh_intervals(intervals);
        me->interval_firmness(SOFT);
      }
      else
        throw Error(MK_INCOMPLETE_MESH_SPECIFICATION,  "Sizing function for edge had neither positive size nor positive intervals.");
    }
  }

  // now call setup_boundary to treat vertices
   // Wrong!!!!!!!!!
  // setup_boundary();
  /* this is not enough to ensure that vertex mesher will be called before
    "this" edge mesher
    the case that fell through the cracks was if the end nodes were already setup
   then the this_op[0] would not be retrieved, and not inserted "before" the edge mesher MeshOp
  */
  int dim=0;
  MeshOp * vm = (MeshOp*) mk_core()->construct_meshop(dim);

  for (MEntSelection::iterator mit = mentSelection.begin(); mit != mentSelection.end(); mit++)
  {
    ModelEnt *me = mit->first;
    MEntVector children;
    me->get_adjacencies(0, children);
    for (unsigned int i=0; i<children.size(); i++)
      if (children[i]->is_meshops_list_empty())
      {
        vm->add_modelent(children[i]);
      }
  }
  // in any case, make sure that the vertex mesher is inserted before this edge mesher
  mk_core()->insert_node(vm, this, mk_core()->root_node());


}

//---------------------------------------------------------------------------//
// execute function: Generate the mesh for edges
void EdgeMesher::execute_this()
{
  std::vector<double> coords;
  std::vector<moab::EntityHandle> nodes;

  for (MEntSelection::iterator mit = mentSelection.begin(); mit != mentSelection.end(); mit++)
  {
    ModelEnt *me = mit -> first;
    if (me->get_meshed_state() >= COMPLETE_MESH)
        continue;
    //resize the coords based on the interval setting
    int num_edges = me->mesh_intervals();
    coords.resize(3*(num_edges+1));
    nodes.clear();
    nodes.reserve(num_edges + 1);

    //get bounding mesh entities, use 1st 2 entries of nodes list temporarily
    //pick up the boundary end nodes
    me->boundary(0, nodes);

    bool periodic = (nodes.size() == 1);
    
    //get coords in list, then move one tuple to the last position
    moab::ErrorCode rval = mk_core()->moab_instance()->get_coords(&nodes[0], nodes.size(), &coords[0]);
    MBERRCHK(rval, mk_core()->moab_instance());

    //move the second node to the endmost position in the node list
    // if periodic, the last node coordinates are also at index 0 in coords array
    // there is only one node, coords[3+i] are not even initialized
    int index2 = (periodic) ? 0 : 3;
    for (int i = 0; i < 3; i++)
      coords[3*num_edges+i] = coords[index2+i];

    EdgeSchemeType scheme = schemeType;
    SizingFunction *sz = mk_core()->sizing_function(me->sizing_function_index());
    if (sz->variable())
      scheme = VARIABLE;

    //choose the scheme for edge mesher
    switch(scheme)
    {
      case EQUAL://equal meshing for edges
          EqualMeshing(me, num_edges, coords);
          break;
      case BIAS://bias meshing for edges
          BiasMeshing(me, num_edges, coords);
          break;
      case DUAL://dual bias meshing for edges
          DualBiasMeshing(me, num_edges, coords);
          break;
      case CURVATURE://curvature-based meshing for edges
          CurvatureMeshing(me, num_edges, coords);
          break;
      case VARIABLE: // use a var size from sizing function
          VariableMeshing(me, num_edges, coords);
          break;
      case EQUIGNOMONIC: // used to generate HOMME type meshes on a sphere
          EquiAngleGnomonic(me, num_edges, coords);
          break;
      default:
          break;                        
    }
    //the variable nodes should be resized, node size may be changed in the different scheme for edge meshing
    me->mesh_intervals(num_edges);
    nodes.resize(num_edges+1);
    
    //move the other nodes to the end of nodes' list
    if (periodic) nodes[num_edges] = nodes[0];
    else nodes[num_edges] = nodes[1];

    //create the vertices' entities on the edge
    if (num_edges > 1) {
      rval = mk_core()->moab_instance()->create_vertices(&coords[3], num_edges - 1, mit -> second);
      MBERRCHK(rval, mk_core()->moab_instance());
    }
    
    //distribute the nodes into vector
    int j = 1;
    for (moab::Range::iterator rit = mit -> second.begin(); rit != mit -> second.end(); rit++)
      nodes[j++] = *rit;

      //get the query interface, which we will use to create the edges directly 
    moab::ReadUtilIface *iface;
    rval = mk_core() -> moab_instance() -> query_interface(iface);
    MBERRCHK(rval, mk_core()->moab_instance());         

      //create the edges, get a direct ptr to connectivity
    moab::EntityHandle starth, *connect, *tmp_connect;
    rval = iface -> get_element_connect(num_edges, 2, moab::MBEDGE, 1, starth, connect);
    MBERRCHK(rval, mk_core()->moab_instance());

      //add edges to range for the MESelection
    mit -> second.insert(starth, starth + num_edges - 1);

      //now set the connectivity array from the nodes
    tmp_connect = &nodes[0];
    for (int i = 0; i < num_edges; i++)
    {
      connect[0] = tmp_connect[0];
      connect[1] = tmp_connect[1];

        //increment connectivity ptr by 2 to go to next edge
      connect += 2;
                        
        //increment tmp_connect by 1, to go to next node
      tmp_connect++;
    }

      //   ok, we are done, commit to ME
    me->commit_mesh(mit->second, COMPLETE_MESH);
   
        
  }

        
}

//---------------------------------------------------------------------------//
// Deconstruction function
EdgeMesher::~EdgeMesher()
{

}

//---------------------------------------------------------------------------//
// Create the mesh for edges with the equal distances
void EdgeMesher::EqualMeshing(ModelEnt *ent, int num_edges, std::vector<double> &coords)
{
  double umin, umax, measure;
  (void) measure;
    //get the u range for the edge
  iGeom::Error gerr =  ent->igeom_instance()->getEntURange(ent->geom_handle(), umin, umax);
  IBERRCHK(gerr, "Trouble get parameter range for edge.");

  if (umin == umax) throw Error(MK_BAD_GEOMETRIC_EVALUATION, "Edge evaluated to some parameter umax and umin.");

  //get the arc length
  measure = ent -> measure();

  double u, du;
  if (!num_edges) throw Error(MK_BAD_INPUT, "Trying to mesh edge with zero edges.");
  du = (umax - umin)/(double)num_edges;
        
  u = umin;
  //distribute the nodes with equal distances
  for (int i = 1; i < num_edges; i++)
  {
    u = umin + i*du;
    gerr = ent->igeom_instance()->getEntUtoXYZ(ent->geom_handle(), u, coords[3*i], coords[3*i+1], coords[3*i+2]);
    IBERRCHK(gerr, "Trouble getting U from XYZ along the edge.");
  }

}

//---------------------------------------------------------------------------//
// Create the mesh for edges based on the curvatures
void EdgeMesher::CurvatureMeshing(ModelEnt *ent, int &num_edges, std::vector<double> &coords)
{
  double umin, umax, measure;
  (void) measure;
  //store the initial edge size, the edge size may be changed during meshing
  int initial_num_edges = num_edges;

  //get the u range for the edge
  iGeom::Error gerr = ent->igeom_instance() ->getEntURange(ent->geom_handle(), umin, umax);
  IBERRCHK(gerr, "Trouble get parameter range for edge.");

  if (umin == umax) throw Error(MK_BAD_GEOMETRIC_EVALUATION, "Edge evaluated to some parameter umax and umin.");

  //get the arc length
  measure = ent -> measure();

  int index = 0;
  double u, du, uMid;
  du = (umax - umin)/(double)num_edges;
        
  std::vector<double> NodeCoordinates;
  std::vector<double> TempNode;
  std::vector<double> URecord;          //record the value of U

  Point3D pts0, pts1, ptsMid;
  double tmp[3];

  NodeCoordinates.resize(3*(num_edges + 1));

  TempNode.resize(3*1);
  URecord.resize(1);    

  gerr = ent -> igeom_instance() -> getEntUtoXYZ(ent->geom_handle(), umin, TempNode[0], TempNode[1], TempNode[2]);
  IBERRCHK(gerr, "Trouble getting U from XYZ along the edge");
        
  NodeCoordinates[3*0] = TempNode[0];
  NodeCoordinates[3*0+1] = TempNode[1];
  NodeCoordinates[3*0+2] = TempNode[2];

  URecord[0] = umin;

  u = umin;
  //distribute the mesh nodes on the edge based on the curvature
  for (int i = 1; i < num_edges; i++)
  {
    //first distribute the nodes evenly
    u = umin + i*du;
    gerr = ent->igeom_instance()->getEntUtoXYZ(ent->geom_handle(), u, NodeCoordinates[3*i], NodeCoordinates[3*i+1], NodeCoordinates[3*i+2]);
    IBERRCHK(gerr, "Trouble getting U from XYZ along the edge.");

    //store the two adjacent mesh nodes on the edge
    pts0.px = NodeCoordinates[3*(i-1)];
    pts0.py = NodeCoordinates[3*(i-1)+1];
    pts0.pz = NodeCoordinates[3*(i-1)+2];

    pts1.px = NodeCoordinates[3*i];
    pts1.py = NodeCoordinates[3*i+1];
    pts1.pz = NodeCoordinates[3*i+2];

    //get the coordinates for mid point between two adjacent mesh nodes on the edge
    uMid = (u-du+u)/2;
    gerr = ent->igeom_instance()->getEntUtoXYZ(ent->geom_handle(), uMid, tmp[0], tmp[1], tmp[2]);
    ptsMid.px = tmp[0];
    ptsMid.py = tmp[1];
    ptsMid.pz = tmp[2];

    //calculate the error and check whether it requires further refinement based on the curvature
    if (!ErrorCalculate(ent, pts0, pts1, ptsMid))
    {
        DivideIntoMore(ent, pts0, ptsMid, pts1, u-du, u, uMid, index, TempNode, URecord);
    }
    
    // add the other end node to the array, get the next two adjacent mesh nodes
    index++;
    TempNode.resize(3*(index + 1));
    URecord.resize(index + 1);
    TempNode[3*index] = pts1.px;
    TempNode[3*index + 1] = pts1.py;
    TempNode[3*index + 2] = pts1.pz;

    URecord[index] = u; 
  }

  //sorting the parametrical coordinate data based on the value of u
  assert(TempNode.size()== (3*URecord.size()) );
        
  QuickSorting(TempNode, URecord, URecord.size());
  num_edges = URecord.size() - 1;
        
  //resize the variable coords
  coords.resize(3*(num_edges+1));       

  //move the other end node to the endmost of the list
  for (int i = 0; i < 3; i++)
    coords[3*num_edges+i] = coords[3*initial_num_edges+i];

  //return the mesh nodes       
  for (int i = 1; i < num_edges; i++)
  {
    coords[3*i] = TempNode[3*i];
    coords[3*i+1] = TempNode[3*i+1];
    coords[3*i+2] = TempNode[3*i+2];
  }

}

//---------------------------------------------------------------------------//
// Create the mesh for edges with dual bias distances
void EdgeMesher::DualBiasMeshing(ModelEnt *ent, int &num_edges, std::vector<double> &coords)
{
  double umin, umax, measure;

    //get the u range for the edge
  iGeom::Error gerr = ent->igeom_instance()->getEntURange(ent->geom_handle(), umin, umax);
  IBERRCHK(gerr, "Trouble get parameter range for edge.");

  if (umin == umax) throw Error(MK_BAD_GEOMETRIC_EVALUATION, "Edge evaluated to some parameter umax and umin.");

  //get the arc length
  measure = ent -> measure();

  double u, L0, dist, u0, u1;
        
  //if the node # is odd, node # will increase by 1
  if ((num_edges%2)!=0)
  {
    num_edges++;
    coords.resize(3*(num_edges+1));
      //move the other end node's position because the variable coords has been resized.
    for (int k = 0; k < 3; k++)
      coords[3*num_edges + k] = coords[3*num_edges + k - 3];
  }

  //default bias ratio is 1.2
  double q = ratio;//
        
  //get the distance between the first two nodes
  L0 = 0.5 * measure * (1-q) / (1 - pow(q, num_edges/2));
                
        
  u0 = umin;
  u1 = umax;
  //distribute the mesh nodes on the edge with dual-bias distances
  for (int i = 1; i < (num_edges/2 + 1); i++)
  {
      //distribute the mesh nodes on the one side               
    dist = L0*pow(q, i-1);
    u = u0 + (umax - umin) * dist/measure;
    u = getUCoord(ent, u0, dist, u, umin, umax);
    u0 = u;
    gerr = ent->igeom_instance()->getEntUtoXYZ(ent->geom_handle(), u, coords[3*i], coords[3*i+1], coords[3*i+2]);
    IBERRCHK(gerr, "Trouble getting U from XYZ along the edge.");

    //distribute the mesh nodes on the other side
    if (i < num_edges/2)
    {
      u = u1 - (umax-umin) * dist / measure;
      u = getUCoord(ent, u1, dist, u, umin, umax);
      u1 = u;
      gerr = ent->igeom_instance()->getEntUtoXYZ(ent->geom_handle(), u, coords[3*(num_edges-i)], coords[3*(num_edges-i)+1], coords[3*(num_edges-i)+2]);
      IBERRCHK(gerr, "Trouble getting U from XYZ along the edge");
    }
                
  }
        
}

//---------------------------------------------------------------------------//
// Create the mesh for edges with bias distances
void EdgeMesher::BiasMeshing(ModelEnt *ent, int num_edges, std::vector<double> &coords)
{
  double umin, umax, measure;

  //get the u range for the edge
  iGeom::Error gerr = ent->igeom_instance()->getEntURange(ent->geom_handle(), umin, umax);
  IBERRCHK(gerr, "Trouble get parameter range for edge.");

  if (umin == umax) throw Error(MK_BAD_GEOMETRIC_EVALUATION, "Edge evaluated to some parameter umax and umin.");

  //get the arc length
  measure = ent -> measure();

  double u, L0, dist = 0, u0;
        
  // the default bias ratio 1.2
  double q = ratio;
  L0 = measure * (1-q) / (1 - pow(q, num_edges));
                
        
  u0 = umin;
  //distribute the mesh nodes on the edge with bias distances
  for (int i = 1; i < num_edges; i++)
  {
    dist = L0*pow(q, i-1);
    u = u0 + (umax - umin)*dist/measure;
    u = getUCoord(ent, u0, dist, u, umin, umax);
    u0 = u;
    gerr = ent->igeom_instance()->getEntUtoXYZ(ent->geom_handle(), u, coords[3*i], coords[3*i+1], coords[3*i+2]);
    IBERRCHK(gerr, "Trouble getting U from XYZ along the edge.");
  }
}

//---------------------------------------------------------------------------//
// Mesh Refinement function based on the curvature: if the error is too big, refine the mesh on the edge
void EdgeMesher::DivideIntoMore(ModelEnt *ent, Point3D p0, Point3D pMid, Point3D p1, double u0, double u1, double uMid, int &index, vector<double> &nodes, vector<double> &URecord)
{
  //this is a recursive process, the process continues until the error is smaller than what is required.
  //first get two adjacent mesh nodes on the edge and coordinates for mid point between two adjacent nodes.
  //then check the left side and right side whether the error is too big nor not
  double uu0, uu1, uumid, tmp[3];
  Point3D pts0, pts1, ptsMid;
        
  index++;
  nodes.resize(3*(index+1));
  URecord.resize(index+1);
  nodes[3*index] = pMid.px;
  nodes[3*index+1] = pMid.py;
  nodes[3*index+2] = pMid.pz;
  URecord[index] = uMid;
        
  //check the left side
  uu0=u0;
  uu1=uMid;
  uumid=(uu0+uu1)/2;
  pts0=p0;
  pts1=pMid;


  iGeom::Error gerr = ent->igeom_instance()->getEntUtoXYZ(ent->geom_handle(), uumid, tmp[0], tmp[1], tmp[2]);
  IBERRCHK(gerr, "Trouble getting U from XYZ along the edge.");
  ptsMid.px = tmp[0];
  ptsMid.py = tmp[1];
  ptsMid.pz = tmp[2];

  //check the error
  if(!ErrorCalculate(ent, pts0, pts1, ptsMid))
  {
    //further refinement
    DivideIntoMore(ent, pts0, ptsMid, pts1, uu0, uu1, uumid, index, nodes, URecord);
  }
        
  //check the right side
  uu0 = uMid;
  uu1=u1;
  uumid=(uu0+uu1)/2;
  pts0=pMid;
  pts1=p1;
  //get the coorindinates for mid point 
  gerr = ent->igeom_instance()->getEntUtoXYZ(ent->geom_handle(), uumid, tmp[0], tmp[1], tmp[2]);
  IBERRCHK(gerr, "Trouble getting U from XYZ along the edge.");
  ptsMid.px = tmp[0];
  ptsMid.py = tmp[1];
  ptsMid.pz = tmp[2];
        
  //check the error
  if(!ErrorCalculate(ent, pts0, pts1, ptsMid))
  {
    //further refinement
    DivideIntoMore(ent, pts0, ptsMid, pts1, uu0, uu1, uumid, index, nodes, URecord);
  }
                        
}

//create the mesh for edges based on variable size from SizingFunction (var)
void EdgeMesher::VariableMeshing(ModelEnt *ent, int &num_edges, std::vector<double> &coords)
{
  double umin, umax, measure;
  (void) measure;
  // because of that, keep track of the first node position and last node position
  // first node position does not change, but the last node position do change
  // coords will contain all nodes, including umax in Urecord!

  SizingFunction *sf = mk_core()->sizing_function(ent->sizing_function_index());
  //get the u range for the edge
  iGeom::EntityHandle edge = ent->geom_handle();
  iGeom::Error gerr = ent->igeom_instance() ->getEntURange(edge, umin, umax);
  IBERRCHK(gerr, "Trouble get parameter range for edge.");

  if (umin == umax) throw Error(MK_BAD_GEOMETRIC_EVALUATION, "Edge evaluated to some parameter umax and umin.");

  //get the arc length
  measure = ent -> measure();

  // start advancing for each edge mesh, from the first point position
  double currentPar = umin;
  double currentPosition[3];
  gerr = ent->igeom_instance() ->getEntUtoXYZ(edge, umin, currentPosition[0],
      currentPosition[1], currentPosition[2] );

  double endPoint[3];
  gerr = ent->igeom_instance() ->getEntUtoXYZ(edge, umax, endPoint[0],
      endPoint[1], endPoint[2] );
  Vector<3> endpt(endPoint);

  double targetSize = sf->size(currentPosition);
  double startSize = targetSize;

  double endSize = sf->size(endPoint);
  // advance u such that the next point is at "currentSize" distance
  // or close to it
  // try first with a u that is coming from the (umax-umin)/number of edges
  double deltaU = (umax-umin)/num_edges;
  //coords.clear(); we do not want to clear, as the first node is still fine
  std::vector<double> URecord;    //record the values for u; we may have to adjust all
  // of them accordingly, and even add one more if we have some evenify problems.
  // keep in mind that size is just a suggestion, number of intervals is more important for
  // paver mesher
  Vector<3> pt(currentPosition);

  //bool notDone = true;
  double prevU = umin;
  while (currentPar + 1.1*deltaU < umax)
  {
    // do some binary search; actually, better, do Newton-Raphson, which should converge
    // faster
    //
    prevU = currentPar;
    currentPar += deltaU;
    // adjust current par, such that
    double point[3];
    gerr=ent->igeom_instance()->getEntUtoXYZ(edge, currentPar, point[0], point[1], point[2] );
    IBERRCHK(gerr, "Trouble getting position at parameter u.");
    Vector<3> ptCandidate(point);
    double compSize = length(ptCandidate-pt);
    int nit = 0;

    while ( (fabs(1.-compSize/targetSize)> 0.02 ) && (nit < 5))// 2% of the target size
    {
      // do Newton iteration
      double tangent[3];
      gerr=ent->igeom_instance() ->getEntTgntU(edge, currentPar, tangent[0], tangent[1], tangent[2] );
      IBERRCHK(gerr, "Trouble getting tangent at parameter u.");
      Vector<3> tang(tangent);
      double dldu = 1./compSize * ((ptCandidate-pt )%tang);
      nit++;// increase iteration count
      if (dldu!=0.)
      {
        double deu= (targetSize-compSize)/dldu;
        currentPar+=deu;
        if (prevU>currentPar)
        {
          break; // something is wrong
        }
        if (umax < currentPar)
        {
          currentPar = umax;
          break;
        }
        ent->igeom_instance()->getEntUtoXYZ(edge, currentPar, point[0], point[1], point[2]);
        Vector<3> newPt(point);
        compSize = length(newPt-pt);
        ptCandidate = newPt;
      }

    }
    // we have found an acceptable point/param
    URecord.push_back(currentPar);
    deltaU = currentPar-prevU;// should be greater than 0
    pt = ptCandidate;
    targetSize = sf->size(pt.data());// a new target size, at the current point



  }
  // when we are out of here, we need to adjust the URecords, to be more nicely
  // distributed; also, look at the evenify again
  int sizeU = (int)URecord.size();
  if ((sizeU%2==0) && ent->constrain_even() )
  {
    // add one more
    if (sizeU==0)
    {
      // just add one in the middle, and call it done
      URecord.push_back( (umin+umax)/2);
    }
    else
    {
      //at least 2 (maybe 4)
      double lastDelta = URecord[sizeU-1]-URecord[sizeU-2];
      URecord.push_back(URecord[sizeU-1]+lastDelta );
    }
  }
  // now, we have to redistribute, such as the last 2 deltas are about the same
  // so, we should have after a little work,
  // umin, umin+c*(URecord[0]-umin), ... umin+c*(URecord[size-1]-umin), umax
  // what we have now is
  // umin, URecord[0], ... ,URecord[size-1], and umax could be even outside or inside
  // keep the last sizes equal
  //  umin+c(UR[size-2]-umin) + umax = 2*( umin+c*(UR[size-1]-umin))
  //  c ( 2*(UR[size-1]-umin) -(UR[size-2]-umin) ) = umax - umin
  // c ( 2*UR[size-1] - UR[size-2] - umin ) = umax - umin
  // c = (umax-umin)/( 2*UR[size-1] - UR[size-2] - umin)
  sizeU = (int)URecord.size();// it may be bigger by one than the last time
  if (sizeU == 0)
  {
    // nothing to do, only one edge to generate
  }
  else if (sizeU == 1)
  {
    // put it according to the sizes at ends, and assume a nice variation for u
    // (u-umin) / (umax-u) = startSize / endSize
    // (u-umin)*endSize = (umax-u) * startSize
    // u(endSize+startSize)=(umax*startSize+umin*endSize)
    URecord[0] = (umax*startSize+umin*endSize)/(startSize+endSize);

  }
  else // sizeU>=2, so we can spread the param a little more, assuming nice
    // uniform mapping
  {
    double c =  (umax-umin)/( 2*URecord[sizeU-1] - URecord[sizeU-2] - umin);
    for (int i=0; i<sizeU; i++)
      URecord[i] = umin + c*(URecord[i] -umin);// some spreading out
  }
  // now, we can finally get the points for each U, U's should be spread out nicely
  URecord.push_back(umax); // just add the last u, for the end point
  //
  sizeU = (int) URecord.size(); // new size, after pushing the last u, umax
  num_edges = sizeU;// this is the new number of edges; the last one will be the end point
  // of the edge, corresponding to umax
  coords.resize(3*sizeU+3);
  // we already know that at i=0 is the first node, start vertex of edge
  // the rest will be computed from u
  // even the last one, which is an overkill
  for (int i = 1; i <= num_edges; i++)
  {
    double u = URecord[i-1];
    gerr = ent->igeom_instance()->getEntUtoXYZ(edge, u, coords[3*i], coords[3*i+1], coords[3*i+2]);
    IBERRCHK(gerr, "Trouble getting U from XYZ along the edge.");
  }
  return;

}
// number of edges is input here
//  equal angles are formed at the center of the sphere/cube mesh
// it is close to bias meshing, but not quite
void EdgeMesher::EquiAngleGnomonic(ModelEnt *ent, int num_edges, std::vector<double> &coords)
{
  const double MY_PI=3.14159265;
  double deltaAngle=MY_PI/num_edges/2;
 // double length=ent->measure();// this is an edge
  double umin, umax;

  //get the u range for the edge
  iGeom::Error gerr = ent->igeom_instance()->getEntURange(ent->geom_handle(), umin, umax);
  IBERRCHK(gerr, "Trouble get parameter range for edge.");

  if (umin == umax) throw Error(MK_BAD_GEOMETRIC_EVALUATION, "Edge evaluated to some parameter umax and umin.");

  // consider that the parametrization is very linear
  // most of the time u will be from 0 to length of edge, for a cube
  double deltau = umax - umin;

  double u = umin;// u will get different values,
  // start at u
  for (int i = 1; i < num_edges; i++)
    {
      double betak=i*deltaAngle;
      double alfak = MY_PI/4-betak;
      double tang_alfak = tan(alfak);
      u = umin+deltau/2*(1-tang_alfak);

      gerr = ent->igeom_instance()->getEntUtoXYZ(ent->geom_handle(), u, coords[3*i], coords[3*i+1], coords[3*i+2]);
      IBERRCHK(gerr, "Trouble getting U from XYZ along the edge.");
    }
  return;
}
//---------------------------------------------------------------------------//
// Rapid sorting the mesh nodes on the edge based on the parametric coordinates. This is a recursive
// process
void EdgeMesher::RapidSorting(vector<double> &nodes, vector<double> &URecord, int left, int right)
{
  int i, j;
  double middle, iTemp;
  Point3D TempData;
        
  middle=URecord[(left+right)/2];
  i=left;
  j=right;
        
  do
  {
      //search the values which are greater than the middle value from the left side            
    while((URecord[i] < middle)&&(i<right))
    {
      i++;
    }
      //search the values which are greater than the middle value from the right side
    while((URecord[j] > middle)&&(j > left))
    {
      j--;
    }
    if (i<=j)//find a pair of values
    {
      iTemp = URecord[i];
      URecord[i] = URecord[j];
      URecord[j]=iTemp;
                        
                        
      TempData.px = nodes[3*i];
      TempData.py = nodes[3*i+1];
      TempData.pz = nodes[3*i+2];

      nodes[3*i] = nodes[3*j];
      nodes[3*i+1] = nodes[3*j+1];
      nodes[3*i+2] = nodes[3*j+2];
      nodes[3*j] = TempData.px;
      nodes[3*j+1] = TempData.py;
      nodes[3*j+2] = TempData.pz;                       
                        
      i++;
      j--;
    }
  }while(i<=j);
  if (left < j)
    RapidSorting(nodes, URecord, left, j);
  if (right > i)
    RapidSorting(nodes, URecord, i, right);     
}

//---------------------------------------------------------------------------//
// Quick Sorting: this function comes together with the function RapidSorting
void EdgeMesher::QuickSorting(vector<double> &nodes, vector<double> &URecord, int count)
{
  RapidSorting(nodes, URecord, 0, count-1);
}


//---------------------------------------------------------------------------//
// return the x, y, z coordinates from the parametric coordinates
EdgeMesher::Point3D EdgeMesher::getXYZCoords(ModelEnt *ent, double u) const
{
  Point3D pts3D;
  double xyz[3];


  //get the coordinates in the physical space
  iGeom::Error gerr = ent->igeom_instance()->getEntUtoXYZ(ent->geom_handle(), u, xyz[0], xyz[1], xyz[2]);
  IBERRCHK(gerr, "Trouble getting U from XYZ along the edge."); 

  pts3D.px = xyz[0];
  pts3D.py = xyz[1];
  pts3D.pz = xyz[2];
  return pts3D;
}

//---------------------------------------------------------------------------//
// get the parametric coordinate based on first parametric coordinate ustart and distance in the physical space
double EdgeMesher::getUCoord(ModelEnt *ent, double ustart, double dist, double uguess, double umin, double umax) const
{

  Point3D p0 = getXYZCoords(ent, ustart);
  Point3D p1 = getXYZCoords(ent, uguess);


  double dx, dy, dz, dl, u=uguess;
  double tol = 1.0E-7;
  int test=0;

  int ntrials=0;
  while(1)
  {
    dx = p1.px - p0.px;
    dy = p1.py - p0.py;
    dz = p1.pz - p0.pz;
    dl = sqrt(dx * dx + dy * dy + dz * dz);
    if ( fabs(dl-dist) < tol) break;
                
    u = ustart + (u - ustart) * (dist/dl);
    if (u > umax)
    {
      u=umax;
      test++;
      if (test>10) break;
    }           
    if (u < umin)
    {           
      u=umin;
      test++;
      if (test>10) break;
    }           
    p1 = getXYZCoords(ent, u);
                

    if (ntrials++ == 100000)
    {
      cout << " Warning: Searching for U failed " << endl;
    }
  }
  uguess = u;
  return uguess;
}

//---------------------------------------------------------------------------//
// calculate the error: the distance between the mid point of two adjacent 
// mesh nodes (on the mesh line segments) and mid point on the edge
bool EdgeMesher::ErrorCalculate(ModelEnt *ent, Point3D p0, Point3D p1, Point3D pMid)
{
  double lengtha, lengthb, lengthc;
  double deltax, deltay, deltaz;
  double angle, error, tol=1.0E-3, H;
  double cvtr_ijk[3], curvature;
  bool result;

  //calculate the distance between the first mesh node and mid point on the edge
  deltax = pMid.px-p0.px;
  deltay = pMid.py-p0.py;
  deltaz = pMid.pz-p0.pz;       
  lengtha = sqrt(deltax*deltax + deltay*deltay + deltaz*deltaz);

  //calculate the distance between two adjacent mesh nodes
  deltax = p1.px-p0.px;
  deltay = p1.py-p0.py;
  deltaz = p1.pz-p0.pz; 
  lengthb = sqrt(deltax*deltax + deltay*deltay + deltaz*deltaz);

  //calculate the distance between the second mesh node and mid point on the edge
  deltax = pMid.px-p1.px;
  deltay = pMid.py-p1.py;
  deltaz = pMid.pz-p1.pz;       
  lengthc = sqrt(deltax*deltax + deltay*deltay + deltaz*deltaz);

  //calculate the angle
  angle = acos((lengtha*lengtha + lengthb*lengthb - lengthc*lengthc)/(2*lengtha*lengthb));
  H = fabs(lengtha*sin(angle));

  //calculate the curvature     
  iGeom::Error gerr = ent->igeom_instance()->getEgCvtrXYZ(ent->geom_handle(), pMid.px, pMid.py, pMid.pz, cvtr_ijk[0], cvtr_ijk[1], cvtr_ijk[2]);
  IBERRCHK(gerr, "Trouble getting U from XYZ along the edge.");         
  curvature = sqrt(cvtr_ijk[0]*cvtr_ijk[0]+cvtr_ijk[1]*cvtr_ijk[1]+cvtr_ijk[2]*cvtr_ijk[2]);
  error= H*curvature;
        
        
  if (error > tol)
    result = false;
  else
    result = true;
  return result;                
}

}