source: src/tesselationhelpers.cpp@ eaee7f

/*
 * TesselationHelpers.cpp
 *
 *  Created on: Aug 3, 2009
 *      Author: heber
 */

#include "tesselationhelpers.hpp"

double DetGet(gsl_matrix *A, int inPlace) {
  /*
  inPlace = 1 => A is replaced with the LU decomposed copy.
  inPlace = 0 => A is retained, and a copy is used for LU.
  */

  double det;
  int signum;
  gsl_permutation *p = gsl_permutation_alloc(A->size1);
  gsl_matrix *tmpA;

  if (inPlace)
  tmpA = A;
  else {
  gsl_matrix *tmpA = gsl_matrix_alloc(A->size1, A->size2);
  gsl_matrix_memcpy(tmpA , A);
  }


  gsl_linalg_LU_decomp(tmpA , p , &signum);
  det = gsl_linalg_LU_det(tmpA , signum);
  gsl_permutation_free(p);
  if (! inPlace)
  gsl_matrix_free(tmpA);

  return det;
};

void GetSphere(Vector *center, Vector &a, Vector &b, Vector &c, double RADIUS)
{
  gsl_matrix *A = gsl_matrix_calloc(3,3);
  double m11, m12, m13, m14;

  for(int i=0;i<3;i++) {
    gsl_matrix_set(A, i, 0, a.x[i]);
    gsl_matrix_set(A, i, 1, b.x[i]);
    gsl_matrix_set(A, i, 2, c.x[i]);
  }
  m11 = DetGet(A, 1);

  for(int i=0;i<3;i++) {
    gsl_matrix_set(A, i, 0, a.x[i]*a.x[i] + b.x[i]*b.x[i] + c.x[i]*c.x[i]);
    gsl_matrix_set(A, i, 1, b.x[i]);
    gsl_matrix_set(A, i, 2, c.x[i]);
  }
  m12 = DetGet(A, 1);

  for(int i=0;i<3;i++) {
    gsl_matrix_set(A, i, 0, a.x[i]*a.x[i] + b.x[i]*b.x[i] + c.x[i]*c.x[i]);
    gsl_matrix_set(A, i, 1, a.x[i]);
    gsl_matrix_set(A, i, 2, c.x[i]);
  }
  m13 = DetGet(A, 1);

  for(int i=0;i<3;i++) {
    gsl_matrix_set(A, i, 0, a.x[i]*a.x[i] + b.x[i]*b.x[i] + c.x[i]*c.x[i]);
    gsl_matrix_set(A, i, 1, a.x[i]);
    gsl_matrix_set(A, i, 2, b.x[i]);
  }
  m14 = DetGet(A, 1);

  if (fabs(m11) < MYEPSILON)
    cerr << "ERROR: three points are colinear." << endl;

  center->x[0] =  0.5 * m12/ m11;
  center->x[1] = -0.5 * m13/ m11;
  center->x[2] =  0.5 * m14/ m11;

  if (fabs(a.Distance(center) - RADIUS) > MYEPSILON)
    cerr << "ERROR: The given center is further way by " << fabs(a.Distance(center) - RADIUS) << " from a than RADIUS." << endl;

  gsl_matrix_free(A);
};



/**
 * Function returns center of sphere with RADIUS, which rests on points a, b, c
 * @param Center this vector will be used for return
 * @param a vector first point of triangle
 * @param b vector second point of triangle
 * @param c vector third point of triangle
 * @param *Umkreismittelpunkt new cneter point of circumference
 * @param Direction vector indicates up/down
 * @param AlternativeDirection vecotr, needed in case the triangles have 90 deg angle
 * @param Halfplaneindicator double indicates whether Direction is up or down
 * @param AlternativeIndicator doube indicates in case of orthogonal triangles which direction of AlternativeDirection is suitable
 * @param alpha double angle at a
 * @param beta double, angle at b
 * @param gamma, double, angle at c
 * @param Radius, double
 * @param Umkreisradius double radius of circumscribing circle
 */
void GetCenterOfSphere(Vector* Center, Vector a, Vector b, Vector c, Vector *NewUmkreismittelpunkt, Vector* Direction, Vector* AlternativeDirection,
    double HalfplaneIndicator, double AlternativeIndicator, double alpha, double beta, double gamma, double RADIUS, double Umkreisradius)
{
  Vector TempNormal, helper;
  double Restradius;
  Vector OtherCenter;
  cout << Verbose(3) << "Begin of GetCenterOfSphere.\n";
  Center->Zero();
  helper.CopyVector(&a);
  helper.Scale(sin(2.*alpha));
  Center->AddVector(&helper);
  helper.CopyVector(&b);
  helper.Scale(sin(2.*beta));
  Center->AddVector(&helper);
  helper.CopyVector(&c);
  helper.Scale(sin(2.*gamma));
  Center->AddVector(&helper);
  //*Center = a * sin(2.*alpha) + b * sin(2.*beta) + c * sin(2.*gamma) ;
  Center->Scale(1./(sin(2.*alpha) + sin(2.*beta) + sin(2.*gamma)));
  NewUmkreismittelpunkt->CopyVector(Center);
  cout << Verbose(4) << "Center of new circumference is " << *NewUmkreismittelpunkt << ".\n";
  // Here we calculated center of circumscribing circle, using barycentric coordinates
  cout << Verbose(4) << "Center of circumference is " << *Center << " in direction " << *Direction << ".\n";

  TempNormal.CopyVector(&a);
  TempNormal.SubtractVector(&b);
  helper.CopyVector(&a);
  helper.SubtractVector(&c);
  TempNormal.VectorProduct(&helper);
  if (fabs(HalfplaneIndicator) < MYEPSILON)
    {
      if ((TempNormal.ScalarProduct(AlternativeDirection) <0 and AlternativeIndicator >0) or (TempNormal.ScalarProduct(AlternativeDirection) >0 and AlternativeIndicator <0))
        {
          TempNormal.Scale(-1);
        }
    }
  else
    {
      if (TempNormal.ScalarProduct(Direction)<0 && HalfplaneIndicator >0 || TempNormal.ScalarProduct(Direction)>0 && HalfplaneIndicator<0)
        {
          TempNormal.Scale(-1);
        }
    }

  TempNormal.Normalize();
  Restradius = sqrt(RADIUS*RADIUS - Umkreisradius*Umkreisradius);
  cout << Verbose(4) << "Height of center of circumference to center of sphere is " << Restradius << ".\n";
  TempNormal.Scale(Restradius);
  cout << Verbose(4) << "Shift vector to sphere of circumference is " << TempNormal << ".\n";

  Center->AddVector(&TempNormal);
  cout << Verbose(0) << "Center of sphere of circumference is " << *Center << ".\n";
  GetSphere(&OtherCenter, a, b, c, RADIUS);
  cout << Verbose(0) << "OtherCenter of sphere of circumference is " << OtherCenter << ".\n";
  cout << Verbose(3) << "End of GetCenterOfSphere.\n";
};


/** Constructs the center of the circumcircle defined by three points \a *a, \a *b and \a *c.
 * \param *Center new center on return
 * \param *a first point
 * \param *b second point
 * \param *c third point
 */
void GetCenterofCircumcircle(Vector *Center, Vector *a, Vector *b, Vector *c)
{
  Vector helper;
  double alpha, beta, gamma;
  Vector SideA, SideB, SideC;
  SideA.CopyVector(b);
  SideA.SubtractVector(c);
  SideB.CopyVector(c);
  SideB.SubtractVector(a);
  SideC.CopyVector(a);
  SideC.SubtractVector(b);
  alpha = M_PI - SideB.Angle(&SideC);
  beta = M_PI - SideC.Angle(&SideA);
  gamma = M_PI - SideA.Angle(&SideB);
  //cout << Verbose(3) << "INFO: alpha = " << alpha/M_PI*180. << ", beta = " << beta/M_PI*180. << ", gamma = " << gamma/M_PI*180. << "." << endl;
  if (fabs(M_PI - alpha - beta - gamma) > HULLEPSILON)
    cerr << "GetCenterofCircumcircle: Sum of angles " << (alpha+beta+gamma)/M_PI*180. << " > 180 degrees by " << fabs(M_PI - alpha - beta - gamma)/M_PI*180. << "!" << endl;

  Center->Zero();
  helper.CopyVector(a);
  helper.Scale(sin(2.*alpha));
  Center->AddVector(&helper);
  helper.CopyVector(b);
  helper.Scale(sin(2.*beta));
  Center->AddVector(&helper);
  helper.CopyVector(c);
  helper.Scale(sin(2.*gamma));
  Center->AddVector(&helper);
  Center->Scale(1./(sin(2.*alpha) + sin(2.*beta) + sin(2.*gamma)));
};

/** Returns the parameter "path length" for a given \a NewSphereCenter relative to \a OldSphereCenter on a circle on the plane \a CirclePlaneNormal with center \a CircleCenter and radius \a CircleRadius.
 * Test whether the \a NewSphereCenter is really on the given plane and in distance \a CircleRadius from \a CircleCenter.
 * It calculates the angle, making it unique on [0,2.*M_PI) by comparing to SearchDirection.
 * Also the new center is invalid if it the same as the old one and does not lie right above (\a NormalVector) the base line (\a CircleCenter).
 * \param CircleCenter Center of the parameter circle
 * \param CirclePlaneNormal normal vector to plane of the parameter circle
 * \param CircleRadius radius of the parameter circle
 * \param NewSphereCenter new center of a circumcircle
 * \param OldSphereCenter old center of a circumcircle, defining the zero "path length" on the parameter circle
 * \param NormalVector normal vector
 * \param SearchDirection search direction to make angle unique on return.
 * \return Angle between \a NewSphereCenter and \a OldSphereCenter relative to \a CircleCenter, 2.*M_PI if one test fails
 */
double GetPathLengthonCircumCircle(Vector &CircleCenter, Vector &CirclePlaneNormal, double CircleRadius, Vector &NewSphereCenter, Vector &OldSphereCenter, Vector &NormalVector, Vector &SearchDirection)
{
  Vector helper;
  double radius, alpha;

  helper.CopyVector(&NewSphereCenter);
  // test whether new center is on the parameter circle's plane
  if (fabs(helper.ScalarProduct(&CirclePlaneNormal)) > HULLEPSILON) {
    cerr << "ERROR: Something's very wrong here: NewSphereCenter is not on the band's plane as desired by " <<fabs(helper.ScalarProduct(&CirclePlaneNormal))  << "!" << endl;
    helper.ProjectOntoPlane(&CirclePlaneNormal);
  }
  radius = helper.ScalarProduct(&helper);
  // test whether the new center vector has length of CircleRadius
  if (fabs(radius - CircleRadius) > HULLEPSILON)
    cerr << Verbose(1) << "ERROR: The projected center of the new sphere has radius " << radius << " instead of " << CircleRadius << "." << endl;
  alpha = helper.Angle(&OldSphereCenter);
  // make the angle unique by checking the halfplanes/search direction
  if (helper.ScalarProduct(&SearchDirection) < -HULLEPSILON)  // acos is not unique on [0, 2.*M_PI), hence extra check to decide between two half intervals
    alpha = 2.*M_PI - alpha;
  //cout << Verbose(2) << "INFO: RelativeNewSphereCenter is " << helper << ", RelativeOldSphereCenter is " << OldSphereCenter << " and resulting angle is " << alpha << "." << endl;
  radius = helper.Distance(&OldSphereCenter);
  helper.ProjectOntoPlane(&NormalVector);
  // check whether new center is somewhat away or at least right over the current baseline to prevent intersecting triangles
  if ((radius > HULLEPSILON) || (helper.Norm() < HULLEPSILON)) {
    //cout << Verbose(2) << "INFO: Distance between old and new center is " << radius << " and between new center and baseline center is " << helper.Norm() << "." << endl;
    return alpha;
  } else {
    //cout << Verbose(1) << "INFO: NewSphereCenter " << helper << " is too close to OldSphereCenter" << OldSphereCenter << "." << endl;
    return 2.*M_PI;
  }
};

struct Intersection {
  Vector x1;
  Vector x2;
  Vector x3;
  Vector x4;
};

/**
 * Intersection calculation function.
 *
 * @param x to find the result for
 * @param function parameter
 */
double MinIntersectDistance(const gsl_vector * x, void *params)
{
  double retval = 0;
  struct Intersection *I = (struct Intersection *)params;
  Vector intersection;
  Vector SideA,SideB,HeightA, HeightB;
  for (int i=0;i<NDIM;i++)
    intersection.x[i] = gsl_vector_get(x, i);

  SideA.CopyVector(&(I->x1));
  SideA.SubtractVector(&I->x2);
  HeightA.CopyVector(&intersection);
  HeightA.SubtractVector(&I->x1);
  HeightA.ProjectOntoPlane(&SideA);

  SideB.CopyVector(&I->x3);
  SideB.SubtractVector(&I->x4);
  HeightB.CopyVector(&intersection);
  HeightB.SubtractVector(&I->x3);
  HeightB.ProjectOntoPlane(&SideB);

  retval = HeightA.ScalarProduct(&HeightA) + HeightB.ScalarProduct(&HeightB);
  //cout << Verbose(2) << "MinIntersectDistance called, result: " << retval << endl;

  return retval;
};


/**
 * Calculates whether there is an intersection between two lines. The first line
 * always goes through point 1 and point 2 and the second line is given by the
 * connection between point 4 and point 5.
 *
 * @param point 1 of line 1
 * @param point 2 of line 1
 * @param point 1 of line 2
 * @param point 2 of line 2
 *
 * @return true if there is an intersection between the given lines, false otherwise
 */
bool existsIntersection(Vector point1, Vector point2, Vector point3, Vector point4)
{
  bool result;

  struct Intersection par;
    par.x1.CopyVector(&point1);
    par.x2.CopyVector(&point2);
    par.x3.CopyVector(&point3);
    par.x4.CopyVector(&point4);

    const gsl_multimin_fminimizer_type *T = gsl_multimin_fminimizer_nmsimplex;
    gsl_multimin_fminimizer *s = NULL;
    gsl_vector *ss, *x;
    gsl_multimin_function minexFunction;

    size_t iter = 0;
    int status;
    double size;

    /* Starting point */
    x = gsl_vector_alloc(NDIM);
    gsl_vector_set(x, 0, point1.x[0]);
    gsl_vector_set(x, 1, point1.x[1]);
    gsl_vector_set(x, 2, point1.x[2]);

    /* Set initial step sizes to 1 */
    ss = gsl_vector_alloc(NDIM);
    gsl_vector_set_all(ss, 1.0);

    /* Initialize method and iterate */
    minexFunction.n = NDIM;
    minexFunction.f = &MinIntersectDistance;
    minexFunction.params = (void *)&par;

    s = gsl_multimin_fminimizer_alloc(T, NDIM);
    gsl_multimin_fminimizer_set(s, &minexFunction, x, ss);

    do {
        iter++;
        status = gsl_multimin_fminimizer_iterate(s);

        if (status) {
          break;
        }

        size = gsl_multimin_fminimizer_size(s);
        status = gsl_multimin_test_size(size, 1e-2);

        if (status == GSL_SUCCESS) {
          cout << Verbose(2) << "converged to minimum" <<  endl;
        }
    } while (status == GSL_CONTINUE && iter < 100);

    // check whether intersection is in between or not
  Vector intersection, SideA, SideB, HeightA, HeightB;
  double t1, t2;
  for (int i = 0; i < NDIM; i++) {
    intersection.x[i] = gsl_vector_get(s->x, i);
  }

  SideA.CopyVector(&par.x2);
  SideA.SubtractVector(&par.x1);
  HeightA.CopyVector(&intersection);
  HeightA.SubtractVector(&par.x1);

  t1 = HeightA.ScalarProduct(&SideA)/SideA.ScalarProduct(&SideA);

  SideB.CopyVector(&par.x4);
  SideB.SubtractVector(&par.x3);
  HeightB.CopyVector(&intersection);
  HeightB.SubtractVector(&par.x3);

  t2 = HeightB.ScalarProduct(&SideB)/SideB.ScalarProduct(&SideB);

  cout << Verbose(2) << "Intersection " << intersection << " is at "
    << t1 << " for (" << point1 << "," << point2 << ") and at "
    << t2 << " for (" << point3 << "," << point4 << "): ";

  if (((t1 >= 0) && (t1 <= 1)) && ((t2 >= 0) && (t2 <= 1))) {
    cout << "true intersection." << endl;
    result = true;
  } else {
    cout << "intersection out of region of interest." << endl;
    result = false;
  }

  // free minimizer stuff
    gsl_vector_free(x);
    gsl_vector_free(ss);
    gsl_multimin_fminimizer_free(s);

  return result;
};

/** Gets the angle between a point and a reference relative to the provided center.
 * We have two shanks point and reference between which the angle is calculated
 * and by scalar product with OrthogonalVector we decide the interval.
 * @param point to calculate the angle for
 * @param reference to which to calculate the angle
 * @param OrthogonalVector points in direction of [pi,2pi] interval
 *
 * @return angle between point and reference
 */
double GetAngle(const Vector &point, const Vector &reference, const Vector OrthogonalVector)
{
  if (reference.IsZero())
    return M_PI;

  // calculate both angles and correct with in-plane vector
  if (point.IsZero())
    return M_PI;
  double phi = point.Angle(&reference);
  if (OrthogonalVector.ScalarProduct(&point) > 0) {
    phi = 2.*M_PI - phi;
  }

  cout << Verbose(4) << "INFO: " << point << " has angle " << phi << " with respect to reference " << reference << "." << endl;

  return phi;
}


/** Calculates the volume of a general tetraeder.
 * \param *a first vector
 * \param *a first vector
 * \param *a first vector
 * \param *a first vector
 * \return \f$ \frac{1}{6} \cdot ((a-d) \times (a-c) \cdot  (a-b)) \f$
 */
double CalculateVolumeofGeneralTetraeder(Vector *a, Vector *b, Vector *c, Vector *d)
{
  Vector Point, TetraederVector[3];
  double volume;

  TetraederVector[0].CopyVector(a);
  TetraederVector[1].CopyVector(b);
  TetraederVector[2].CopyVector(c);
  for (int j=0;j<3;j++)
    TetraederVector[j].SubtractVector(d);
  Point.CopyVector(&TetraederVector[0]);
  Point.VectorProduct(&TetraederVector[1]);
  volume = 1./6. * fabs(Point.ScalarProduct(&TetraederVector[2]));
  return volume;
};


/** Checks for a new special triangle whether one of its edges is already present with one one triangle connected.
 * This enforces that special triangles (i.e. degenerated ones) should at last close the open-edge frontier and not
 * make it bigger (i.e. closing one (the baseline) and opening two new ones).
 * \param TPS[3] nodes of the triangle
 * \return true - there is such a line (i.e. creation of degenerated triangle is valid), false - no such line (don't create)
 */
bool CheckLineCriteriaForDegeneratedTriangle(class BoundaryPointSet *nodes[3])
{
  bool result = false;
  int counter = 0;

  // check all three points
  for (int i=0;i<3;i++)
    for (int j=i+1; j<3; j++) {
      if (nodes[i]->lines.find(nodes[j]->node->nr) != nodes[i]->lines.end()) {  // there already is a line
        LineMap::iterator FindLine;
        pair<LineMap::iterator,LineMap::iterator> FindPair;
        FindPair = nodes[i]->lines.equal_range(nodes[j]->node->nr);
        for (FindLine = FindPair.first; FindLine != FindPair.second; ++FindLine) {
          // If there is a line with less than two attached triangles, we don't need a new line.
          if (FindLine->second->triangles.size() < 2) {
            counter++;
            break;  // increase counter only once per edge
          }
        }
      } else { // no line
        cout << Verbose(1) << "The line between " << *nodes[i] << " and " << *nodes[j] << " is not yet present, hence no need for a degenerate triangle." << endl;
        result = true;
      }
    }
  if ((!result) && (counter > 1)) {
    cout << Verbose(2) << "INFO: Degenerate triangle is ok, at least two, here " << counter << ", existing lines are used." << endl;
    result = true;
  }
  return result;
};


/** Sort function for the candidate list.
 */
bool SortCandidates(CandidateForTesselation* candidate1, CandidateForTesselation* candidate2)
{
  Vector BaseLineVector, OrthogonalVector, helper;
  if (candidate1->BaseLine != candidate2->BaseLine) {  // sanity check
    cout << Verbose(0) << "ERROR: sortCandidates was called for two different baselines: " << candidate1->BaseLine << " and " << candidate2->BaseLine << "." << endl;
    //return false;
    exit(1);
  }
  // create baseline vector
  BaseLineVector.CopyVector(candidate1->BaseLine->endpoints[1]->node->node);
  BaseLineVector.SubtractVector(candidate1->BaseLine->endpoints[0]->node->node);
  BaseLineVector.Normalize();

  // create normal in-plane vector to cope with acos() non-uniqueness on [0,2pi] (note that is pointing in the "right" direction already, hence ">0" test!)
  helper.CopyVector(candidate1->BaseLine->endpoints[0]->node->node);
  helper.SubtractVector(candidate1->point->node);
  OrthogonalVector.CopyVector(&helper);
  helper.VectorProduct(&BaseLineVector);
  OrthogonalVector.SubtractVector(&helper);
  OrthogonalVector.Normalize();

  // calculate both angles and correct with in-plane vector
  helper.CopyVector(candidate1->point->node);
  helper.SubtractVector(candidate1->BaseLine->endpoints[0]->node->node);
  double phi = BaseLineVector.Angle(&helper);
  if (OrthogonalVector.ScalarProduct(&helper) > 0) {
    phi = 2.*M_PI - phi;
  }
  helper.CopyVector(candidate2->point->node);
  helper.SubtractVector(candidate1->BaseLine->endpoints[0]->node->node);
  double psi = BaseLineVector.Angle(&helper);
  if (OrthogonalVector.ScalarProduct(&helper) > 0) {
    psi = 2.*M_PI - psi;
  }

  cout << Verbose(2) << *candidate1->point << " has angle " << phi << endl;
  cout << Verbose(2) << *candidate2->point << " has angle " << psi << endl;

  // return comparison
  return phi < psi;
};

/**
 * Finds the point which is second closest to the provided one.
 *
 * @param Point to which to find the second closest other point
 * @param linked cell structure
 *
 * @return point which is second closest to the provided one
 */
TesselPoint* FindSecondClosestPoint(const Vector* Point, LinkedCell* LC)
{
  LinkedNodes *List = NULL;
  TesselPoint* closestPoint = NULL;
  TesselPoint* secondClosestPoint = NULL;
  double distance = 1e16;
  double secondDistance = 1e16;
  Vector helper;
  int N[NDIM], Nlower[NDIM], Nupper[NDIM];

  LC->SetIndexToVector(Point); // ignore status as we calculate bounds below sensibly
  for(int i=0;i<NDIM;i++) // store indices of this cell
    N[i] = LC->n[i];
  cout << Verbose(2) << "INFO: Center cell is " << N[0] << ", " << N[1] << ", " << N[2] << " with No. " << LC->index << "." << endl;

  LC->GetNeighbourBounds(Nlower, Nupper);
  //cout << endl;
  for (LC->n[0] = Nlower[0]; LC->n[0] <= Nupper[0]; LC->n[0]++)
    for (LC->n[1] = Nlower[1]; LC->n[1] <= Nupper[1]; LC->n[1]++)
      for (LC->n[2] = Nlower[2]; LC->n[2] <= Nupper[2]; LC->n[2]++) {
        List = LC->GetCurrentCell();
        //cout << Verbose(3) << "The current cell " << LC->n[0] << "," << LC->n[1] << "," << LC->n[2] << endl;
        if (List != NULL) {
          for (LinkedNodes::iterator Runner = List->begin(); Runner != List->end(); Runner++) {
            helper.CopyVector(Point);
            helper.SubtractVector((*Runner)->node);
            double currentNorm = helper. Norm();
            if (currentNorm < distance) {
              // remember second point
              secondDistance = distance;
              secondClosestPoint = closestPoint;
              // mark down new closest point
              distance = currentNorm;
              closestPoint = (*Runner);
              //cout << Verbose(2) << "INFO: New Second Nearest Neighbour is " << *secondClosestPoint << "." << endl;
            }
          }
        } else {
          cerr << "ERROR: The current cell " << LC->n[0] << "," << LC->n[1] << ","
            << LC->n[2] << " is invalid!" << endl;
        }
      }

  return secondClosestPoint;
};

/**
 * Finds the point which is closest to the provided one.
 *
 * @param Point to which to find the closest other point
 * @param SecondPoint the second closest other point on return, NULL if none found
 * @param linked cell structure
 *
 * @return point which is closest to the provided one, NULL if none found
 */
TesselPoint* FindClosestPoint(const Vector* Point, TesselPoint *&SecondPoint, LinkedCell* LC)
{
  LinkedNodes *List = NULL;
  TesselPoint* closestPoint = NULL;
  SecondPoint = NULL;
  double distance = 1e16;
  double secondDistance = 1e16;
  Vector helper;
  int N[NDIM], Nlower[NDIM], Nupper[NDIM];

  LC->SetIndexToVector(Point); // ignore status as we calculate bounds below sensibly
  for(int i=0;i<NDIM;i++) // store indices of this cell
    N[i] = LC->n[i];
  cout << Verbose(3) << "INFO: Center cell is " << N[0] << ", " << N[1] << ", " << N[2] << " with No. " << LC->index << "." << endl;

  LC->GetNeighbourBounds(Nlower, Nupper);
  //cout << endl;
  for (LC->n[0] = Nlower[0]; LC->n[0] <= Nupper[0]; LC->n[0]++)
    for (LC->n[1] = Nlower[1]; LC->n[1] <= Nupper[1]; LC->n[1]++)
      for (LC->n[2] = Nlower[2]; LC->n[2] <= Nupper[2]; LC->n[2]++) {
        List = LC->GetCurrentCell();
        //cout << Verbose(3) << "The current cell " << LC->n[0] << "," << LC->n[1] << "," << LC->n[2] << endl;
        if (List != NULL) {
          for (LinkedNodes::iterator Runner = List->begin(); Runner != List->end(); Runner++) {
            helper.CopyVector(Point);
            helper.SubtractVector((*Runner)->node);
            double currentNorm = helper. Norm();
            if (currentNorm < distance) {
              secondDistance = distance;
              SecondPoint = closestPoint;
              distance = currentNorm;
              closestPoint = (*Runner);
              //cout << Verbose(2) << "INFO: New Nearest Neighbour is " << *closestPoint << "." << endl;
            } else if (currentNorm < secondDistance) {
              secondDistance = currentNorm;
              SecondPoint = (*Runner);
              //cout << Verbose(2) << "INFO: New Second Nearest Neighbour is " << *SecondPoint << "." << endl;
            }
          }
        } else {
          cerr << "ERROR: The current cell " << LC->n[0] << "," << LC->n[1] << ","
            << LC->n[2] << " is invalid!" << endl;
        }
      }

  return closestPoint;
};

/** Returns the closest point on \a *Base with respect to \a *OtherBase.
 * \param *out output stream for debugging
 * \param *Base reference line
 * \param *OtherBase other base line
 * \return Vector on reference line that has closest distance
 */
Vector * GetClosestPointBetweenLine(ofstream *out, class BoundaryLineSet *Base, class BoundaryLineSet *OtherBase)
{
  // construct the plane of the two baselines (i.e. take both their directional vectors)
  Vector Normal;
  Vector Baseline, OtherBaseline;
  Baseline.CopyVector(Base->endpoints[1]->node->node);
  Baseline.SubtractVector(Base->endpoints[0]->node->node);
  OtherBaseline.CopyVector(OtherBase->endpoints[1]->node->node);
  OtherBaseline.SubtractVector(OtherBase->endpoints[0]->node->node);
  Normal.CopyVector(&Baseline);
  Normal.VectorProduct(&OtherBaseline);
  Normal.Normalize();
  *out << Verbose(4) << "First direction is " << Baseline << ", second direction is " << OtherBaseline << ", normal of intersection plane is " << Normal << "." << endl;

  // project one offset point of OtherBase onto this plane (and add plane offset vector)
  Vector NewOffset;
  NewOffset.CopyVector(OtherBase->endpoints[0]->node->node);
  NewOffset.SubtractVector(Base->endpoints[0]->node->node);
  NewOffset.ProjectOntoPlane(&Normal);
  NewOffset.AddVector(Base->endpoints[0]->node->node);
  Vector NewDirection;
  NewDirection.CopyVector(&NewOffset);
  NewDirection.AddVector(&OtherBaseline);

  // calculate the intersection between this projected baseline and Base
  Vector *Intersection = new Vector;
  Intersection->GetIntersectionOfTwoLinesOnPlane(out, Base->endpoints[0]->node->node, Base->endpoints[1]->node->node, &NewOffset, &NewDirection, &Normal);
  Normal.CopyVector(Intersection);
  Normal.SubtractVector(Base->endpoints[0]->node->node);
  *out << Verbose(3) << "Found closest point on " << *Base << " at " << *Intersection << ", factor in line is " << fabs(Normal.ScalarProduct(&Baseline)/Baseline.NormSquared()) << "." << endl;

  return Intersection;
};


/** Creates the objects in a VRML file.
 * \param *out output stream for debugging
 * \param *vrmlfile output stream for tecplot data
 * \param *Tess Tesselation structure with constructed triangles
 * \param *mol molecule structure with atom positions
 */
void WriteVrmlFile(ofstream *out, ofstream *vrmlfile, class Tesselation *Tess, PointCloud *cloud)
{
  TesselPoint *Walker = NULL;
  int i;
  Vector *center = cloud->GetCenter(out);
  if (vrmlfile != NULL) {
    //cout << Verbose(1) << "Writing Raster3D file ... ";
    *vrmlfile << "#VRML V2.0 utf8" << endl;
    *vrmlfile << "#Created by molecuilder" << endl;
    *vrmlfile << "#All atoms as spheres" << endl;
    cloud->GoToFirst();
    while (!cloud->IsEnd()) {
      Walker = cloud->GetPoint();
      *vrmlfile << "Sphere {" << endl << "  "; // 2 is sphere type
      for (i=0;i<NDIM;i++)
        *vrmlfile << Walker->node->x[i]-center->x[i] << " ";
      *vrmlfile << "\t0.1\t1. 1. 1." << endl; // radius 0.05 and white as colour
      cloud->GoToNext();
    }

    *vrmlfile << "# All tesselation triangles" << endl;
    for (TriangleMap::iterator TriangleRunner = Tess->TrianglesOnBoundary.begin(); TriangleRunner != Tess->TrianglesOnBoundary.end(); TriangleRunner++) {
      *vrmlfile << "1" << endl << "  "; // 1 is triangle type
      for (i=0;i<3;i++) { // print each node
        for (int j=0;j<NDIM;j++)  // and for each node all NDIM coordinates
          *vrmlfile << TriangleRunner->second->endpoints[i]->node->node->x[j]-center->x[j] << " ";
        *vrmlfile << "\t";
      }
      *vrmlfile << "1. 0. 0." << endl;  // red as colour
      *vrmlfile << "18" << endl << "  0.5 0.5 0.5" << endl; // 18 is transparency type for previous object
    }
  } else {
    cerr << "ERROR: Given vrmlfile is " << vrmlfile << "." << endl;
  }
  delete(center);
};

/** Writes additionally the current sphere (i.e. the last triangle to file).
 * \param *out output stream for debugging
 * \param *rasterfile output stream for tecplot data
 * \param *Tess Tesselation structure with constructed triangles
 * \param *mol molecule structure with atom positions
 */
void IncludeSphereinRaster3D(ofstream *out, ofstream *rasterfile, class Tesselation *Tess, PointCloud *cloud)
{
  Vector helper;
  // include the current position of the virtual sphere in the temporary raster3d file
  Vector *center = cloud->GetCenter(out);
  // make the circumsphere's center absolute again
  helper.CopyVector(Tess->LastTriangle->endpoints[0]->node->node);
  helper.AddVector(Tess->LastTriangle->endpoints[1]->node->node);
  helper.AddVector(Tess->LastTriangle->endpoints[2]->node->node);
  helper.Scale(1./3.);
  helper.SubtractVector(center);
  // and add to file plus translucency object
  *rasterfile << "# current virtual sphere\n";
  *rasterfile << "8\n  25.0    0.6     -1.0 -1.0 -1.0     0.2        0 0 0 0\n";
  *rasterfile << "2\n  " << helper.x[0] << " " << helper.x[1] << " " << helper.x[2] << "\t" << 5. << "\t1 0 0\n";
  *rasterfile << "9\n  terminating special property\n";
  delete(center);
};

/** Creates the objects in a raster3d file (renderable with a header.r3d).
 * \param *out output stream for debugging
 * \param *rasterfile output stream for tecplot data
 * \param *Tess Tesselation structure with constructed triangles
 * \param *mol molecule structure with atom positions
 */
void WriteRaster3dFile(ofstream *out, ofstream *rasterfile, class Tesselation *Tess, PointCloud *cloud)
{
  TesselPoint *Walker = NULL;
  int i;
  Vector *center = cloud->GetCenter(out);
  if (rasterfile != NULL) {
    //cout << Verbose(1) << "Writing Raster3D file ... ";
    *rasterfile << "# Raster3D object description, created by MoleCuilder" << endl;
    *rasterfile << "@header.r3d" << endl;
    *rasterfile << "# All atoms as spheres" << endl;
    cloud->GoToFirst();
    while (!cloud->IsEnd()) {
      Walker = cloud->GetPoint();
      *rasterfile << "2" << endl << "  ";  // 2 is sphere type
      for (i=0;i<NDIM;i++)
        *rasterfile << Walker->node->x[i]-center->x[i] << " ";
      *rasterfile << "\t0.1\t1. 1. 1." << endl; // radius 0.05 and white as colour
      cloud->GoToNext();
    }

    *rasterfile << "# All tesselation triangles" << endl;
    *rasterfile << "8\n  25. -1.   1. 1. 1.   0.0    0 0 0 2\n  SOLID     1.0 0.0 0.0\n  BACKFACE  0.3 0.3 1.0   0 0\n";
    for (TriangleMap::iterator TriangleRunner = Tess->TrianglesOnBoundary.begin(); TriangleRunner != Tess->TrianglesOnBoundary.end(); TriangleRunner++) {
      *rasterfile << "1" << endl << "  ";  // 1 is triangle type
      for (i=0;i<3;i++) {  // print each node
        for (int j=0;j<NDIM;j++)  // and for each node all NDIM coordinates
          *rasterfile << TriangleRunner->second->endpoints[i]->node->node->x[j]-center->x[j] << " ";
        *rasterfile << "\t";
      }
      *rasterfile << "1. 0. 0." << endl;  // red as colour
      //*rasterfile << "18" << endl << "  0.5 0.5 0.5" << endl;  // 18 is transparency type for previous object
    }
    *rasterfile << "9\n#  terminating special property\n";
  } else {
    cerr << "ERROR: Given rasterfile is " << rasterfile << "." << endl;
  }
  IncludeSphereinRaster3D(out, rasterfile, Tess, cloud);
  delete(center);
};

/** This function creates the tecplot file, displaying the tesselation of the hull.
 * \param *out output stream for debugging
 * \param *tecplot output stream for tecplot data
 * \param N arbitrary number to differentiate various zones in the tecplot format
 */
void WriteTecplotFile(ofstream *out, ofstream *tecplot, class Tesselation *TesselStruct, PointCloud *cloud, int N)
{
  if ((tecplot != NULL) && (TesselStruct != NULL)) {
    // write header
    *tecplot << "TITLE = \"3D CONVEX SHELL\"" << endl;
    *tecplot << "VARIABLES = \"X\" \"Y\" \"Z\" \"U\"" << endl;
    *tecplot << "ZONE T=\"" << N << "-";
    for (int i=0;i<3;i++)
      *tecplot << (i==0 ? "" : "_") << TesselStruct->LastTriangle->endpoints[i]->node->Name;
    *tecplot << "\", N=" << TesselStruct->PointsOnBoundary.size() << ", E=" << TesselStruct->TrianglesOnBoundary.size() << ", DATAPACKING=POINT, ZONETYPE=FETRIANGLE" << endl;
    int i=0;
    for (cloud->GoToFirst(); !cloud->IsEnd(); cloud->GoToNext(), i++);
    int *LookupList = new int[i];
    for (cloud->GoToFirst(), i=0; !cloud->IsEnd(); cloud->GoToNext(), i++)
      LookupList[i] = -1;

    // print atom coordinates
    *out << Verbose(2) << "The following triangles were created:";
    int Counter = 1;
    TesselPoint *Walker = NULL;
    for (PointMap::iterator target = TesselStruct->PointsOnBoundary.begin(); target != TesselStruct->PointsOnBoundary.end(); target++) {
      Walker = target->second->node;
      LookupList[Walker->nr] = Counter++;
      *tecplot << Walker->node->x[0] << " " << Walker->node->x[1] << " " << Walker->node->x[2] << " " << target->second->value << endl;
    }
    *tecplot << endl;
    // print connectivity
    for (TriangleMap::iterator runner = TesselStruct->TrianglesOnBoundary.begin(); runner != TesselStruct->TrianglesOnBoundary.end(); runner++) {
      *out << " " << runner->second->endpoints[0]->node->Name << "<->" << runner->second->endpoints[1]->node->Name << "<->" << runner->second->endpoints[2]->node->Name;
      *tecplot << LookupList[runner->second->endpoints[0]->node->nr] << " " << LookupList[runner->second->endpoints[1]->node->nr] << " " << LookupList[runner->second->endpoints[2]->node->nr] << endl;
    }
    delete[] (LookupList);
    *out << endl;
  }
};

/** Calculates the concavity for each of the BoundaryPointSet's in a Tesselation.
 * Sets BoundaryPointSet::value equal to the number of connected lines that are not convex.
 * \param *out output stream for debugging
 * \param *TesselStruct pointer to Tesselation structure
 */
void CalculateConcavityPerBoundaryPoint(ofstream *out, class Tesselation *TesselStruct)
{
  class BoundaryPointSet *point = NULL;
  class BoundaryLineSet *line = NULL;

  //*out << Verbose(2) << "Begin of CalculateConcavityPerBoundaryPoint" << endl;
  // calculate remaining concavity
  for (PointMap::iterator PointRunner = TesselStruct->PointsOnBoundary.begin(); PointRunner != TesselStruct->PointsOnBoundary.end(); PointRunner++) {
    point = PointRunner->second;
    *out << Verbose(1) << "INFO: Current point is " << *point << "." << endl;
    point->value = 0;
    for (LineMap::iterator LineRunner = point->lines.begin(); LineRunner != point->lines.end(); LineRunner++) {
      line = LineRunner->second;
      //*out << Verbose(2) << "INFO: Current line of point " << *point << " is " << *line << "." << endl;
      if (!line->CheckConvexityCriterion(out))
        point->value += 1;
    }
  }
  //*out << Verbose(2) << "End of CalculateConcavityPerBoundaryPoint" << endl;
};


/** Checks whether each BoundaryLineSet in the Tesselation has two triangles.
 * \param *out output stream for debugging
 * \param *TesselStruct
 * \return true - all have exactly two triangles, false - some not, list is printed to screen
 */
bool CheckListOfBaselines(ofstream *out, Tesselation *TesselStruct)
{
  LineMap::iterator testline;
  bool result = false;
  int counter = 0;

  *out << Verbose(1) << "Check: List of Baselines with not two connected triangles:" << endl;
  for (testline = TesselStruct->LinesOnBoundary.begin(); testline != TesselStruct->LinesOnBoundary.end(); testline++) {
    if (testline->second->triangles.size() != 2) {
      *out << Verbose(1) << *testline->second << "\t" << testline->second->triangles.size() << endl;
      counter++;
    }
  }
  if (counter == 0) {
    *out << Verbose(1) << "None." << endl;
    result = true;
  }
  return result;
}