source: src/boundary.cpp@ 6ac7ee

#include "boundary.hpp"

#define DEBUG 1
#define DoSingleStepOutput 0
#define DoTecplotOutput 1
#define DoRaster3DOutput 1
#define DoVRMLOutput 1
#define TecplotSuffix ".dat"
#define Raster3DSuffix ".r3d"
#define VRMLSUffix ".wrl"
#define HULLEPSILON MYEPSILON

// ======================================== Points on Boundary =================================

BoundaryPointSet::BoundaryPointSet()
{
        LinesCount = 0;
        Nr = -1;
}
;

BoundaryPointSet::BoundaryPointSet(atom *Walker)
{
        node = Walker;
        LinesCount = 0;
        Nr = Walker->nr;
}
;

BoundaryPointSet::~BoundaryPointSet()
{
        cout << Verbose(5) << "Erasing point nr. " << Nr << "." << endl;
        if (!lines.empty())
                cerr << "WARNING: Memory Leak! I " << *this << " am still connected to some lines." << endl;
        node = NULL;
}
;

void BoundaryPointSet::AddLine(class BoundaryLineSet *line)
{
        cout << Verbose(6) << "Adding " << *this << " to line " << *line << "."
                        << endl;
        if (line->endpoints[0] == this)
                {
                        lines.insert(LinePair(line->endpoints[1]->Nr, line));
                }
        else
                {
                        lines.insert(LinePair(line->endpoints[0]->Nr, line));
                }
        LinesCount++;
}
;

ostream &
operator <<(ostream &ost, BoundaryPointSet &a)
{
        ost << "[" << a.Nr << "|" << a.node->Name << "]";
        return ost;
}
;

// ======================================== Lines on Boundary =================================

BoundaryLineSet::BoundaryLineSet()
{
        for (int i = 0; i < 2; i++)
                endpoints[i] = NULL;
        TrianglesCount = 0;
        Nr = -1;
}
;

BoundaryLineSet::BoundaryLineSet(class BoundaryPointSet *Point[2], int number)
{
        // set number
        Nr = number;
        // set endpoints in ascending order
        SetEndpointsOrdered(endpoints, Point[0], Point[1]);
        // add this line to the hash maps of both endpoints
        Point[0]->AddLine(this); //Taken out, to check whether we can avoid unwanted double adding.
        Point[1]->AddLine(this); //
        // clear triangles list
        TrianglesCount = 0;
        cout << Verbose(5) << "New Line with endpoints " << *this << "." << endl;
}
;

BoundaryLineSet::~BoundaryLineSet()
{
        int Numbers[2];
        Numbers[0] = endpoints[1]->Nr;
        Numbers[1] = endpoints[0]->Nr;
        for (int i = 0; i < 2; i++) {
                cout << Verbose(5) << "Erasing Line Nr. " << Nr << " in boundary point " << *endpoints[i] << "." << endl;
                endpoints[i]->lines.erase(Numbers[i]);
                if (endpoints[i]->lines.empty()) {
                        cout << Verbose(5) << *endpoints[i] << " has no more lines it's attached to, erasing." << endl;
                        if (endpoints[i] != NULL) {
                                delete(endpoints[i]);
                                endpoints[i] = NULL;
                        } else
                                cerr << "ERROR: Endpoint " << i << " has already been free'd." << endl;
                } else
                        cout << Verbose(5) << *endpoints[i] << " has still lines it's attached to." << endl;
        }
        if (!triangles.empty())
                cerr << "WARNING: Memory Leak! I " << *this << " am still connected to some triangles." << endl;
}
;

void
BoundaryLineSet::AddTriangle(class BoundaryTriangleSet *triangle)
{
        cout << Verbose(6) << "Add " << triangle->Nr << " to line " << *this << "."
                        << endl;
        triangles.insert(TrianglePair(triangle->Nr, triangle));
        TrianglesCount++;
}
;

ostream &
operator <<(ostream &ost, BoundaryLineSet &a)
{
        ost << "[" << a.Nr << "|" << a.endpoints[0]->node->Name << ","
                        << a.endpoints[1]->node->Name << "]";
        return ost;
}
;

// ======================================== Triangles on Boundary =================================


BoundaryTriangleSet::BoundaryTriangleSet()
{
        for (int i = 0; i < 3; i++)
                {
                        endpoints[i] = NULL;
                        lines[i] = NULL;
                }
        Nr = -1;
}
;

BoundaryTriangleSet::BoundaryTriangleSet(class BoundaryLineSet *line[3],
                int number)
{
        // set number
        Nr = number;
        // set lines
        cout << Verbose(5) << "New triangle " << Nr << ":" << endl;
        for (int i = 0; i < 3; i++)
                {
                        lines[i] = line[i];
                        lines[i]->AddTriangle(this);
                }
        // get ascending order of endpoints
        map<int, class BoundaryPointSet *> OrderMap;
        for (int i = 0; i < 3; i++)
                // for all three lines
                for (int j = 0; j < 2; j++)
                        { // for both endpoints
                                OrderMap.insert(pair<int, class BoundaryPointSet *> (
                                                line[i]->endpoints[j]->Nr, line[i]->endpoints[j]));
                                // and we don't care whether insertion fails
                        }
        // set endpoints
        int Counter = 0;
        cout << Verbose(6) << " with end points ";
        for (map<int, class BoundaryPointSet *>::iterator runner = OrderMap.begin(); runner
                        != OrderMap.end(); runner++)
                {
                        endpoints[Counter] = runner->second;
                        cout << " " << *endpoints[Counter];
                        Counter++;
                }
        if (Counter < 3)
                {
                        cerr << "ERROR! We have a triangle with only two distinct endpoints!"
                                        << endl;
                        //exit(1);
                }
        cout << "." << endl;
}
;

BoundaryTriangleSet::~BoundaryTriangleSet()
{
        for (int i = 0; i < 3; i++) {
                cout << Verbose(5) << "Erasing triangle Nr." << Nr << endl;
                lines[i]->triangles.erase(Nr);
                if (lines[i]->triangles.empty()) {
                        cout << Verbose(5) << *lines[i] << " is no more attached to any triangle, erasing." << endl;
                        if (lines[i] != NULL) {
                                delete (lines[i]);
                                lines[i] = NULL;
                        } else
                                cerr << "ERROR: This line " << i << " has already been free'd." << endl;
                } else
                        cout << Verbose(5) << *lines[i] << " is still attached to a triangle." << endl;
        }
}
;

void
BoundaryTriangleSet::GetNormalVector(Vector &OtherVector)
{
        // get normal vector
        NormalVector.MakeNormalVector(&endpoints[0]->node->x, &endpoints[1]->node->x,
                        &endpoints[2]->node->x);

        // make it always point inward (any offset vector onto plane projected onto normal vector suffices)
        if (NormalVector.Projection(&OtherVector) > 0)
                NormalVector.Scale(-1.);
}
;

ostream &
operator <<(ostream &ost, BoundaryTriangleSet &a)
{
        ost << "[" << a.Nr << "|" << a.endpoints[0]->node->Name << ","
                        << a.endpoints[1]->node->Name << "," << a.endpoints[2]->node->Name << "]";
        return ost;
}
;

// ========================================== F U N C T I O N S =================================

/** Finds the endpoint two lines are sharing.
 * \param *line1 first line
 * \param *line2 second line
 * \return point which is shared or NULL if none
 */
class BoundaryPointSet *
GetCommonEndpoint(class BoundaryLineSet * line1, class BoundaryLineSet * line2)
{
        class BoundaryLineSet * lines[2] =
                { line1, line2 };
        class BoundaryPointSet *node = NULL;
        map<int, class BoundaryPointSet *> OrderMap;
        pair<map<int, class BoundaryPointSet *>::iterator, bool> OrderTest;
        for (int i = 0; i < 2; i++)
                // for both lines
                for (int j = 0; j < 2; j++)
                        { // for both endpoints
                                OrderTest = OrderMap.insert(pair<int, class BoundaryPointSet *> (
                                                lines[i]->endpoints[j]->Nr, lines[i]->endpoints[j]));
                                if (!OrderTest.second)
                                        { // if insertion fails, we have common endpoint
                                                node = OrderTest.first->second;
                                                cout << Verbose(5) << "Common endpoint of lines " << *line1
                                                                << " and " << *line2 << " is: " << *node << "." << endl;
                                                j = 2;
                                                i = 2;
                                                break;
                                        }
                        }
        return node;
}
;

/** Determines the boundary points of a cluster.
 * Does a projection per axis onto the orthogonal plane, transforms into spherical coordinates, sorts them by the angle
 * and looks at triples: if the middle has less a distance than the allowed maximum height of the triangle formed by the plane's
 * center and first and last point in the triple, it is thrown out.
 * \param *out output stream for debugging
 * \param *mol molecule structure representing the cluster
 */
Boundaries *
GetBoundaryPoints(ofstream *out, molecule *mol)
{
        atom *Walker = NULL;
        PointMap PointsOnBoundary;
        LineMap LinesOnBoundary;
        TriangleMap TrianglesOnBoundary;

        *out << Verbose(1) << "Finding all boundary points." << endl;
        Boundaries *BoundaryPoints = new Boundaries[NDIM]; // first is alpha, second is (r, nr)
        BoundariesTestPair BoundaryTestPair;
        Vector AxisVector, AngleReferenceVector, AngleReferenceNormalVector;
        double radius, angle;
        // 3a. Go through every axis
        for (int axis = 0; axis < NDIM; axis++)
                {
                        AxisVector.Zero();
                        AngleReferenceVector.Zero();
                        AngleReferenceNormalVector.Zero();
                        AxisVector.x[axis] = 1.;
                        AngleReferenceVector.x[(axis + 1) % NDIM] = 1.;
                        AngleReferenceNormalVector.x[(axis + 2) % NDIM] = 1.;
                        //              *out << Verbose(1) << "Axisvector is ";
                        //              AxisVector.Output(out);
                        //              *out << " and AngleReferenceVector is ";
                        //              AngleReferenceVector.Output(out);
                        //              *out << "." << endl;
                        //              *out << " and AngleReferenceNormalVector is ";
                        //              AngleReferenceNormalVector.Output(out);
                        //              *out << "." << endl;
                        // 3b. construct set of all points, transformed into cylindrical system and with left and right neighbours
                        Walker = mol->start;
                        while (Walker->next != mol->end)
                                {
                                        Walker = Walker->next;
                                        Vector ProjectedVector;
                                        ProjectedVector.CopyVector(&Walker->x);
                                        ProjectedVector.ProjectOntoPlane(&AxisVector);
                                        // correct for negative side
                                        //if (Projection(y) < 0)
                                        //angle = 2.*M_PI - angle;
                                        radius = ProjectedVector.Norm();
                                        if (fabs(radius) > MYEPSILON)
                                                angle = ProjectedVector.Angle(&AngleReferenceVector);
                                        else
                                                angle = 0.; // otherwise it's a vector in Axis Direction and unimportant for boundary issues

                                        //*out << "Checking sign in quadrant : " << ProjectedVector.Projection(&AngleReferenceNormalVector) << "." << endl;
                                        if (ProjectedVector.Projection(&AngleReferenceNormalVector) > 0)
                                                {
                                                        angle = 2. * M_PI - angle;
                                                }
                                        //*out << Verbose(2) << "Inserting " << *Walker << ": (r, alpha) = (" << radius << "," << angle << "): ";
                                        //ProjectedVector.Output(out);
                                        //*out << endl;
                                        BoundaryTestPair = BoundaryPoints[axis].insert(BoundariesPair(angle,
                                                        DistancePair (radius, Walker)));
                                        if (BoundaryTestPair.second)
                                                { // successfully inserted
                                                }
                                        else
                                                { // same point exists, check first r, then distance of original vectors to center of gravity
                                                        *out << Verbose(2)
                                                                        << "Encountered two vectors whose projection onto axis "
                                                                        << axis << " is equal: " << endl;
                                                        *out << Verbose(2) << "Present vector: ";
                                                        BoundaryTestPair.first->second.second->x.Output(out);
                                                        *out << endl;
                                                        *out << Verbose(2) << "New vector: ";
                                                        Walker->x.Output(out);
                                                        *out << endl;
                                                        double tmp = ProjectedVector.Norm();
                                                        if (tmp > BoundaryTestPair.first->second.first)
                                                                {
                                                                        BoundaryTestPair.first->second.first = tmp;
                                                                        BoundaryTestPair.first->second.second = Walker;
                                                                        *out << Verbose(2) << "Keeping new vector." << endl;
                                                                }
                                                        else if (tmp == BoundaryTestPair.first->second.first)
                                                                {
                                                                        if (BoundaryTestPair.first->second.second->x.ScalarProduct(
                                                                                        &BoundaryTestPair.first->second.second->x)
                                                                                        < Walker->x.ScalarProduct(&Walker->x))
                                                                                { // Norm() does a sqrt, which makes it a lot slower
                                                                                        BoundaryTestPair.first->second.second = Walker;
                                                                                        *out << Verbose(2) << "Keeping new vector." << endl;
                                                                                }
                                                                        else
                                                                                {
                                                                                        *out << Verbose(2) << "Keeping present vector." << endl;
                                                                                }
                                                                }
                                                        else
                                                                {
                                                                        *out << Verbose(2) << "Keeping present vector." << endl;
                                                                }
                                                }
                                }
                        // printing all inserted for debugging
                        //              {
                        //                      *out << Verbose(2) << "Printing list of candidates for axis " << axis << " which we have inserted so far." << endl;
                        //                      int i=0;
                        //                      for(Boundaries::iterator runner = BoundaryPoints[axis].begin(); runner != BoundaryPoints[axis].end(); runner++) {
                        //                              if (runner != BoundaryPoints[axis].begin())
                        //                                      *out << ", " << i << ": " << *runner->second.second;
                        //                              else
                        //                                      *out << i << ": " << *runner->second.second;
                        //                              i++;
                        //                      }
                        //                      *out << endl;
                        //              }
                        // 3c. throw out points whose distance is less than the mean of left and right neighbours
                        bool flag = false;
                        do
                                { // do as long as we still throw one out per round
                                        *out << Verbose(1)
                                                        << "Looking for candidates to kick out by convex condition ... "
                                                        << endl;
                                        flag = false;
                                        Boundaries::iterator left = BoundaryPoints[axis].end();
                                        Boundaries::iterator right = BoundaryPoints[axis].end();
                                        for (Boundaries::iterator runner = BoundaryPoints[axis].begin(); runner
                                                        != BoundaryPoints[axis].end(); runner++)
                                                {
                                                        // set neighbours correctly
                                                        if (runner == BoundaryPoints[axis].begin())
                                                                {
                                                                        left = BoundaryPoints[axis].end();
                                                                }
                                                        else
                                                                {
                                                                        left = runner;
                                                                }
                                                        left--;
                                                        right = runner;
                                                        right++;
                                                        if (right == BoundaryPoints[axis].end())
                                                                {
                                                                        right = BoundaryPoints[axis].begin();
                                                                }
                                                        // check distance

                                                        // construct the vector of each side of the triangle on the projected plane (defined by normal vector AxisVector)
                                                                {
                                                                        Vector SideA, SideB, SideC, SideH;
                                                                        SideA.CopyVector(&left->second.second->x);
                                                                        SideA.ProjectOntoPlane(&AxisVector);
                                                                        //                                      *out << "SideA: ";
                                                                        //                                      SideA.Output(out);
                                                                        //                                      *out << endl;

                                                                        SideB.CopyVector(&right->second.second->x);
                                                                        SideB.ProjectOntoPlane(&AxisVector);
                                                                        //                                      *out << "SideB: ";
                                                                        //                                      SideB.Output(out);
                                                                        //                                      *out << endl;

                                                                        SideC.CopyVector(&left->second.second->x);
                                                                        SideC.SubtractVector(&right->second.second->x);
                                                                        SideC.ProjectOntoPlane(&AxisVector);
                                                                        //                                      *out << "SideC: ";
                                                                        //                                      SideC.Output(out);
                                                                        //                                      *out << endl;

                                                                        SideH.CopyVector(&runner->second.second->x);
                                                                        SideH.ProjectOntoPlane(&AxisVector);
                                                                        //                                      *out << "SideH: ";
                                                                        //                                      SideH.Output(out);
                                                                        //                                      *out << endl;

                                                                        // calculate each length
                                                                        double a = SideA.Norm();
                                                                        //double b = SideB.Norm();
                                                                        //double c = SideC.Norm();
                                                                        double h = SideH.Norm();
                                                                        // calculate the angles
                                                                        double alpha = SideA.Angle(&SideH);
                                                                        double beta = SideA.Angle(&SideC);
                                                                        double gamma = SideB.Angle(&SideH);
                                                                        double delta = SideC.Angle(&SideH);
                                                                        double MinDistance = a * sin(beta) / (sin(delta)) * (((alpha
                                                                                        < M_PI / 2.) || (gamma < M_PI / 2.)) ? 1. : -1.);
                                                                        //                                      *out << Verbose(2) << " I calculated: a = " << a << ", h = " << h << ", beta(" << left->second.second->Name << "," << left->second.second->Name << "-" << right->second.second->Name << ") = " << beta << ", delta(" << left->second.second->Name << "," << runner->second.second->Name << ") = " << delta << ", Min = " << MinDistance << "." << endl;
                                                                        //*out << Verbose(1) << "Checking CoG distance of runner " << *runner->second.second << " " << h << " against triangle's side length spanned by (" << *left->second.second << "," << *right->second.second << ") of " << MinDistance << "." << endl;
                                                                        if ((fabs(h / fabs(h) - MinDistance / fabs(MinDistance))
                                                                                        < MYEPSILON) && (h < MinDistance))
                                                                                {
                                                                                        // throw out point
                                                                                        //*out << Verbose(1) << "Throwing out " << *runner->second.second << "." << endl;
                                                                                        BoundaryPoints[axis].erase(runner);
                                                                                        flag = true;
                                                                                }
                                                                }
                                                }
                                }
                        while (flag);
                }
        return BoundaryPoints;
}
;

/** Determines greatest diameters of a cluster defined by its convex envelope.
 * Looks at lines parallel to one axis and where they intersect on the projected planes
 * \param *out output stream for debugging
 * \param *BoundaryPoints NDIM set of boundary points defining the convex envelope on each projected plane
 * \param *mol molecule structure representing the cluster
 * \param IsAngstroem whether we have angstroem or atomic units
 * \return NDIM array of the diameters
 */
double *
GetDiametersOfCluster(ofstream *out, Boundaries *BoundaryPtr, molecule *mol,
                bool IsAngstroem)
{
        // get points on boundary of NULL was given as parameter
        bool BoundaryFreeFlag = false;
        Boundaries *BoundaryPoints = BoundaryPtr;
        if (BoundaryPoints == NULL)
                {
                        BoundaryFreeFlag = true;
                        BoundaryPoints = GetBoundaryPoints(out, mol);
                }
        else
                {
                        *out << Verbose(1) << "Using given boundary points set." << endl;
                }
        // determine biggest "diameter" of cluster for each axis
        Boundaries::iterator Neighbour, OtherNeighbour;
        double *GreatestDiameter = new double[NDIM];
        for (int i = 0; i < NDIM; i++)
                GreatestDiameter[i] = 0.;
        double OldComponent, tmp, w1, w2;
        Vector DistanceVector, OtherVector;
        int component, Othercomponent;
        for (int axis = 0; axis < NDIM; axis++)
                { // regard each projected plane
                        //*out << Verbose(1) << "Current axis is " << axis << "." << endl;
                        for (int j = 0; j < 2; j++)
                                { // and for both axis on the current plane
                                        component = (axis + j + 1) % NDIM;
                                        Othercomponent = (axis + 1 + ((j + 1) & 1)) % NDIM;
                                        //*out << Verbose(1) << "Current component is " << component << ", Othercomponent is " << Othercomponent << "." << endl;
                                        for (Boundaries::iterator runner = BoundaryPoints[axis].begin(); runner
                                                        != BoundaryPoints[axis].end(); runner++)
                                                {
                                                        //*out << Verbose(2) << "Current runner is " << *(runner->second.second) << "." << endl;
                                                        // seek for the neighbours pair where the Othercomponent sign flips
                                                        Neighbour = runner;
                                                        Neighbour++;
                                                        if (Neighbour == BoundaryPoints[axis].end()) // make it wrap around
                                                                Neighbour = BoundaryPoints[axis].begin();
                                                        DistanceVector.CopyVector(&runner->second.second->x);
                                                        DistanceVector.SubtractVector(&Neighbour->second.second->x);
                                                        do
                                                                { // seek for neighbour pair where it flips
                                                                        OldComponent = DistanceVector.x[Othercomponent];
                                                                        Neighbour++;
                                                                        if (Neighbour == BoundaryPoints[axis].end()) // make it wrap around
                                                                                Neighbour = BoundaryPoints[axis].begin();
                                                                        DistanceVector.CopyVector(&runner->second.second->x);
                                                                        DistanceVector.SubtractVector(&Neighbour->second.second->x);
                                                                        //*out << Verbose(3) << "OldComponent is " << OldComponent << ", new one is " << DistanceVector.x[Othercomponent] << "." << endl;
                                                                }
                                                        while ((runner != Neighbour) && (fabs(OldComponent / fabs(
                                                                        OldComponent) - DistanceVector.x[Othercomponent] / fabs(
                                                                        DistanceVector.x[Othercomponent])) < MYEPSILON)); // as long as sign does not flip
                                                        if (runner != Neighbour)
                                                                {
                                                                        OtherNeighbour = Neighbour;
                                                                        if (OtherNeighbour == BoundaryPoints[axis].begin()) // make it wrap around
                                                                                OtherNeighbour = BoundaryPoints[axis].end();
                                                                        OtherNeighbour--;
                                                                        //*out << Verbose(2) << "The pair, where the sign of OtherComponent flips, is: " << *(Neighbour->second.second) << " and " << *(OtherNeighbour->second.second) << "." << endl;
                                                                        // now we have found the pair: Neighbour and OtherNeighbour
                                                                        OtherVector.CopyVector(&runner->second.second->x);
                                                                        OtherVector.SubtractVector(&OtherNeighbour->second.second->x);
                                                                        //*out << Verbose(2) << "Distances to Neighbour and OtherNeighbour are " << DistanceVector.x[component] << " and " << OtherVector.x[component] << "." << endl;
                                                                        //*out << Verbose(2) << "OtherComponents to Neighbour and OtherNeighbour are " << DistanceVector.x[Othercomponent] << " and " << OtherVector.x[Othercomponent] << "." << endl;
                                                                        // do linear interpolation between points (is exact) to extract exact intersection between Neighbour and OtherNeighbour
                                                                        w1 = fabs(OtherVector.x[Othercomponent]);
                                                                        w2 = fabs(DistanceVector.x[Othercomponent]);
                                                                        tmp = fabs((w1 * DistanceVector.x[component] + w2
                                                                                        * OtherVector.x[component]) / (w1 + w2));
                                                                        // mark if it has greater diameter
                                                                        //*out << Verbose(2) << "Comparing current greatest " << GreatestDiameter[component] << " to new " << tmp << "." << endl;
                                                                        GreatestDiameter[component] = (GreatestDiameter[component]
                                                                                        > tmp) ? GreatestDiameter[component] : tmp;
                                                                } //else
                                                        //*out << Verbose(2) << "Saw no sign flip, probably top or bottom node." << endl;
                                                }
                                }
                }
        *out << Verbose(0) << "RESULT: The biggest diameters are "
                        << GreatestDiameter[0] << " and " << GreatestDiameter[1] << " and "
                        << GreatestDiameter[2] << " " << (IsAngstroem ? "angstrom"
                        : "atomiclength") << "." << endl;

        // free reference lists
        if (BoundaryFreeFlag)
                delete[] (BoundaryPoints);

        return GreatestDiameter;
}
;

/** 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 write_vrml_file(ofstream *out, ofstream *vrmlfile, class Tesselation *Tess, class molecule *mol)
{
        atom *Walker = mol->start;
        bond *Binder = mol->first;
        int i;
        Vector *center = mol->DetermineCenterOfAll(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;
                while (Walker->next != mol->end) {
                        Walker = Walker->next;
                        *vrmlfile << "Sphere {" << endl << "    "; // 2 is sphere type
                        for (i=0;i<NDIM;i++)
                                *vrmlfile << Walker->x.x[i]+center->x[i] << " ";
                        *vrmlfile << "\t0.1\t1. 1. 1." << endl; // radius 0.05 and white as colour
                }

                *vrmlfile << "# All bonds as vertices" << endl;
                while (Binder->next != mol->last) {
                        Binder = Binder->next;
                        *vrmlfile << "3" << endl << "   "; // 2 is round-ended cylinder type
                        for (i=0;i<NDIM;i++)
                                *vrmlfile << Binder->leftatom->x.x[i]+center->x[i] << " ";
                        *vrmlfile << "\t0.03\t";
                        for (i=0;i<NDIM;i++)
                                *vrmlfile << Binder->rightatom->x.x[i]+center->x[i] << " ";
                        *vrmlfile << "\t0.03\t0. 0. 1." << endl; // radius 0.05 and blue as colour
                }

                *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->x.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);
};

/** 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 write_raster3d_file(ofstream *out, ofstream *rasterfile, class Tesselation *Tess, class molecule *mol)
{
        atom *Walker = mol->start;
        bond *Binder = mol->first;
        int i;
        Vector *center = mol->DetermineCenterOfAll(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;
                while (Walker->next != mol->end) {
                        Walker = Walker->next;
                        *rasterfile << "2" << endl << " ";      // 2 is sphere type
                        for (i=0;i<NDIM;i++)
                                *rasterfile << Walker->x.x[i]+center->x[i] << " ";
                        *rasterfile << "\t0.1\t1. 1. 1." << endl; // radius 0.05 and white as colour
                }

                *rasterfile << "# All bonds as vertices" << endl;
                while (Binder->next != mol->last) {
                        Binder = Binder->next;
                        *rasterfile << "3" << endl << " ";      // 2 is round-ended cylinder type
                        for (i=0;i<NDIM;i++)
                                *rasterfile << Binder->leftatom->x.x[i]+center->x[i] << " ";
                        *rasterfile << "\t0.03\t";
                        for (i=0;i<NDIM;i++)
                                *rasterfile << Binder->rightatom->x.x[i]+center->x[i] << " ";
                        *rasterfile << "\t0.03\t0. 0. 1." << endl; // radius 0.05 and blue as colour
                }

                *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->x.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;
        }
        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
write_tecplot_file(ofstream *out, ofstream *tecplot,
                class Tesselation *TesselStruct, class molecule *mol, int N)
{
        if (tecplot != NULL)
                {
                        *tecplot << "TITLE = \"3D CONVEX SHELL\"" << endl;
                        *tecplot << "VARIABLES = \"X\" \"Y\" \"Z\"" << endl;
                        *tecplot << "ZONE T=\"TRIANGLES" << N << "\", N="
                                        << TesselStruct->PointsOnBoundaryCount << ", E="
                                        << TesselStruct->TrianglesOnBoundaryCount
                                        << ", DATAPACKING=POINT, ZONETYPE=FETRIANGLE" << endl;
                        int *LookupList = new int[mol->AtomCount];
                        for (int i = 0; i < mol->AtomCount; i++)
                                LookupList[i] = -1;

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

/** Determines the volume of a cluster.
 * Determines first the convex envelope, then tesselates it and calculates its volume.
 * \param *out output stream for debugging
 * \param *filename filename prefix for output of vertex data
 * \param *configuration needed for path to store convex envelope file
 * \param *BoundaryPoints NDIM set of boundary points on the projected plane per axis, on return if desired
 * \param *mol molecule structure representing the cluster
 * \return determined volume of the cluster in cubed config:GetIsAngstroem()
 */
double
VolumeOfConvexEnvelope(ofstream *out, const char *filename, config *configuration,
                Boundaries *BoundaryPtr, molecule *mol)
{
        bool IsAngstroem = configuration->GetIsAngstroem();
        atom *Walker = NULL;
        struct Tesselation *TesselStruct = new Tesselation;
        bool BoundaryFreeFlag = false;
        Boundaries *BoundaryPoints = BoundaryPtr;
        double volume = 0.;
        double PyramidVolume = 0.;
        double G, h;
        Vector x, y;
        double a, b, c;

        //Find_non_convex_border(out, tecplot, *TesselStruct, mol); // Is now called from command line.

        // 1. calculate center of gravity
        *out << endl;
        Vector *CenterOfGravity = mol->DetermineCenterOfGravity(out);

        // 2. translate all points into CoG
        *out << Verbose(1) << "Translating system to Center of Gravity." << endl;
        Walker = mol->start;
        while (Walker->next != mol->end)
                {
                        Walker = Walker->next;
                        Walker->x.Translate(CenterOfGravity);
                }

        // 3. Find all points on the boundary
        if (BoundaryPoints == NULL)
                {
                        BoundaryFreeFlag = true;
                        BoundaryPoints = GetBoundaryPoints(out, mol);
                }
        else
                {
                        *out << Verbose(1) << "Using given boundary points set." << endl;
                }

        // 4. fill the boundary point list
        for (int axis = 0; axis < NDIM; axis++)
                for (Boundaries::iterator runner = BoundaryPoints[axis].begin(); runner
                                != BoundaryPoints[axis].end(); runner++)
                        {
                                TesselStruct->AddPoint(runner->second.second);
                        }

        *out << Verbose(2) << "I found " << TesselStruct->PointsOnBoundaryCount
                        << " points on the convex boundary." << endl;
        // now we have the whole set of edge points in the BoundaryList

        // listing for debugging
        //      *out << Verbose(1) << "Listing PointsOnBoundary:";
        //      for(PointMap::iterator runner = PointsOnBoundary.begin(); runner != PointsOnBoundary.end(); runner++) {
        //              *out << " " << *runner->second;
        //      }
        //      *out << endl;

        // 5a. guess starting triangle
        TesselStruct->GuessStartingTriangle(out);

        // 5b. go through all lines, that are not yet part of two triangles (only of one so far)
        TesselStruct->TesselateOnBoundary(out, configuration, mol);

        *out << Verbose(2) << "I created " << TesselStruct->TrianglesOnBoundaryCount
                        << " triangles with " << TesselStruct->LinesOnBoundaryCount
                        << " lines and " << TesselStruct->PointsOnBoundaryCount << " points."
                        << endl;

        // 6a. Every triangle forms a pyramid with the center of gravity as its peak, sum up the volumes
        *out << Verbose(1)
                        << "Calculating the volume of the pyramids formed out of triangles and center of gravity."
                        << endl;
        for (TriangleMap::iterator runner = TesselStruct->TrianglesOnBoundary.begin(); runner
                        != TesselStruct->TrianglesOnBoundary.end(); runner++)
                { // go through every triangle, calculate volume of its pyramid with CoG as peak
                        x.CopyVector(&runner->second->endpoints[0]->node->x);
                        x.SubtractVector(&runner->second->endpoints[1]->node->x);
                        y.CopyVector(&runner->second->endpoints[0]->node->x);
                        y.SubtractVector(&runner->second->endpoints[2]->node->x);
                        a = sqrt(runner->second->endpoints[0]->node->x.DistanceSquared(
                                        &runner->second->endpoints[1]->node->x));
                        b = sqrt(runner->second->endpoints[0]->node->x.DistanceSquared(
                                        &runner->second->endpoints[2]->node->x));
                        c = sqrt(runner->second->endpoints[2]->node->x.DistanceSquared(
                                        &runner->second->endpoints[1]->node->x));
                        G = sqrt(((a + b + c) * (a + b + c) - 2 * (a * a + b * b + c * c)) / 16.); // area of tesselated triangle
                        x.MakeNormalVector(&runner->second->endpoints[0]->node->x,
                                        &runner->second->endpoints[1]->node->x,
                                        &runner->second->endpoints[2]->node->x);
                        x.Scale(runner->second->endpoints[1]->node->x.Projection(&x));
                        h = x.Norm(); // distance of CoG to triangle
                        PyramidVolume = (1. / 3.) * G * h; // this formula holds for _all_ pyramids (independent of n-edge base or (not) centered peak)
                        *out << Verbose(2) << "Area of triangle is " << G << " "
                                        << (IsAngstroem ? "angstrom" : "atomiclength") << "^2, height is "
                                        << h << " and the volume is " << PyramidVolume << " "
                                        << (IsAngstroem ? "angstrom" : "atomiclength") << "^3." << endl;
                        volume += PyramidVolume;
                }
        *out << Verbose(0) << "RESULT: The summed volume is " << setprecision(10)
                        << volume << " " << (IsAngstroem ? "angstrom" : "atomiclength") << "^3."
                        << endl;

        // 7. translate all points back from CoG
        *out << Verbose(1) << "Translating system back from Center of Gravity."
                        << endl;
        CenterOfGravity->Scale(-1);
        Walker = mol->start;
        while (Walker->next != mol->end)
                {
                        Walker = Walker->next;
                        Walker->x.Translate(CenterOfGravity);
                }

        // 8. Store triangles in tecplot file
        string OutputName(filename);
        OutputName.append(TecplotSuffix);
        ofstream *tecplot = new ofstream(OutputName.c_str());
        write_tecplot_file(out, tecplot, TesselStruct, mol, 0);
        tecplot->close();
        delete(tecplot);

        // free reference lists
        if (BoundaryFreeFlag)
                delete[] (BoundaryPoints);

        return volume;
}
;

/** Creates multiples of the by \a *mol given cluster and suspends them in water with a given final density.
 * We get cluster volume by VolumeOfConvexEnvelope() and its diameters by GetDiametersOfCluster()
 * \param *out output stream for debugging
 * \param *configuration needed for path to store convex envelope file
 * \param *mol molecule structure representing the cluster
 * \param ClusterVolume guesstimated cluster volume, if equal 0 we used VolumeOfConvexEnvelope() instead.
 * \param celldensity desired average density in final cell
 */
void
PrepareClustersinWater(ofstream *out, config *configuration, molecule *mol,
                double ClusterVolume, double celldensity)
{
        // transform to PAS
        mol->PrincipalAxisSystem(out, true);

        // some preparations beforehand
        bool IsAngstroem = configuration->GetIsAngstroem();
        Boundaries *BoundaryPoints = GetBoundaryPoints(out, mol);
        double clustervolume;
        if (ClusterVolume == 0)
                clustervolume = VolumeOfConvexEnvelope(out, NULL, configuration,
                                BoundaryPoints, mol);
        else
                clustervolume = ClusterVolume;
        double *GreatestDiameter = GetDiametersOfCluster(out, BoundaryPoints, mol,
                        IsAngstroem);
        Vector BoxLengths;
        int repetition[NDIM] =
                { 1, 1, 1 };
        int TotalNoClusters = 1;
        for (int i = 0; i < NDIM; i++)
                TotalNoClusters *= repetition[i];

        // sum up the atomic masses
        double totalmass = 0.;
        atom *Walker = mol->start;
        while (Walker->next != mol->end)
                {
                        Walker = Walker->next;
                        totalmass += Walker->type->mass;
                }
        *out << Verbose(0) << "RESULT: The summed mass is " << setprecision(10)
                        << totalmass << " atomicmassunit." << endl;

        *out << Verbose(0) << "RESULT: The average density is " << setprecision(10)
                        << totalmass / clustervolume << " atomicmassunit/"
                        << (IsAngstroem ? "angstrom" : "atomiclength") << "^3." << endl;

        // solve cubic polynomial
        *out << Verbose(1) << "Solving equidistant suspension in water problem ..."
                        << endl;
        double cellvolume;
        if (IsAngstroem)
                cellvolume = (TotalNoClusters * totalmass / SOLVENTDENSITY_A - (totalmass
                                / clustervolume)) / (celldensity - 1);
        else
                cellvolume = (TotalNoClusters * totalmass / SOLVENTDENSITY_a0 - (totalmass
                                / clustervolume)) / (celldensity - 1);
        *out << Verbose(1) << "Cellvolume needed for a density of " << celldensity
                        << " g/cm^3 is " << cellvolume << " " << (IsAngstroem ? "angstrom"
                        : "atomiclength") << "^3." << endl;

        double minimumvolume = TotalNoClusters * (GreatestDiameter[0]
                        * GreatestDiameter[1] * GreatestDiameter[2]);
        *out << Verbose(1)
                        << "Minimum volume of the convex envelope contained in a rectangular box is "
                        << minimumvolume << " atomicmassunit/" << (IsAngstroem ? "angstrom"
                        : "atomiclength") << "^3." << endl;
        if (minimumvolume > cellvolume)
                {
                        cerr << Verbose(0)
                                        << "ERROR: the containing box already has a greater volume than the envisaged cell volume!"
                                        << endl;
                        cout << Verbose(0)
                                        << "Setting Box dimensions to minimum possible, the greatest diameters."
                                        << endl;
                        for (int i = 0; i < NDIM; i++)
                                BoxLengths.x[i] = GreatestDiameter[i];
                        mol->CenterEdge(out, &BoxLengths);
                }
        else
                {
                        BoxLengths.x[0] = (repetition[0] * GreatestDiameter[0] + repetition[1]
                                        * GreatestDiameter[1] + repetition[2] * GreatestDiameter[2]);
                        BoxLengths.x[1] = (repetition[0] * repetition[1] * GreatestDiameter[0]
                                        * GreatestDiameter[1] + repetition[0] * repetition[2]
                                        * GreatestDiameter[0] * GreatestDiameter[2] + repetition[1]
                                        * repetition[2] * GreatestDiameter[1] * GreatestDiameter[2]);
                        BoxLengths.x[2] = minimumvolume - cellvolume;
                        double x0 = 0., x1 = 0., x2 = 0.;
                        if (gsl_poly_solve_cubic(BoxLengths.x[0], BoxLengths.x[1],
                                        BoxLengths.x[2], &x0, &x1, &x2) == 1) // either 1 or 3 on return
                                *out << Verbose(0) << "RESULT: The resulting spacing is: " << x0
                                                << " ." << endl;
                        else
                                {
                                        *out << Verbose(0) << "RESULT: The resulting spacings are: " << x0
                                                        << " and " << x1 << " and " << x2 << " ." << endl;
                                        x0 = x2; // sorted in ascending order
                                }

                        cellvolume = 1;
                        for (int i = 0; i < NDIM; i++)
                                {
                                        BoxLengths.x[i] = repetition[i] * (x0 + GreatestDiameter[i]);
                                        cellvolume *= BoxLengths.x[i];
                                }

                        // set new box dimensions
                        *out << Verbose(0) << "Translating to box with these boundaries." << endl;
                        mol->CenterInBox((ofstream *) &cout, &BoxLengths);
                }
        // update Box of atoms by boundary
        mol->SetBoxDimension(&BoxLengths);
        *out << Verbose(0) << "RESULT: The resulting cell dimensions are: "
                        << BoxLengths.x[0] << " and " << BoxLengths.x[1] << " and "
                        << BoxLengths.x[2] << " with total volume of " << cellvolume << " "
                        << (IsAngstroem ? "angstrom" : "atomiclength") << "^3." << endl;
}
;

// =========================================================== class TESSELATION ===========================================

/** Constructor of class Tesselation.
 */
Tesselation::Tesselation()
{
        PointsOnBoundaryCount = 0;
        LinesOnBoundaryCount = 0;
        TrianglesOnBoundaryCount = 0;
        TriangleFilesWritten = 0;
}
;

/** Constructor of class Tesselation.
 * We have to free all points, lines and triangles.
 */
Tesselation::~Tesselation()
{
        cout << Verbose(1) << "Free'ing TesselStruct ... " << endl;
        for (TriangleMap::iterator runner = TrianglesOnBoundary.begin(); runner != TrianglesOnBoundary.end(); runner++) {
                if (runner->second != NULL) {
                        delete (runner->second);
                        runner->second = NULL;
                } else
                        cerr << "ERROR: The triangle " << runner->first << " has already been free'd." << endl;
        }
}
;

/** Gueses first starting triangle of the convex envelope.
 * We guess the starting triangle by taking the smallest distance between two points and looking for a fitting third.
 * \param *out output stream for debugging
 * \param PointsOnBoundary set of boundary points defining the convex envelope of the cluster
 */
void
Tesselation::GuessStartingTriangle(ofstream *out)
{
        // 4b. create a starting triangle
        // 4b1. create all distances
        DistanceMultiMap DistanceMMap;
        double distance, tmp;
        Vector PlaneVector, TrialVector;
        PointMap::iterator A, B, C; // three nodes of the first triangle
        A = PointsOnBoundary.begin(); // the first may be chosen arbitrarily

        // with A chosen, take each pair B,C and sort
        if (A != PointsOnBoundary.end())
                {
                        B = A;
                        B++;
                        for (; B != PointsOnBoundary.end(); B++)
                                {
                                        C = B;
                                        C++;
                                        for (; C != PointsOnBoundary.end(); C++)
                                                {
                                                        tmp = A->second->node->x.DistanceSquared(&B->second->node->x);
                                                        distance = tmp * tmp;
                                                        tmp = A->second->node->x.DistanceSquared(&C->second->node->x);
                                                        distance += tmp * tmp;
                                                        tmp = B->second->node->x.DistanceSquared(&C->second->node->x);
                                                        distance += tmp * tmp;
                                                        DistanceMMap.insert(DistanceMultiMapPair(distance, pair<
                                                                        PointMap::iterator, PointMap::iterator> (B, C)));
                                                }
                                }
                }
        //              // listing distances
        //              *out << Verbose(1) << "Listing DistanceMMap:";
        //              for(DistanceMultiMap::iterator runner = DistanceMMap.begin(); runner != DistanceMMap.end(); runner++) {
        //                      *out << " " << runner->first << "(" << *runner->second.first->second << ", " << *runner->second.second->second << ")";
        //              }
        //              *out << endl;
        // 4b2. pick three baselines forming a triangle
        // 1. we take from the smallest sum of squared distance as the base line BC (with peak A) onward as the triangle candidate
        DistanceMultiMap::iterator baseline = DistanceMMap.begin();
        for (; baseline != DistanceMMap.end(); baseline++)
                {
                        // we take from the smallest sum of squared distance as the base line BC (with peak A) onward as the triangle candidate
                        // 2. next, we have to check whether all points reside on only one side of the triangle
                        // 3. construct plane vector
                        PlaneVector.MakeNormalVector(&A->second->node->x,
                                        &baseline->second.first->second->node->x,
                                        &baseline->second.second->second->node->x);
                        *out << Verbose(2) << "Plane vector of candidate triangle is ";
                        PlaneVector.Output(out);
                        *out << endl;
                        // 4. loop over all points
                        double sign = 0.;
                        PointMap::iterator checker = PointsOnBoundary.begin();
                        for (; checker != PointsOnBoundary.end(); checker++)
                                {
                                        // (neglecting A,B,C)
                                        if ((checker == A) || (checker == baseline->second.first) || (checker
                                                        == baseline->second.second))
                                                continue;
                                        // 4a. project onto plane vector
                                        TrialVector.CopyVector(&checker->second->node->x);
                                        TrialVector.SubtractVector(&A->second->node->x);
                                        distance = TrialVector.Projection(&PlaneVector);
                                        if (fabs(distance) < 1e-4) // we need to have a small epsilon around 0 which is still ok
                                                continue;
                                        *out << Verbose(3) << "Projection of " << checker->second->node->Name
                                                        << " yields distance of " << distance << "." << endl;
                                        tmp = distance / fabs(distance);
                                        // 4b. Any have different sign to than before? (i.e. would lie outside convex hull with this starting triangle)
                                        if ((sign != 0) && (tmp != sign))
                                                {
                                                        // 4c. If so, break 4. loop and continue with next candidate in 1. loop
                                                        *out << Verbose(2) << "Current candidates: "
                                                                        << A->second->node->Name << ","
                                                                        << baseline->second.first->second->node->Name << ","
                                                                        << baseline->second.second->second->node->Name << " leave "
                                                                        << checker->second->node->Name << " outside the convex hull."
                                                                        << endl;
                                                        break;
                                                }
                                        else
                                                { // note the sign for later
                                                        *out << Verbose(2) << "Current candidates: "
                                                                        << A->second->node->Name << ","
                                                                        << baseline->second.first->second->node->Name << ","
                                                                        << baseline->second.second->second->node->Name << " leave "
                                                                        << checker->second->node->Name << " inside the convex hull."
                                                                        << endl;
                                                        sign = tmp;
                                                }
                                        // 4d. Check whether the point is inside the triangle (check distance to each node
                                        tmp = checker->second->node->x.DistanceSquared(&A->second->node->x);
                                        int innerpoint = 0;
                                        if ((tmp < A->second->node->x.DistanceSquared(
                                                        &baseline->second.first->second->node->x)) && (tmp
                                                        < A->second->node->x.DistanceSquared(
                                                                        &baseline->second.second->second->node->x)))
                                                innerpoint++;
                                        tmp = checker->second->node->x.DistanceSquared(
                                                        &baseline->second.first->second->node->x);
                                        if ((tmp < baseline->second.first->second->node->x.DistanceSquared(
                                                        &A->second->node->x)) && (tmp
                                                        < baseline->second.first->second->node->x.DistanceSquared(
                                                                        &baseline->second.second->second->node->x)))
                                                innerpoint++;
                                        tmp = checker->second->node->x.DistanceSquared(
                                                        &baseline->second.second->second->node->x);
                                        if ((tmp < baseline->second.second->second->node->x.DistanceSquared(
                                                        &baseline->second.first->second->node->x)) && (tmp
                                                        < baseline->second.second->second->node->x.DistanceSquared(
                                                                        &A->second->node->x)))
                                                innerpoint++;
                                        // 4e. If so, break 4. loop and continue with next candidate in 1. loop
                                        if (innerpoint == 3)
                                                break;
                                }
                        // 5. come this far, all on same side? Then break 1. loop and construct triangle
                        if (checker == PointsOnBoundary.end())
                                {
                                        *out << "Looks like we have a candidate!" << endl;
                                        break;
                                }
                }
        if (baseline != DistanceMMap.end())
                {
                        BPS[0] = baseline->second.first->second;
                        BPS[1] = baseline->second.second->second;
                        BLS[0] = new class BoundaryLineSet(BPS, LinesOnBoundaryCount);
                        BPS[0] = A->second;
                        BPS[1] = baseline->second.second->second;
                        BLS[1] = new class BoundaryLineSet(BPS, LinesOnBoundaryCount);
                        BPS[0] = baseline->second.first->second;
                        BPS[1] = A->second;
                        BLS[2] = new class BoundaryLineSet(BPS, LinesOnBoundaryCount);

                        // 4b3. insert created triangle
                        BTS = new class BoundaryTriangleSet(BLS, TrianglesOnBoundaryCount);
                        TrianglesOnBoundary.insert(TrianglePair(TrianglesOnBoundaryCount, BTS));
                        TrianglesOnBoundaryCount++;
                        for (int i = 0; i < NDIM; i++)
                                {
                                        LinesOnBoundary.insert(LinePair(LinesOnBoundaryCount, BTS->lines[i]));
                                        LinesOnBoundaryCount++;
                                }

                        *out << Verbose(1) << "Starting triangle is " << *BTS << "." << endl;
                }
        else
                {
                        *out << Verbose(1) << "No starting triangle found." << endl;
                        exit(255);
                }
}
;

/** Tesselates the convex envelope of a cluster from a single starting triangle.
 * The starting triangle is made out of three baselines. Each line in the final tesselated cluster may belong to at most
 * 2 triangles. Hence, we go through all current lines:
 * -# if the lines contains to only one triangle
 * -# We search all points in the boundary
 *              -# if the triangle with the baseline and the current point has the smallest of angles (comparison between normal vectors
 *              -# if the triangle is in forward direction of the baseline (at most 90 degrees angle between vector orthogonal to
 *                       baseline in triangle plane pointing out of the triangle and normal vector of new triangle)
 *              -# then we have a new triangle, whose baselines we again add (or increase their TriangleCount)
 * \param *out output stream for debugging
 * \param *configuration for IsAngstroem
 * \param *mol the cluster as a molecule structure
 */
void
Tesselation::TesselateOnBoundary(ofstream *out, config *configuration,
                molecule *mol)
{
        bool flag;
        PointMap::iterator winner;
        class BoundaryPointSet *peak = NULL;
        double SmallestAngle, TempAngle;
        Vector NormalVector, VirtualNormalVector, CenterVector, TempVector,
                        PropagationVector;
        LineMap::iterator LineChecker[2];
        do
                {
                        flag = false;
                        for (LineMap::iterator baseline = LinesOnBoundary.begin(); baseline
                                        != LinesOnBoundary.end(); baseline++)
                                if (baseline->second->TrianglesCount == 1)
                                        {
                                                *out << Verbose(2) << "Current baseline is between "
                                                                << *(baseline->second) << "." << endl;
                                                // 5a. go through each boundary point if not _both_ edges between either endpoint of the current line and this point exist (and belong to 2 triangles)
                                                SmallestAngle = M_PI;
                                                BTS = baseline->second->triangles.begin()->second; // there is only one triangle so far
                                                // get peak point with respect to this base line's only triangle
                                                for (int i = 0; i < 3; i++)
                                                        if ((BTS->endpoints[i] != baseline->second->endpoints[0])
                                                                        && (BTS->endpoints[i] != baseline->second->endpoints[1]))
                                                                peak = BTS->endpoints[i];
                                                *out << Verbose(3) << " and has peak " << *peak << "." << endl;
                                                // normal vector of triangle
                                                BTS->GetNormalVector(NormalVector);
                                                *out << Verbose(4) << "NormalVector of base triangle is ";
                                                NormalVector.Output(out);
                                                *out << endl;
                                                // offset to center of triangle
                                                CenterVector.Zero();
                                                for (int i = 0; i < 3; i++)
                                                        CenterVector.AddVector(&BTS->endpoints[i]->node->x);
                                                CenterVector.Scale(1. / 3.);
                                                *out << Verbose(4) << "CenterVector of base triangle is ";
                                                CenterVector.Output(out);
                                                *out << endl;
                                                // vector in propagation direction (out of triangle)
                                                // project center vector onto triangle plane (points from intersection plane-NormalVector to plane-CenterVector intersection)
                                                TempVector.CopyVector(&baseline->second->endpoints[0]->node->x);
                                                TempVector.SubtractVector(&baseline->second->endpoints[1]->node->x);
                                                PropagationVector.MakeNormalVector(&TempVector, &NormalVector);
                                                TempVector.CopyVector(&CenterVector);
                                                TempVector.SubtractVector(&baseline->second->endpoints[0]->node->x); // TempVector is vector on triangle plane pointing from one baseline egde towards center!
                                                //*out << Verbose(2) << "Projection of propagation onto temp: " << PropagationVector.Projection(&TempVector) << "." << endl;
                                                if (PropagationVector.Projection(&TempVector) > 0) // make sure normal propagation vector points outward from baseline
                                                        PropagationVector.Scale(-1.);
                                                *out << Verbose(4) << "PropagationVector of base triangle is ";
                                                PropagationVector.Output(out);
                                                *out << endl;
                                                winner = PointsOnBoundary.end();
                                                for (PointMap::iterator target = PointsOnBoundary.begin(); target
                                                                != PointsOnBoundary.end(); target++)
                                                        if ((target->second != baseline->second->endpoints[0])
                                                                        && (target->second != baseline->second->endpoints[1]))
                                                                { // don't take the same endpoints
                                                                        *out << Verbose(3) << "Target point is " << *(target->second)
                                                                                        << ":";
                                                                        bool continueflag = true;

                                                                        VirtualNormalVector.CopyVector(
                                                                                        &baseline->second->endpoints[0]->node->x);
                                                                        VirtualNormalVector.AddVector(
                                                                                        &baseline->second->endpoints[0]->node->x);
                                                                        VirtualNormalVector.Scale(-1. / 2.); // points now to center of base line
                                                                        VirtualNormalVector.AddVector(&target->second->node->x); // points from center of base line to target
                                                                        TempAngle = VirtualNormalVector.Angle(&PropagationVector);
                                                                        continueflag = continueflag && (TempAngle < (M_PI/2.)); // no bends bigger than Pi/2 (90 degrees)
                                                                        if (!continueflag)
                                                                                {
                                                                                        *out << Verbose(4)
                                                                                                        << "Angle between propagation direction and base line to "
                                                                                                        << *(target->second) << " is " << TempAngle
                                                                                                        << ", bad direction!" << endl;
                                                                                        continue;
                                                                                }
                                                                        else
                                                                                *out << Verbose(4)
                                                                                                << "Angle between propagation direction and base line to "
                                                                                                << *(target->second) << " is " << TempAngle
                                                                                                << ", good direction!" << endl;
                                                                        LineChecker[0] = baseline->second->endpoints[0]->lines.find(
                                                                                        target->first);
                                                                        LineChecker[1] = baseline->second->endpoints[1]->lines.find(
                                                                                        target->first);
                                                                        //                                              if (LineChecker[0] != baseline->second->endpoints[0]->lines.end())
                                                                        //                                                      *out << Verbose(4) << *(baseline->second->endpoints[0]) << " has line " << *(LineChecker[0]->second) << " to " << *(target->second) << " as endpoint with " << LineChecker[0]->second->TrianglesCount << " triangles." << endl;
                                                                        //                                              else
                                                                        //                                                      *out << Verbose(4) << *(baseline->second->endpoints[0]) << " has no line to " << *(target->second) << " as endpoint." << endl;
                                                                        //                                              if (LineChecker[1] != baseline->second->endpoints[1]->lines.end())
                                                                        //                                                      *out << Verbose(4) << *(baseline->second->endpoints[1]) << " has line " << *(LineChecker[1]->second) << " to " << *(target->second) << " as endpoint with " << LineChecker[1]->second->TrianglesCount << " triangles." << endl;
                                                                        //                                              else
                                                                        //                                                      *out << Verbose(4) << *(baseline->second->endpoints[1]) << " has no line to " << *(target->second) << " as endpoint." << endl;
                                                                        // check first endpoint (if any connecting line goes to target or at least not more than 1)
                                                                        continueflag = continueflag && (((LineChecker[0]
                                                                                        == baseline->second->endpoints[0]->lines.end())
                                                                                        || (LineChecker[0]->second->TrianglesCount == 1)));
                                                                        if (!continueflag)
                                                                                {
                                                                                        *out << Verbose(4) << *(baseline->second->endpoints[0])
                                                                                                        << " has line " << *(LineChecker[0]->second)
                                                                                                        << " to " << *(target->second)
                                                                                                        << " as endpoint with "
                                                                                                        << LineChecker[0]->second->TrianglesCount
                                                                                                        << " triangles." << endl;
                                                                                        continue;
                                                                                }
                                                                        // check second endpoint (if any connecting line goes to target or at least not more than 1)
                                                                        continueflag = continueflag && (((LineChecker[1]
                                                                                        == baseline->second->endpoints[1]->lines.end())
                                                                                        || (LineChecker[1]->second->TrianglesCount == 1)));
                                                                        if (!continueflag)
                                                                                {
                                                                                        *out << Verbose(4) << *(baseline->second->endpoints[1])
                                                                                                        << " has line " << *(LineChecker[1]->second)
                                                                                                        << " to " << *(target->second)
                                                                                                        << " as endpoint with "
                                                                                                        << LineChecker[1]->second->TrianglesCount
                                                                                                        << " triangles." << endl;
                                                                                        continue;
                                                                                }
                                                                        // check whether the envisaged triangle does not already exist (if both lines exist and have same endpoint)
                                                                        continueflag = continueflag && (!(((LineChecker[0]
                                                                                        != baseline->second->endpoints[0]->lines.end())
                                                                                        && (LineChecker[1]
                                                                                                        != baseline->second->endpoints[1]->lines.end())
                                                                                        && (GetCommonEndpoint(LineChecker[0]->second,
                                                                                                        LineChecker[1]->second) == peak))));
                                                                        if (!continueflag)
                                                                                {
                                                                                        *out << Verbose(4) << "Current target is peak!" << endl;
                                                                                        continue;
                                                                                }
                                                                        // in case NOT both were found
                                                                        if (continueflag)
                                                                                { // create virtually this triangle, get its normal vector, calculate angle
                                                                                        flag = true;
                                                                                        VirtualNormalVector.MakeNormalVector(
                                                                                                        &baseline->second->endpoints[0]->node->x,
                                                                                                        &baseline->second->endpoints[1]->node->x,
                                                                                                        &target->second->node->x);
                                                                                        // make it always point inward
                                                                                        if (baseline->second->endpoints[0]->node->x.Projection(
                                                                                                        &VirtualNormalVector) > 0)
                                                                                                VirtualNormalVector.Scale(-1.);
                                                                                        // calculate angle
                                                                                        TempAngle = NormalVector.Angle(&VirtualNormalVector);
                                                                                        *out << Verbose(4) << "NormalVector is ";
                                                                                        VirtualNormalVector.Output(out);
                                                                                        *out << " and the angle is " << TempAngle << "." << endl;
                                                                                        if (SmallestAngle > TempAngle)
                                                                                                { // set to new possible winner
                                                                                                        SmallestAngle = TempAngle;
                                                                                                        winner = target;
                                                                                                }
                                                                                }
                                                                }
                                                // 5b. The point of the above whose triangle has the greatest angle with the triangle the current line belongs to (it only belongs to one, remember!): New triangle
                                                if (winner != PointsOnBoundary.end())
                                                        {
                                                                *out << Verbose(2) << "Winning target point is "
                                                                                << *(winner->second) << " with angle " << SmallestAngle
                                                                                << "." << endl;
                                                                // create the lins of not yet present
                                                                BLS[0] = baseline->second;
                                                                // 5c. add lines to the line set if those were new (not yet part of a triangle), delete lines that belong to two triangles)
                                                                LineChecker[0] = baseline->second->endpoints[0]->lines.find(
                                                                                winner->first);
                                                                LineChecker[1] = baseline->second->endpoints[1]->lines.find(
                                                                                winner->first);
                                                                if (LineChecker[0]
                                                                                == baseline->second->endpoints[0]->lines.end())
                                                                        { // create
                                                                                BPS[0] = baseline->second->endpoints[0];
                                                                                BPS[1] = winner->second;
                                                                                BLS[1] = new class BoundaryLineSet(BPS,
                                                                                                LinesOnBoundaryCount);
                                                                                LinesOnBoundary.insert(LinePair(LinesOnBoundaryCount,
                                                                                                BLS[1]));
                                                                                LinesOnBoundaryCount++;
                                                                        }
                                                                else
                                                                        BLS[1] = LineChecker[0]->second;
                                                                if (LineChecker[1]
                                                                                == baseline->second->endpoints[1]->lines.end())
                                                                        { // create
                                                                                BPS[0] = baseline->second->endpoints[1];
                                                                                BPS[1] = winner->second;
                                                                                BLS[2] = new class BoundaryLineSet(BPS,
                                                                                                LinesOnBoundaryCount);
                                                                                LinesOnBoundary.insert(LinePair(LinesOnBoundaryCount,
                                                                                                BLS[2]));
                                                                                LinesOnBoundaryCount++;
                                                                        }
                                                                else
                                                                        BLS[2] = LineChecker[1]->second;
                                                                BTS = new class BoundaryTriangleSet(BLS,
                                                                                TrianglesOnBoundaryCount);
                                                                TrianglesOnBoundary.insert(TrianglePair(
                                                                                TrianglesOnBoundaryCount, BTS));
                                                                TrianglesOnBoundaryCount++;
                                                        }
                                                else
                                                        {
                                                                *out << Verbose(1)
                                                                                << "I could not determine a winner for this baseline "
                                                                                << *(baseline->second) << "." << endl;
                                                        }

                                                // 5d. If the set of lines is not yet empty, go to 5. and continue
                                        }
                                else
                                        *out << Verbose(2) << "Baseline candidate " << *(baseline->second)
                                                        << " has a triangle count of "
                                                        << baseline->second->TrianglesCount << "." << endl;
                }
        while (flag);

}
;

/** Adds an atom to the tesselation::PointsOnBoundary list.
 * \param *Walker atom to add
 */
void
Tesselation::AddPoint(atom *Walker)
{
        PointTestPair InsertUnique;
        BPS[0] = new class BoundaryPointSet(Walker);
        InsertUnique = PointsOnBoundary.insert(PointPair(Walker->nr, BPS[0]));
        if (InsertUnique.second) // if new point was not present before, increase counter
                PointsOnBoundaryCount++;
}
;

/** Adds point to Tesselation::PointsOnBoundary if not yet present.
 * Tesselation::TPS is set to either this new BoundaryPointSet or to the existing one of not unique.
 * @param Candidate point to add
 * @param n index for this point in Tesselation::TPS array
 */
void
Tesselation::AddTrianglePoint(atom* Candidate, int n)
{
        PointTestPair InsertUnique;
        TPS[n] = new class BoundaryPointSet(Candidate);
        InsertUnique = PointsOnBoundary.insert(PointPair(Candidate->nr, TPS[n]));
        if (InsertUnique.second) // if new point was not present before, increase counter
                {
                        PointsOnBoundaryCount++;
                }
        else
                {
                        delete TPS[n];
                        cout << Verbose(2) << "Atom " << *((InsertUnique.first)->second->node)
                                        << " gibt's schon in der PointMap." << endl;
                        TPS[n] = (InsertUnique.first)->second;
                }
}
;

/** Function tries to add line from current Points in BPS to BoundaryLineSet.
 * If succesful it raises the line count and inserts the new line into the BLS,
 * if unsuccesful, it writes the line which had been present into the BLS, deleting the new constructed one.
 * @param *a first endpoint
 * @param *b second endpoint
 * @param n index of Tesselation::BLS giving the line with both endpoints
 */
void
Tesselation::AddTriangleLine(class BoundaryPointSet *a,
                class BoundaryPointSet *b, int n)
{
        LineMap::iterator LineWalker;
        //cout << "Manually checking endpoints for line." << endl;
        if ((a->lines.find(b->node->nr)) != a->lines.end()) // ->first == b->node->nr)
        //If a line is there, how do I recognize that beyond a shadow of a doubt?
                {
                        //cout << Verbose(2) << "Line exists already, retrieving it from LinesOnBoundarySet" << endl;

                        LineWalker = LinesOnBoundary.end();
                        LineWalker--;

                        while (LineWalker->second->endpoints[0]->node->nr != min(a->node->nr,
                                        b->node->nr) or LineWalker->second->endpoints[1]->node->nr != max(
                                        a->node->nr, b->node->nr))
                                {
                                        //cout << Verbose(1) << "Looking for line which already exists"<< endl;
                                        LineWalker--;
                                }
                        BPS[0] = LineWalker->second->endpoints[0];
                        BPS[1] = LineWalker->second->endpoints[1];
                        BLS[n] = LineWalker->second;

                }
        else
                {
                        cout << Verbose(2)
                                        << "Adding line which has not been used before between "
                                        << *(a->node) << " and " << *(b->node) << "." << endl;
                        BPS[0] = a;
                        BPS[1] = b;
                        BLS[n] = new class BoundaryLineSet(BPS, LinesOnBoundaryCount);

                        LinesOnBoundary.insert(LinePair(LinesOnBoundaryCount, BLS[n]));
                        LinesOnBoundaryCount++;

                }
}
;

/** Function tries to add Triangle just created to Triangle and remarks if already existent (Failure of algorithm).
 * Furthermore it adds the triangle to all of its lines, in order to recognize those which are saturated later.
 */
void
Tesselation::AddTriangleToLines()
{

        cout << Verbose(1) << "Adding triangle to its lines" << endl;
        int i = 0;
        TrianglesOnBoundary.insert(TrianglePair(TrianglesOnBoundaryCount, BTS));
        TrianglesOnBoundaryCount++;

        /*
         * this is apparently done when constructing triangle

         for (i=0; i<3; i++)
         {
         BLS[i]->AddTriangle(BTS);
         }
         */
}
;


double det_get(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 get_sphere(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 = det_get(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 = det_get(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 = det_get(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 = det_get(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 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 Get_center_of_sphere(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 Get_center_of_sphere.\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";
        get_sphere(&OtherCenter, a, b, c, RADIUS);
        cout << Verbose(0) << "OtherCenter of sphere of circumference is " << OtherCenter << ".\n";
        cout << Verbose(3) << "End of Get_center_of_sphere.\n";
};

/** This recursive function finds a third point, to form a triangle with two given ones.
 * Two atoms are fixed, a candidate is supplied, additionally two vectors for direction distinction, a Storage area to \
 *      supply results to the calling function, the radius of the sphere which the triangle shall support and the molecule \
 *      upon which we operate.
 *      If the candidate is more fitting to support the sphere than the already stored atom is, then we write its general \
 *      direction and angle into Storage.
 *      We the determine the recursive level we have reached and if this is not on the threshold yet, call this function again, \
 *      with all neighbours of the candidate.
 * @param a first point
 * @param b second point
 * *param c atom old third point of old triangle
 * @param Candidate base point along whose bonds to start looking from
 * @param Parent point to avoid during search as its wrong direction
 * @param RecursionLevel contains current recursion depth
 * @param Chord baseline vector of first and second point
 * @param direction1 second in plane vector (along with \a Chord) of the triangle the baseline belongs to
 * @param OldNormal normal of the triangle which the baseline belongs to
 * @param ReferencePoint Vector of center of circumscribing circle from which we look towards center of sphere
 * @param Opt_Candidate candidate reference to return
 * @param Storage array containing two angles of current Opt_Candidate
 * @param RADIUS radius of ball
 * @param mol molecule structure with atoms and bonds
 */
void Tesselation::Find_next_suitable_point_via_Angle_of_Sphere(atom* a, atom* b, atom* c, atom* Candidate, atom* Parent,
                int RecursionLevel, Vector *Chord, Vector *direction1, Vector *OldNormal, Vector ReferencePoint,
                atom*& Opt_Candidate, double *Storage, const double RADIUS, molecule* mol)
{
        cout << Verbose(2) << "Begin of Find_next_suitable_point_via_Angle_of_Sphere, recursion level " << RecursionLevel << ".\n";
        cout << Verbose(3) << "Candidate is "<< *Candidate << endl;
        cout << Verbose(4) << "Baseline vector is " << *Chord << "." << endl;
        cout << Verbose(4) << "ReferencePoint is " << ReferencePoint << "." << endl;
        cout << Verbose(4) << "Normal of base triangle is " << *OldNormal << "." << endl;
        cout << Verbose(4) << "Search direction is " << *direction1 << "." << endl;
        /* OldNormal is normal vector on the old triangle
         * direction1 is normal on the triangle line, from which we come, as well as on OldNormal.
         * Chord points from b to a!!!
         */
        Vector dif_a; //Vector from a to candidate
        Vector dif_b; //Vector from b to candidate
        Vector AngleCheck;
        Vector TempNormal, Umkreismittelpunkt;
        Vector Mittelpunkt;

        double CurrentEpsilon = 0.1;
        double alpha, beta, gamma, SideA, SideB, SideC, sign, Umkreisradius, Restradius, Distance;
        double BallAngle, AlternativeSign;
        atom *Walker; // variable atom point

        Vector NewUmkreismittelpunkt;

        if (a != Candidate and b != Candidate and c != Candidate) {
                cout << Verbose(3) << "We have a unique candidate!" << endl;
                dif_a.CopyVector(&(a->x));
                dif_a.SubtractVector(&(Candidate->x));
                dif_b.CopyVector(&(b->x));
                dif_b.SubtractVector(&(Candidate->x));
                AngleCheck.CopyVector(&(Candidate->x));
                AngleCheck.SubtractVector(&(a->x));
                AngleCheck.ProjectOntoPlane(Chord);

                SideA = dif_b.Norm();
                SideB = dif_a.Norm();
                SideC = Chord->Norm();
                //Chord->Scale(-1);

                alpha = Chord->Angle(&dif_a);
                beta = M_PI - Chord->Angle(&dif_b);
                gamma = dif_a.Angle(&dif_b);

                cout << Verbose(2) << "Base triangle has sides " << dif_a << ", " << dif_b << ", " << *Chord << " with angles " << alpha/M_PI*180. << ", " << beta/M_PI*180. << ", " << gamma/M_PI*180. << "." << endl;

                if (fabs(M_PI - alpha - beta - gamma) > MYEPSILON) {
                        cerr << Verbose(0) << "WARNING: sum of angles for base triangle " << (alpha + beta + gamma)/M_PI*180. << " != 180.\n";
                        cout << Verbose(1) << "Base triangle has sides " << dif_a << ", " << dif_b << ", " << *Chord << " with angles " << alpha/M_PI*180. << ", " << beta/M_PI*180. << ", " << gamma/M_PI*180. << "." << endl;
                }

                if ((M_PI*179./180. > alpha) && (M_PI*179./180. > beta) && (M_PI*179./180. > gamma)) {
                        Umkreisradius = SideA / 2.0 / sin(alpha);
                        //cout << Umkreisradius << endl;
                        //cout << SideB / 2.0 / sin(beta) << endl;
                        //cout << SideC / 2.0 / sin(gamma) << endl;

                        if (Umkreisradius < RADIUS) { //Checking whether ball will at least rest on points.
                                cout << Verbose(3) << "Circle of circumference would fit: " << Umkreisradius << " < " << RADIUS << "." << endl;
                                cout << Verbose(2) << "Candidate is "<< *Candidate << endl;
                                sign = AngleCheck.ScalarProduct(direction1);
                                if (fabs(sign)<MYEPSILON) {
                                        if (AngleCheck.ScalarProduct(OldNormal)<0) {
                                                sign =0;
                                                AlternativeSign=1;
                                        } else {
                                                sign =0;
                                                AlternativeSign=-1;
                                        }
                                } else {
                                        sign /= fabs(sign);
                                }
                                if (sign >= 0) {
                                        cout << Verbose(3) << "Candidate is in search direction: " << sign << "." << endl;
                                        Get_center_of_sphere(&Mittelpunkt, (a->x), (b->x), (Candidate->x), &NewUmkreismittelpunkt, OldNormal, direction1, sign, AlternativeSign, alpha, beta, gamma, RADIUS, Umkreisradius);
                                        Mittelpunkt.SubtractVector(&ReferencePoint);
                                        cout << Verbose(3) << "Reference vector to sphere's center is " << Mittelpunkt << "." << endl;
                                        BallAngle = Mittelpunkt.Angle(OldNormal);
                                        cout << Verbose(3) << "Angle between normal of base triangle and center of ball sphere is :" << BallAngle << "." << endl;

                                        //cout << "direction1 is " << *direction1 << "." << endl;
                                        //cout << "Mittelpunkt is " << Mittelpunkt << "."<< endl;
                                        //cout << Verbose(3) << "BallAngle is " << BallAngle << " Sign is " << sign << endl;

                                        NewUmkreismittelpunkt.SubtractVector(&ReferencePoint);

                                        if ((Mittelpunkt.ScalarProduct(direction1) >=0) || (fabs(NewUmkreismittelpunkt.Norm()) < MYEPSILON)) {
                                                if (Storage[0]< -1.5) { // first Candidate at all
                                                        if (1) {//if (CheckPresenceOfTriangle((ofstream *)&cout,a,b,Candidate)) {
                                                                cout << Verbose(2) << "First good candidate is " << *Candidate << " with ";
                                                                Opt_Candidate = Candidate;
                                                                Storage[0] = sign;
                                                                Storage[1] = AlternativeSign;
                                                                Storage[2] = BallAngle;
                                                                cout << " angle " << Storage[2] << " and Up/Down " << Storage[0] << endl;
                                                        } else
                                                                cout << "Candidate " << *Candidate << " does not belong to a valid triangle." << endl;
                                                } else {
                                                        if ( Storage[2] > BallAngle) {
                                                                if (1) { //if (CheckPresenceOfTriangle((ofstream *)&cout,a,b,Candidate)) {
                                                                        cout << Verbose(2) << "Next better candidate is " << *Candidate << " with ";
                                                                        Opt_Candidate = Candidate;
                                                                        Storage[0] = sign;
                                                                        Storage[1] = AlternativeSign;
                                                                        Storage[2] = BallAngle;
                                                                        cout << " angle " << Storage[2] << " and Up/Down " << Storage[0] << endl;
                                                                } else
                                                                        cout << "Candidate " << *Candidate << " does not belong to a valid triangle." << endl;
                                                        } else {
                                                                if (DEBUG) {
                                                                        cout << Verbose(3) << *Candidate << " looses against better candidate " << *Opt_Candidate << "." << endl;
                                                                }
                                                        }
                                                }
                                        } else {
                                                if (DEBUG) {
                                                        cout << Verbose(3) << *Candidate << " refused due to Up/Down sign which is " << sign << endl;
                                                }
                                        }
                                } else {
                                        if (DEBUG) {
                                                cout << Verbose(3) << *Candidate << " is not in search direction." << endl;
                                        }
                                }
                        } else {
                                if (DEBUG) {
                                        cout << Verbose(3) << *Candidate << " would have circumference of " << Umkreisradius << " bigger than ball's radius " << RADIUS << "." << endl;
                                }
                        }
                } else {
                        if (DEBUG) {
                                cout << Verbose(0) << "Triangle consisting of " << *Candidate << ", " << *a << " and " << *b << " has an angle >150!" << endl;
                        }
                }
        } else {
                if (DEBUG) {
                        cout << Verbose(3) << *Candidate << " is either " << *a << " or " << *b << "." << endl;
                }
        }

        if (RecursionLevel < 5) { // Seven is the recursion level threshold.
                for (int i = 0; i < mol->NumberOfBondsPerAtom[Candidate->nr]; i++) { // go through all bond
                        Walker = mol->ListOfBondsPerAtom[Candidate->nr][i]->GetOtherAtom(Candidate);
                        if (Walker == Parent) { // don't go back the same bond
                                continue;
                        } else {
                                Find_next_suitable_point_via_Angle_of_Sphere(a, b, c, Walker, Candidate, RecursionLevel+1, Chord, direction1, OldNormal, ReferencePoint, Opt_Candidate, Storage, RADIUS, mol); //call function again
                        }
                }
        }
        cout << Verbose(2) << "End of Find_next_suitable_point_via_Angle_of_Sphere, recursion level " << RecursionLevel << ".\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 << "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) << "ERROR: NewSphereCenter " << helper << " is too close to OldSphereCenter" << OldSphereCenter << "." << endl;
                return 2.*M_PI;
        }
};


/** This recursive function finds a third point, to form a triangle with two given ones.
 * The idea is as follows: A sphere with fixed radius is (almost) uniquely defined in space by three points
 * that sit on its boundary. Hence, when two points are given and we look for the (next) third point, then
 * the center of the sphere is still fixed up to a single parameter. The band of possible values
 * describes a circle in 3D-space. The old center of the sphere for the current base triangle gives
 * us the "null" on this circle, the new center of the candidate point will be some way along this
 * circle. The shorter the way the better is the candidate. Note that the direction is clearly given
 * by the normal vector of the base triangle that always points outwards by construction.
 * Hence, we construct a Center of this circle which sits right in the middle of the current base line.
 * We construct the normal vector that defines the plane this circle lies in, it is just in the
 * direction of the baseline. And finally, we need the radius of the circle, which is given by the rest
 * with respect to the length of the baseline and the sphere's fixed \a RADIUS.
 * Note that there is one difficulty: The circumcircle is uniquely defined, but for the circumsphere's center
 * there are two possibilities which becomes clear from the construction as seen below. Hence, we must check
 * both.
 * Note also that the acos() function is not unique on [0, 2.*M_PI). Hence, we need an additional check
 * to decide for one of the two possible angles. Therefore we need a SearchDirection and to make this check
 * sensible we need OldSphereCenter to be orthogonal to it. Either we construct SearchDirection orthogonal
 * right away, or -- what we do here -- we rotate the relative sphere centers such that this orthogonality
 * holds. Then, the normalized projection onto the SearchDirection is either +1 or -1 and thus states whether
 * the angle is uniquely in either (0,M_PI] or [M_PI, 2.*M_PI).
 * @param BaseTriangle BoundaryTriangleSet of the current base triangle with all three points
 * @param BaseLine BoundaryLineSet of BaseTriangle with the current base line
 * @param OptCandidate candidate reference on return
 * @param OptCandidateCenter candidate's sphere center on return
 * @param ShortestAngle the current path length on this circle band for the current Opt_Candidate
 * @param RADIUS radius of sphere
 * @param *LC LinkedCell structure with neighbouring atoms
 */
// void Find_next_suitable_point(class BoundaryTriangleSet *BaseTriangle, class BoundaryLineSet *BaseLine, atom*& OptCandidate, Vector *OptCandidateCenter, double *ShortestAngle, const double RADIUS, LinkedCell *LC)
// {
//       atom *Walker = NULL;
//       Vector CircleCenter;   // center of the circle, i.e. of the band of sphere's centers
//       Vector CirclePlaneNormal; // normal vector defining the plane this circle lives in
//       Vector OldSphereCenter;         // center of the sphere defined by the three points of BaseTriangle
//       Vector NewSphereCenter;         // center of the sphere defined by the two points of BaseLine and the one of Candidate, first possibility
//       Vector OtherNewSphereCenter;    // center of the sphere defined by the two points of BaseLine and the one of Candidate, second possibility
//       Vector NewNormalVector;         // normal vector of the Candidate's triangle
//       Vector SearchDirection;         // vector that points out of BaseTriangle and is orthonormal to its NormalVector (i.e. the desired direction for the best Candidate)
//       Vector helper;
//       LinkedAtoms *List = NULL;
//       double CircleRadius; // radius of this circle
//       double radius;
//       double alpha, Otheralpha; // angles (i.e. parameter for the circle).
//       double Nullalpha; // angle between OldSphereCenter and NormalVector of base triangle
//       int N[NDIM], Nlower[NDIM], Nupper[NDIM];
//       atom *Candidate = NULL;
//
//       cout << Verbose(1) << "Begin of Find_next_suitable_point" << endl;
//
//       cout << Verbose(2) << "INFO: NormalVector of BaseTriangle is " << BaseTriangle->NormalVector << "." << endl;
//
//       // construct center of circle
//       CircleCenter.CopyVector(&(BaseLine->endpoints[0]->node->x));
//       CircleCenter.AddVector(&BaseLine->endpoints[1]->node->x);
//       CircleCenter.Scale(0.5);
//
//       // construct normal vector of circle
//       CirclePlaneNormal.CopyVector(&BaseLine->endpoints[0]->node->x);
//       CirclePlaneNormal.SubtractVector(&BaseLine->endpoints[1]->node->x);
//
//       // calculate squared radius of circle
//       radius = CirclePlaneNormal.ScalarProduct(&CirclePlaneNormal);
//       if (radius/4. < RADIUS*RADIUS) {
//               CircleRadius = RADIUS*RADIUS - radius/4.;
//               CirclePlaneNormal.Normalize();
//               cout << Verbose(2) << "INFO: CircleCenter is at " << CircleCenter << ", CirclePlaneNormal is " << CirclePlaneNormal << " with circle radius " << sqrt(CircleRadius) << "." << endl;
//
//               // construct old center
//               GetCenterofCircumcircle(&OldSphereCenter, &(BaseTriangle->endpoints[0]->node->x), &(BaseTriangle->endpoints[1]->node->x), &(BaseTriangle->endpoints[2]->node->x));
//               helper.CopyVector(&BaseTriangle->NormalVector);        // normal vector ensures that this is correct center of the two possible ones
//               radius = BaseLine->endpoints[0]->node->x.DistanceSquared(&OldSphereCenter);
//               helper.Scale(sqrt(RADIUS*RADIUS - radius));
//               OldSphereCenter.AddVector(&helper);
//               OldSphereCenter.SubtractVector(&CircleCenter);
//               cout << Verbose(2) << "INFO: OldSphereCenter is at " << OldSphereCenter << "." << endl;
//
//               // test whether old center is on the band's plane
//               if (fabs(OldSphereCenter.ScalarProduct(&CirclePlaneNormal)) > HULLEPSILON) {
//                       cerr << "ERROR: Something's very wrong here: OldSphereCenter is not on the band's plane as desired by " << fabs(OldSphereCenter.ScalarProduct(&CirclePlaneNormal)) << "!" << endl;
//                       OldSphereCenter.ProjectOntoPlane(&CirclePlaneNormal);
//               }
//               radius = OldSphereCenter.ScalarProduct(&OldSphereCenter);
//               if (fabs(radius - CircleRadius) < HULLEPSILON) {
//
//                       // construct SearchDirection
//                       SearchDirection.MakeNormalVector(&BaseTriangle->NormalVector, &CirclePlaneNormal);
//                       helper.CopyVector(&BaseLine->endpoints[0]->node->x);
//                       for(int i=0;i<3;i++)   // just take next different endpoint
//                               if ((BaseTriangle->endpoints[i]->node != BaseLine->endpoints[0]->node) && (BaseTriangle->endpoints[i]->node != BaseLine->endpoints[1]->node)) {
//                                       helper.SubtractVector(&BaseTriangle->endpoints[i]->node->x);
//                               }
//                       if (helper.ScalarProduct(&SearchDirection) < -HULLEPSILON)     // ohoh, SearchDirection points inwards!
//                               SearchDirection.Scale(-1.);
//                       SearchDirection.ProjectOntoPlane(&OldSphereCenter);
//                       SearchDirection.Normalize();
//                       cout << Verbose(2) << "INFO: SearchDirection is " << SearchDirection << "." << endl;
//                       if (fabs(OldSphereCenter.ScalarProduct(&SearchDirection)) > HULLEPSILON) {     // rotated the wrong way!
//                               cerr << "ERROR: SearchDirection and RelativeOldSphereCenter are still not orthogonal!" << endl;
//                       }
//
//                       if (LC->SetIndexToVector(&CircleCenter)) {     // get cell for the starting atom
//                               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;
//                       } else {
//                               cerr << "ERROR: Vector " << CircleCenter << " is outside of LinkedCell's bounding box." << endl;
//                               return;
//                       }
//                       // then go through the current and all neighbouring cells and check the contained atoms for possible candidates
//                       cout << Verbose(2) << "LC Intervals:";
//                       for (int i=0;i<NDIM;i++) {
//                               Nlower[i] = ((N[i]-1) >= 0) ? N[i]-1 : 0;
//                               Nupper[i] = ((N[i]+1) < LC->N[i]) ? N[i]+1 : LC->N[i]-1;
//                               cout << " [" << Nlower[i] << "," << Nupper[i] << "] ";
//                       }
//                       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(2) << "Current cell is " << LC->n[0] << ", " << LC->n[1] << ", " << LC->n[2] << " with No. " << LC->index << "." << endl;
//                                               if (List != NULL) {
//                                                       for (LinkedAtoms::iterator Runner = List->begin(); Runner != List->end(); Runner++) {
//                                                               Candidate = (*Runner);
//
//                                                               // check for three unique points
//                                                               if ((Candidate != BaseTriangle->endpoints[0]->node) && (Candidate != BaseTriangle->endpoints[1]->node) && (Candidate != BaseTriangle->endpoints[2]->node)) {
//                                                                       cout << Verbose(1) << "INFO: Current Candidate is " << *Candidate << " at " << Candidate->x << "." << endl;
//
//                                                                       // construct both new centers
//                                                                       GetCenterofCircumcircle(&NewSphereCenter, &(BaseLine->endpoints[0]->node->x), &(BaseLine->endpoints[1]->node->x), &(Candidate->x));
//                                                                       OtherNewSphereCenter.CopyVector(&NewSphereCenter);
//
//                                                                       if ((NewNormalVector.MakeNormalVector(&(BaseLine->endpoints[0]->node->x), &(BaseLine->endpoints[1]->node->x), &(Candidate->x))) && (fabs(NewNormalVector.ScalarProduct(&NewNormalVector)) > HULLEPSILON)) {
//                                                                               helper.CopyVector(&NewNormalVector);
//                                                                               cout << Verbose(2) << "INFO: NewNormalVector is " << NewNormalVector << "." << endl;
//                                                                               radius = BaseLine->endpoints[0]->node->x.DistanceSquared(&NewSphereCenter);
//                                                                               if (radius < RADIUS*RADIUS) {
//                                                                                       helper.Scale(sqrt(RADIUS*RADIUS - radius));
//                                                                                       cout << Verbose(3) << "INFO: Distance of NewCircleCenter to NewSphereCenter is " << helper.Norm() << "." << endl;
//                                                                                       NewSphereCenter.AddVector(&helper);
//                                                                                       NewSphereCenter.SubtractVector(&CircleCenter);
//                                                                                       cout << Verbose(2) << "INFO: NewSphereCenter is at " << NewSphereCenter << "." << endl;
//
//                                                                                       helper.Scale(-1.); // OtherNewSphereCenter is created by the same vector just in the other direction
//                                                                                       OtherNewSphereCenter.AddVector(&helper);
//                                                                                       OtherNewSphereCenter.SubtractVector(&CircleCenter);
//                                                                                       cout << Verbose(2) << "INFO: OtherNewSphereCenter is at " << OtherNewSphereCenter << "." << endl;
//
//                                                                                       // check both possible centers
//                                                                                       alpha = GetPathLengthonCircumCircle(CircleCenter, CirclePlaneNormal, CircleRadius, NewSphereCenter, OldSphereCenter, BaseTriangle->NormalVector, SearchDirection);
//                                                                                       Otheralpha = GetPathLengthonCircumCircle(CircleCenter, CirclePlaneNormal, CircleRadius, OtherNewSphereCenter, OldSphereCenter, BaseTriangle->NormalVector, SearchDirection);
//                                                                                       alpha = min(alpha, Otheralpha);
//                                                                                       if (*ShortestAngle > alpha) {
//                                                                                                       OptCandidate = Candidate;
//                                                                                                       *ShortestAngle = alpha;
//                                                                                                       if (alpha != Otheralpha)
//                                                                                                               OptCandidateCenter->CopyVector(&NewSphereCenter);
//                                                                                                       else
//                                                                                                               OptCandidateCenter->CopyVector(&OtherNewSphereCenter);
//                                                                                                       cout << Verbose(1) << "We have found a better candidate: " << *OptCandidate << " with " << alpha << " and circumsphere's center at " << *OptCandidateCenter << "." << endl;
//                                                                                       } else {
//                                                                                               if (OptCandidate != NULL)
//                                                                                                       cout << Verbose(1) << "REJECT: Old candidate: " << *OptCandidate << " is better than " << alpha << " with " << *ShortestAngle << "." << endl;
//                                                                                               else
//                                                                                                       cout << Verbose(2) << "REJECT: Candidate " << *Candidate << " with " << alpha << " was rejected." << endl;
//                                                                                       }
//
//                                                                               } else {
//                                                                                       cout << Verbose(1) << "REJECT: NewSphereCenter " << NewSphereCenter << " is too far away: " << radius << "." << endl;
//                                                                               }
//                                                                       } else {
//                                                                               cout << Verbose(1) << "REJECT: Three points from " << *BaseLine << " and Candidate " << *Candidate << " are linear-dependent." << endl;
//                                                                       }
//                                                               } else {
//                                                                       cout << Verbose(1) << "REJECT: Base triangle " << *BaseTriangle << " contains Candidate " << *Candidate << "." << endl;
//                                                               }
//                                                       }
//                                               }
//                                       }
//               } else {
//                       cerr << Verbose(1) << "ERROR: The projected center of the old sphere has radius " << radius << " instead of " << CircleRadius << "." << endl;
//               }
//       } else {
//               cout << Verbose(1) << "Circumcircle for base line " << *BaseLine << " and base triangle " << *BaseTriangle << " is too big!" << endl;
//       }
//
//       cout << Verbose(1) << "End of Find_next_suitable_point" << endl;
// };


/** Checks whether the triangle consisting of the three atoms is already present.
 * Searches for the points in Tesselation::PointsOnBoundary and checks their
 * lines. If any of the three edges already has two triangles attached, false is
 * returned.
 * \param *out output stream for debugging
 * \param *Candidates endpoints of the triangle candidate
 * \return false - triangle invalid due to edge criteria, true - triangle may be added.
 */
bool Tesselation::CheckPresenceOfTriangle(ofstream *out, atom *Candidates[3]) {
        LineMap::iterator FindLine;
        PointMap::iterator FindPoint;
        bool Present[3];

        *out << Verbose(2) << "Begin of CheckPresenceOfTriangle" << endl;
        for (int i=0;i<3;i++) { // check through all endpoints
                FindPoint = PointsOnBoundary.find(Candidates[i]->nr);
                if (FindPoint != PointsOnBoundary.end())
                        TPS[i] = FindPoint->second;
                else
                        TPS[i] = NULL;
        }

        // check lines
        for (int i=0;i<3;i++)
                if (TPS[i] != NULL)
                        for (int j=i;j<3;j++)
                                if (TPS[j] != NULL) {
                                        FindLine = TPS[i]->lines.find(TPS[j]->node->nr);
                                        if ((FindLine != TPS[i]->lines.end()) && (FindLine->second->TrianglesCount > 1)) {
                                                *out << "WARNING: Line " << *FindLine->second << " already present with " << FindLine->second->TrianglesCount << " triangles attached." << endl;
                                                *out << Verbose(2) << "End of CheckPresenceOfTriangle" << endl;
                                                return false;
                                        }
                                }
        *out << Verbose(2) << "End of CheckPresenceOfTriangle" << endl;
        return true;
};

/** This recursive function finds a third point, to form a triangle with two given ones.
 * Note that this function is for the starting triangle.
 * The idea is as follows: A sphere with fixed radius is (almost) uniquely defined in space by three points
 * that sit on its boundary. Hence, when two points are given and we look for the (next) third point, then
 * the center of the sphere is still fixed up to a single parameter. The band of possible values
 * describes a circle in 3D-space. The old center of the sphere for the current base triangle gives
 * us the "null" on this circle, the new center of the candidate point will be some way along this
 * circle. The shorter the way the better is the candidate. Note that the direction is clearly given
 * by the normal vector of the base triangle that always points outwards by construction.
 * Hence, we construct a Center of this circle which sits right in the middle of the current base line.
 * We construct the normal vector that defines the plane this circle lies in, it is just in the
 * direction of the baseline. And finally, we need the radius of the circle, which is given by the rest
 * with respect to the length of the baseline and the sphere's fixed \a RADIUS.
 * Note that there is one difficulty: The circumcircle is uniquely defined, but for the circumsphere's center
 * there are two possibilities which becomes clear from the construction as seen below. Hence, we must check
 * both.
 * Note also that the acos() function is not unique on [0, 2.*M_PI). Hence, we need an additional check
 * to decide for one of the two possible angles. Therefore we need a SearchDirection and to make this check
 * sensible we need OldSphereCenter to be orthogonal to it. Either we construct SearchDirection orthogonal
 * right away, or -- what we do here -- we rotate the relative sphere centers such that this orthogonality
 * holds. Then, the normalized projection onto the SearchDirection is either +1 or -1 and thus states whether
 * the angle is uniquely in either (0,M_PI] or [M_PI, 2.*M_PI).
 * @param NormalVector normal direction of the base triangle (here the unit axis vector, \sa Find_starting_triangle())
 * @param SearchDirection general direction where to search for the next point, relative to center of BaseLine
 * @param OldSphereCenter center of sphere for base triangle, relative to center of BaseLine, giving null angle for the parameter circle
 * @param BaseLine BoundaryLineSet with the current base line
 * @param ThirdNode third atom to avoid in search
 * @param OptCandidate candidate reference on return
 * @param OptCandidateCenter candidate's sphere center on return
 * @param ShortestAngle the current path length on this circle band for the current Opt_Candidate
 * @param RADIUS radius of sphere
 * @param *LC LinkedCell structure with neighbouring atoms
 */
void Find_third_point_for_Tesselation(Vector NormalVector, Vector SearchDirection, Vector OldSphereCenter, class BoundaryLineSet *BaseLine, atom *ThirdNode, atom*& OptCandidate, Vector *OptCandidateCenter, double *ShortestAngle, const double RADIUS, LinkedCell *LC)
{
        atom *Walker = NULL;
        Vector CircleCenter;    // center of the circle, i.e. of the band of sphere's centers
        Vector CirclePlaneNormal; // normal vector defining the plane this circle lives in
        Vector NewSphereCenter;  // center of the sphere defined by the two points of BaseLine and the one of Candidate, first possibility
        Vector OtherNewSphereCenter;     // center of the sphere defined by the two points of BaseLine and the one of Candidate, second possibility
        Vector NewNormalVector;  // normal vector of the Candidate's triangle
        Vector helper;
        LinkedAtoms *List = NULL;
        double CircleRadius; // radius of this circle
        double radius;
        double alpha, Otheralpha; // angles (i.e. parameter for the circle).
        double Nullalpha; // angle between OldSphereCenter and NormalVector of base triangle
        int N[NDIM], Nlower[NDIM], Nupper[NDIM];
        atom *Candidate = NULL;

        cout << Verbose(1) << "Begin of Find_third_point_for_Tesselation" << endl;

        cout << Verbose(2) << "INFO: NormalVector of BaseTriangle is " << NormalVector << "." << endl;

        // construct center of circle
        CircleCenter.CopyVector(&(BaseLine->endpoints[0]->node->x));
        CircleCenter.AddVector(&BaseLine->endpoints[1]->node->x);
        CircleCenter.Scale(0.5);

        // construct normal vector of circle
        CirclePlaneNormal.CopyVector(&BaseLine->endpoints[0]->node->x);
        CirclePlaneNormal.SubtractVector(&BaseLine->endpoints[1]->node->x);

        // calculate squared radius of circle
        radius = CirclePlaneNormal.ScalarProduct(&CirclePlaneNormal);
        if (radius/4. < RADIUS*RADIUS) {
                CircleRadius = RADIUS*RADIUS - radius/4.;
                CirclePlaneNormal.Normalize();
                cout << Verbose(2) << "INFO: CircleCenter is at " << CircleCenter << ", CirclePlaneNormal is " << CirclePlaneNormal << " with circle radius " << sqrt(CircleRadius) << "." << endl;

                // test whether old center is on the band's plane
                if (fabs(OldSphereCenter.ScalarProduct(&CirclePlaneNormal)) > HULLEPSILON) {
                        cerr << "ERROR: Something's very wrong here: OldSphereCenter is not on the band's plane as desired by " << fabs(OldSphereCenter.ScalarProduct(&CirclePlaneNormal)) << "!" << endl;
                        OldSphereCenter.ProjectOntoPlane(&CirclePlaneNormal);
                }
                radius = OldSphereCenter.ScalarProduct(&OldSphereCenter);
                if (fabs(radius - CircleRadius) < HULLEPSILON) {

                        // check SearchDirection
                        cout << Verbose(2) << "INFO: SearchDirection is " << SearchDirection << "." << endl;
                        if (fabs(OldSphereCenter.ScalarProduct(&SearchDirection)) > HULLEPSILON) {      // rotated the wrong way!
                                cerr << "ERROR: SearchDirection and RelativeOldSphereCenter are not orthogonal!" << endl;
                        }
                        // get cell for the starting atom
                        if (LC->SetIndexToVector(&CircleCenter)) {
                                        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;
                        } else {
                                cerr << "ERROR: Vector " << CircleCenter << " is outside of LinkedCell's bounding box." << endl;
                                return;
                        }
                        // then go through the current and all neighbouring cells and check the contained atoms for possible candidates
                        cout << Verbose(2) << "LC Intervals:";
                        for (int i=0;i<NDIM;i++) {
                                Nlower[i] = ((N[i]-1) >= 0) ? N[i]-1 : 0;
                                Nupper[i] = ((N[i]+1) < LC->N[i]) ? N[i]+1 : LC->N[i]-1;
                                cout << " [" << Nlower[i] << "," << Nupper[i] << "] ";
                        }
                        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(2) << "Current cell is " << LC->n[0] << ", " << LC->n[1] << ", " << LC->n[2] << " with No. " << LC->index << "." << endl;
                                                if (List != NULL) {
                                                        for (LinkedAtoms::iterator Runner = List->begin(); Runner != List->end(); Runner++) {
                                                                Candidate = (*Runner);

                                                                // check for three unique points
                                                                if ((Candidate != BaseLine->endpoints[0]->node) && (Candidate != BaseLine->endpoints[1]->node) && (Candidate != ThirdNode)) {
                                                                        cout << Verbose(1) << "INFO: Current Candidate is " << *Candidate << " at " << Candidate->x << "." << endl;

                                                                        // construct both new centers
                                                                        GetCenterofCircumcircle(&NewSphereCenter, &(BaseLine->endpoints[0]->node->x), &(BaseLine->endpoints[1]->node->x), &(Candidate->x));
                                                                        OtherNewSphereCenter.CopyVector(&NewSphereCenter);

                                                                        if ((NewNormalVector.MakeNormalVector(&(BaseLine->endpoints[0]->node->x), &(BaseLine->endpoints[1]->node->x), &(Candidate->x))) && (fabs(NewNormalVector.ScalarProduct(&NewNormalVector)) > HULLEPSILON)) {
                                                                                helper.CopyVector(&NewNormalVector);
                                                                                cout << Verbose(2) << "INFO: NewNormalVector is " << NewNormalVector << "." << endl;
                                                                                radius = BaseLine->endpoints[0]->node->x.DistanceSquared(&NewSphereCenter);
                                                                                if (radius < RADIUS*RADIUS) {
                                                                                        helper.Scale(sqrt(RADIUS*RADIUS - radius));
                                                                                        cout << Verbose(3) << "INFO: Distance of NewCircleCenter to NewSphereCenter is " << helper.Norm() << "." << endl;
                                                                                        NewSphereCenter.AddVector(&helper);
                                                                                        NewSphereCenter.SubtractVector(&CircleCenter);
                                                                                        cout << Verbose(2) << "INFO: NewSphereCenter is at " << NewSphereCenter << "." << endl;

                                                                                        helper.Scale(-1.); // OtherNewSphereCenter is created by the same vector just in the other direction
                                                                                        OtherNewSphereCenter.AddVector(&helper);
                                                                                        OtherNewSphereCenter.SubtractVector(&CircleCenter);
                                                                                        cout << Verbose(2) << "INFO: OtherNewSphereCenter is at " << OtherNewSphereCenter << "." << endl;

                                                                                        // check both possible centers
                                                                                        alpha = GetPathLengthonCircumCircle(CircleCenter, CirclePlaneNormal, CircleRadius, NewSphereCenter, OldSphereCenter, NormalVector, SearchDirection);
                                                                                        Otheralpha = GetPathLengthonCircumCircle(CircleCenter, CirclePlaneNormal, CircleRadius, OtherNewSphereCenter, OldSphereCenter, NormalVector, SearchDirection);
                                                                                        alpha = min(alpha, Otheralpha);
                                                                                        if (*ShortestAngle > alpha) {
                                                                                                        OptCandidate = Candidate;
                                                                                                        *ShortestAngle = alpha;
                                                                                                        if (alpha != Otheralpha)
                                                                                                                OptCandidateCenter->CopyVector(&NewSphereCenter);
                                                                                                        else
                                                                                                                OptCandidateCenter->CopyVector(&OtherNewSphereCenter);
                                                                                                        cout << Verbose(1) << "ACCEPT: We have found a better candidate: " << *OptCandidate << " with " << alpha << " and circumsphere's center at " << *OptCandidateCenter << "." << endl;
                                                                                        } else {
                                                                                                if (OptCandidate != NULL)
                                                                                                        cout << Verbose(1) << "REJECT: Old candidate: " << *OptCandidate << " is better than " << alpha << " with " << *ShortestAngle << "." << endl;
                                                                                                else
                                                                                                        cout << Verbose(2) << "REJECT: Candidate " << *Candidate << " with " << alpha << " was rejected." << endl;
                                                                                        }

                                                                                } else {
                                                                                        cout << Verbose(1) << "REJECT: NewSphereCenter " << NewSphereCenter << " is too far away: " << radius << "." << endl;
                                                                                }
                                                                        } else {
                                                                                cout << Verbose(1) << "REJECT: Three points from " << *BaseLine << " and Candidate " << *Candidate << " are linear-dependent." << endl;
                                                                        }
                                                                } else {
                                                                        if (ThirdNode != NULL)
                                                                                cout << Verbose(1) << "REJECT: Base triangle " << *BaseLine << " and " << *ThirdNode << " contains Candidate " << *Candidate << "." << endl;
                                                                        else
                                                                                cout << Verbose(1) << "REJECT: Base triangle " << *BaseLine << " contains Candidate " << *Candidate << "." << endl;
                                                                }
                                                        }
                                                }
                                        }
                } else {
                        cerr << Verbose(1) << "ERROR: The projected center of the old sphere has radius " << radius << " instead of " << CircleRadius << "." << endl;
                }
        } else {
                if (ThirdNode != NULL)
                        cout << Verbose(1) << "Circumcircle for base line " << *BaseLine << " and third node " << *ThirdNode << " is too big!" << endl;
                else
                        cout << Verbose(1) << "Circumcircle for base line " << *BaseLine << " is too big!" << endl;
        }

        cout << Verbose(1) << "End of Find_third_point_for_Tesselation" << endl;
};

/** Finds the second point of starting triangle.
 * \param *a first atom
 * \param *Candidate pointer to candidate atom on return
 * \param Oben vector indicating the outside
 * \param Opt_Candidate reference to recommended candidate on return
 * \param Storage[3] array storing angles and other candidate information
 * \param RADIUS radius of virtual sphere
 * \param *LC LinkedCell structure with neighbouring atoms
 */
void Find_second_point_for_Tesselation(atom* a, atom* Candidate, Vector Oben, atom*& Opt_Candidate, double Storage[3], double RADIUS, LinkedCell *LC)
{
        cout << Verbose(2) << "Begin of Find_second_point_for_Tesselation" << endl;
        int i;
        Vector AngleCheck;
        atom* Walker;
        double norm = -1., angle;
        LinkedAtoms *List = NULL;
        int N[NDIM], Nlower[NDIM], Nupper[NDIM];

        if (LC->SetIndexToAtom(a)) {    // get cell for the starting atom
                for(int i=0;i<NDIM;i++) // store indices of this cell
                        N[i] = LC->n[i];
        } else {
                cerr << "ERROR: Atom " << *a << " is not found in cell " << LC->index << "." << endl;
                return;
        }
        // then go through the current and all neighbouring cells and check the contained atoms for possible candidates
        cout << Verbose(2) << "LC Intervals:";
        for (int i=0;i<NDIM;i++) {
                Nlower[i] = ((N[i]-1) >= 0) ? N[i]-1 : 0;
                Nupper[i] = ((N[i]+1) < LC->N[i]) ? N[i]+1 : LC->N[i]-1;
                cout << " [" << Nlower[i] << "," << Nupper[i] << "] ";
        }
        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(2) << "Current cell is " << LC->n[0] << ", " << LC->n[1] << ", " << LC->n[2] << " with No. " << LC->index << "." << endl;
                                if (List != NULL) {
                                        for (LinkedAtoms::iterator Runner = List->begin(); Runner != List->end(); Runner++) {
                                                Candidate = (*Runner);
                                                                // check if we only have one unique point yet ...
                                                                if (a != Candidate) {
                        cout << Verbose(3) << "Current candidate is " << *Candidate << ": ";
                        AngleCheck.CopyVector(&(Candidate->x));
                        AngleCheck.SubtractVector(&(a->x));
                        norm = AngleCheck.Norm();
                                                        // second point shall have smallest angle with respect to Oben vector
                                                        if (norm < RADIUS) {
                                                                angle = AngleCheck.Angle(&Oben);
                                                                if (angle < Storage[0]) {
                                                                        //cout << Verbose(1) << "Old values of Storage: %lf %lf \n", Storage[0], Storage[1]);
                                                                        cout << "Is a better candidate with distance " << norm << " and " << angle << ".\n";
                                                                        Opt_Candidate = Candidate;
                                                                        Storage[0] = AngleCheck.Angle(&Oben);
                                                                        //cout << Verbose(1) << "Changing something in Storage: %lf %lf. \n", Storage[0], Storage[2]);
                                                                } else {
                                                                        cout << "Looses with angle " << angle << " to a better candidate " << *Opt_Candidate << endl;
                                                                }
                                                        } else {
                                                                cout << "Refused due to Radius " << norm << endl;
                                                        }
                                                }
                                        }
                                }
                        }
        cout << Verbose(2) << "End of Find_second_point_for_Tesselation" << endl;
};

/** Finds the starting triangle for find_non_convex_border().
 * Looks at the outermost atom per axis, then Find_second_point_for_Tesselation()
 * for the second and Find_next_suitable_point_via_Angle_of_Sphere() for the third
 * point are called.
 * \param RADIUS radius of virtual rolling sphere
 * \param *LC LinkedCell structure with neighbouring atoms
 */
void Tesselation::Find_starting_triangle(ofstream *out, molecule *mol, const double RADIUS, LinkedCell *LC)
{
        cout << Verbose(1) << "Begin of Find_starting_triangle\n";
        int i = 0;
        LinkedAtoms *List = NULL;
        atom* Walker;
        atom* FirstPoint;
        atom* SecondPoint;
        atom* MaxAtom[NDIM];
        double max_coordinate[NDIM];
        Vector Oben;
        Vector helper;
        Vector Chord;
        Vector SearchDirection;
        Vector OptCandidateCenter;

        Oben.Zero();

        for (i = 0; i < 3; i++) {
                MaxAtom[i] = NULL;
                max_coordinate[i] = -1;
        }

        // 1. searching topmost atom with respect to each axis
        for (int i=0;i<NDIM;i++) { // each axis
                LC->n[i] = LC->N[i]-1; // current axis is topmost cell
                for (LC->n[(i+1)%NDIM]=0;LC->n[(i+1)%NDIM]<LC->N[(i+1)%NDIM];LC->n[(i+1)%NDIM]++)
                        for (LC->n[(i+2)%NDIM]=0;LC->n[(i+2)%NDIM]<LC->N[(i+2)%NDIM];LC->n[(i+2)%NDIM]++) {
                                List = LC->GetCurrentCell();
                                //cout << Verbose(2) << "Current cell is " << LC->n[0] << ", " << LC->n[1] << ", " << LC->n[2] << " with No. " << LC->index << "." << endl;
                                if (List != NULL) {
                                        for (LinkedAtoms::iterator Runner = List->begin();Runner != List->end();Runner++) {
                                                cout << Verbose(2) << "Current atom is " << *(*Runner) << "." << endl;
                                                if ((*Runner)->x.x[i] > max_coordinate[i]) {
                                                        max_coordinate[i] = (*Runner)->x.x[i];
                                                        MaxAtom[i] = (*Runner);
                                                }
                                        }
                                } else {
                                        cerr << "ERROR: The current cell " << LC->n[0] << "," << LC->n[1] << "," << LC->n[2] << " is invalid!" << endl;
                                }
                        }
        }

        cout << Verbose(2) << "Found maximum coordinates: ";
        for (int i=0;i<NDIM;i++)
                cout << i << ": " << *MaxAtom[i] << "\t";
        cout << endl;
        const int k = 1;                // arbitrary choice
        Oben.x[k] = 1.;
        FirstPoint = MaxAtom[k];
        cout << Verbose(1) << "Coordinates of start atom " << *FirstPoint << " at " << FirstPoint->x << "." << endl;

        // add first point
        AddTrianglePoint(FirstPoint, 0);

        double ShortestAngle;
        atom* Opt_Candidate = NULL;
        ShortestAngle = 999999.; // This will contain the angle, which will be always positive (when looking for second point), when looking for third point this will be the quadrant.

        Find_second_point_for_Tesselation(FirstPoint, NULL, Oben, Opt_Candidate, &ShortestAngle, RADIUS, LC); // we give same point as next candidate as its bonds are looked into in find_second_...
        SecondPoint = Opt_Candidate;
        cout << Verbose(1) << "Found second point is " << *SecondPoint << " at " << SecondPoint->x << ".\n";

        // add second point and first baseline
        AddTrianglePoint(SecondPoint, 1);
        AddTriangleLine(TPS[0], TPS[1], 0);

        helper.CopyVector(&(FirstPoint->x));
        helper.SubtractVector(&(SecondPoint->x));
        helper.Normalize();
        Oben.ProjectOntoPlane(&helper);
        Oben.Normalize();
        helper.VectorProduct(&Oben);
        ShortestAngle = 2.*M_PI; // This will indicate the quadrant.

        Chord.CopyVector(&(FirstPoint->x)); // bring into calling function
        Chord.SubtractVector(&(SecondPoint->x));
        double radius = Chord.ScalarProduct(&Chord);
        double CircleRadius = sqrt(RADIUS*RADIUS - radius/4.);
        helper.CopyVector(&Oben);
        helper.Scale(CircleRadius);
        // Now, oben and helper are two orthonormalized vectors in the plane defined by Chord (not normalized)

        cout << Verbose(2) << "Looking for third point candidates \n";
        // look in one direction of baseline for initial candidate
        Opt_Candidate = NULL;
        SearchDirection.MakeNormalVector(&Chord, &Oben);        // whether we look "left" first or "right" first is not important ...

        cout << Verbose(1) << "Looking for third point candidates ...\n";
        Find_third_point_for_Tesselation(Oben, SearchDirection, helper, BLS[0], NULL, Opt_Candidate, &OptCandidateCenter, &ShortestAngle, RADIUS, LC);
        cout << Verbose(1) << "Third Point is " << *Opt_Candidate << endl;

        // add third point
        AddTrianglePoint(Opt_Candidate, 2);

        // FOUND Starting Triangle: FirstPoint, SecondPoint, Opt_Candidate

        // Finally, we only have to add the found further lines
        AddTriangleLine(TPS[1], TPS[2], 1);
        AddTriangleLine(TPS[0], TPS[2], 2);
        // ... and triangles to the Maps of the Tesselation class
        BTS = new class BoundaryTriangleSet(BLS, TrianglesOnBoundaryCount);
        AddTriangleToLines();
        // ... and calculate its normal vector (with correct orientation)
        OptCandidateCenter.Scale(-1.);
        cout << Verbose(2) << "Oben is currently " << OptCandidateCenter << "." << endl;
        BTS->GetNormalVector(OptCandidateCenter);
        cout << Verbose(0) << "==> The found starting triangle consists of " << *FirstPoint << ", " << *SecondPoint << " and " << *Opt_Candidate << " with normal vector " << BTS->NormalVector << ".\n";
        cout << Verbose(2) << "Projection is " << BTS->NormalVector.Projection(&Oben) << "." << endl;
        cout << Verbose(1) << "End of Find_starting_triangle\n";
};

/** This function finds a triangle to a line, adjacent to an existing one.
 * @param out output stream for debugging
 * @param *mol molecule with Atom's and Bond's
 * @param Line current baseline to search from
 * @param T current triangle which \a Line is edge of
 * @param RADIUS radius of the rolling ball
 * @param N number of found triangles
 * @param *filename filename base for intermediate envelopes
 * @param *LC LinkedCell structure with neighbouring atoms
 */
bool Tesselation::Find_next_suitable_triangle(ofstream *out,
                molecule *mol, BoundaryLineSet &Line, BoundaryTriangleSet &T,
                const double& RADIUS, int N, const char *tempbasename, LinkedCell *LC)
{
        cout << Verbose(1) << "Begin of Find_next_suitable_triangle\n";
        ofstream *tempstream = NULL;
        char NumberName[255];
        double tmp;

        atom* Opt_Candidate = NULL;
        Vector OptCandidateCenter;

        Vector CircleCenter;
        Vector CirclePlaneNormal;
        Vector OldSphereCenter;
        Vector SearchDirection;
        Vector helper;
        atom *ThirdNode = NULL;
        double ShortestAngle = 2.*M_PI; // This will indicate the quadrant.
        double radius, CircleRadius;

        cout << Verbose(1) << "Current baseline is " << Line << " of triangle " << T << "." << endl;
        for (int i=0;i<3;i++)
                if ((T.endpoints[i]->node != Line.endpoints[0]->node) && (T.endpoints[i]->node != Line.endpoints[1]->node))
                        ThirdNode = T.endpoints[i]->node;

        // construct center of circle
        CircleCenter.CopyVector(&Line.endpoints[0]->node->x);
        CircleCenter.AddVector(&Line.endpoints[1]->node->x);
        CircleCenter.Scale(0.5);

        // construct normal vector of circle
        CirclePlaneNormal.CopyVector(&Line.endpoints[0]->node->x);
        CirclePlaneNormal.SubtractVector(&Line.endpoints[1]->node->x);

        // calculate squared radius of circle
        radius = CirclePlaneNormal.ScalarProduct(&CirclePlaneNormal);
        if (radius/4. < RADIUS*RADIUS) {
                CircleRadius = RADIUS*RADIUS - radius/4.;
                CirclePlaneNormal.Normalize();
                cout << Verbose(2) << "INFO: CircleCenter is at " << CircleCenter << ", CirclePlaneNormal is " << CirclePlaneNormal << " with circle radius " << sqrt(CircleRadius) << "." << endl;

                // construct old center
                GetCenterofCircumcircle(&OldSphereCenter, &(T.endpoints[0]->node->x), &(T.endpoints[1]->node->x), &(T.endpoints[2]->node->x));
                helper.CopyVector(&T.NormalVector);     // normal vector ensures that this is correct center of the two possible ones
                radius = Line.endpoints[0]->node->x.DistanceSquared(&OldSphereCenter);
                helper.Scale(sqrt(RADIUS*RADIUS - radius));
                OldSphereCenter.AddVector(&helper);
                OldSphereCenter.SubtractVector(&CircleCenter);
                cout << Verbose(2) << "INFO: OldSphereCenter is at " << OldSphereCenter << "." << endl;

                // construct SearchDirection
                SearchDirection.MakeNormalVector(&T.NormalVector, &CirclePlaneNormal);
                helper.CopyVector(&Line.endpoints[0]->node->x);
                helper.SubtractVector(&ThirdNode->x);
                if (helper.ScalarProduct(&SearchDirection) < -HULLEPSILON)      // ohoh, SearchDirection points inwards!
                        SearchDirection.Scale(-1.);
                SearchDirection.ProjectOntoPlane(&OldSphereCenter);
                SearchDirection.Normalize();
                cout << Verbose(2) << "INFO: SearchDirection is " << SearchDirection << "." << endl;
                if (fabs(OldSphereCenter.ScalarProduct(&SearchDirection)) > HULLEPSILON) {      // rotated the wrong way!
                        cerr << "ERROR: SearchDirection and RelativeOldSphereCenter are still not orthogonal!" << endl;
                }

                // add third point
                cout << Verbose(1) << "Looking for third point candidates for triangle ... " << endl;
                Find_third_point_for_Tesselation(T.NormalVector, SearchDirection, OldSphereCenter, &Line, ThirdNode, Opt_Candidate, &OptCandidateCenter, &ShortestAngle, RADIUS, LC);

        } else {
                cout << Verbose(1) << "Circumcircle for base line " << Line << " and base triangle " << T << " is too big!" << endl;
        }

        if (Opt_Candidate == NULL) {
                cerr << "WARNING: Could not find a suitable candidate." << endl;
                return false;
        }
        cout << Verbose(1) << " Optimal candidate is " << *Opt_Candidate << " with circumsphere's center at " << OptCandidateCenter << "." << endl;

        // check whether all edges of the new triangle still have space for one more triangle (i.e. TriangleCount <2)
        atom *AtomCandidates[3];
        AtomCandidates[0] = Opt_Candidate;
        AtomCandidates[1] = Line.endpoints[0]->node;
        AtomCandidates[2] = Line.endpoints[1]->node;
        bool flag = CheckPresenceOfTriangle(out, AtomCandidates);

        if (flag) { // if so, add
                AddTrianglePoint(Opt_Candidate, 0);
                AddTrianglePoint(Line.endpoints[0]->node, 1);
                AddTrianglePoint(Line.endpoints[1]->node, 2);

                AddTriangleLine(TPS[0], TPS[1], 0);
                AddTriangleLine(TPS[0], TPS[2], 1);
                AddTriangleLine(TPS[1], TPS[2], 2);

                BTS = new class BoundaryTriangleSet(BLS, TrianglesOnBoundaryCount);
                AddTriangleToLines();

                OptCandidateCenter.Scale(-1.);
                BTS->GetNormalVector(OptCandidateCenter);
                OptCandidateCenter.Scale(-1.);

                cout << "--> New triangle with " << *BTS << " and normal vector " << BTS->NormalVector << " for this triangle ... " << endl;
                cout << Verbose(1) << "We have "<< TrianglesOnBoundaryCount << " for line " << Line << "." << endl;
        } else { // else, yell and do nothing
                cout << Verbose(1) << "This triangle consisting of ";
                cout << *Opt_Candidate << ", ";
                cout << *Line.endpoints[0]->node << " and ";
                cout << *Line.endpoints[1]->node << " ";
                cout << "is invalid!" << endl;
                return false;
        }

        if (flag && (DoSingleStepOutput && (TrianglesOnBoundaryCount % 10 == 0))) { // if we have a new triangle and want to output each new triangle configuration
                sprintf(NumberName, "-%04d-%s_%s_%s", TriangleFilesWritten, BTS->endpoints[0]->node->Name, BTS->endpoints[1]->node->Name, BTS->endpoints[2]->node->Name);
                if (DoTecplotOutput) {
                        string NameofTempFile(tempbasename);
                        NameofTempFile.append(NumberName);
                        for(size_t npos = NameofTempFile.find_first_of(' '); npos != -1; npos = NameofTempFile.find(' ', npos))
                                NameofTempFile.erase(npos, 1);
                        NameofTempFile.append(TecplotSuffix);
                        cout << Verbose(1) << "Writing temporary non convex hull to file " << NameofTempFile << ".\n";
                        tempstream = new ofstream(NameofTempFile.c_str(), ios::trunc);
                        write_tecplot_file(out, tempstream, this, mol, TriangleFilesWritten);
                        tempstream->close();
                        tempstream->flush();
                        delete(tempstream);
                }

                if (DoRaster3DOutput) {
                        string NameofTempFile(tempbasename);
                        NameofTempFile.append(NumberName);
                        for(size_t npos = NameofTempFile.find_first_of(' '); npos != -1; npos = NameofTempFile.find(' ', npos))
                                NameofTempFile.erase(npos, 1);
                        NameofTempFile.append(Raster3DSuffix);
                        cout << Verbose(1) << "Writing temporary non convex hull to file " << NameofTempFile << ".\n";
                        tempstream = new ofstream(NameofTempFile.c_str(), ios::trunc);
                        write_raster3d_file(out, tempstream, this, mol);
                        // include the current position of the virtual sphere in the temporary raster3d file
                        // make the circumsphere's center absolute again
                        helper.CopyVector(&Line.endpoints[0]->node->x);
                        helper.AddVector(&Line.endpoints[1]->node->x);
                        helper.Scale(0.5);
                        OptCandidateCenter.AddVector(&helper);
                        Vector *center = mol->DetermineCenterOfAll(out);
                        OptCandidateCenter.AddVector(center);
                        delete(center);
                        // and add to file plus translucency object
                        *tempstream << "# current virtual sphere\n";
                        *tempstream << "8\n     25.0            0.6              -1.0 -1.0 -1.0          0.2                            0 0 0 0\n";
                        *tempstream << "2\n     " << OptCandidateCenter.x[0] << " " << OptCandidateCenter.x[1] << " " << OptCandidateCenter.x[2] << "\t" << RADIUS << "\t1 0 0\n";
                        *tempstream << "9\n     terminating special property\n";
                        tempstream->close();
                        tempstream->flush();
                        delete(tempstream);
                }
                if (DoTecplotOutput || DoRaster3DOutput)
                        TriangleFilesWritten++;
        }

        cout << Verbose(1) << "End of Find_next_suitable_triangle\n";
        return true;
};

/** Tesselates the non convex boundary by rolling a virtual sphere along the surface of the molecule.
 * \param *out output stream for debugging
 * \param *mol molecule structure with Atom's and Bond's
 * \param *Tess Tesselation filled with points, lines and triangles on boundary on return
 * \param *LCList linked cell list of all atoms
 * \param *filename filename prefix for output of vertex data
 * \para RADIUS radius of the virtual sphere
 */
void Find_non_convex_border(ofstream *out, molecule* mol, class Tesselation *Tess, class LinkedCell *LCList, const char *filename, const double RADIUS)
{
        int N = 0;
        bool freeTess = false;
        *out << Verbose(1) << "Entering search for non convex hull. " << endl;
        if (Tess == NULL) {
                *out << Verbose(1) << "Allocating Tesselation struct ..." << endl;
                Tess = new Tesselation;
                freeTess = true;
        }
        bool freeLC = false;
        LineMap::iterator baseline;
        *out << Verbose(0) << "Begin of Find_non_convex_border\n";
        bool flag = false;      // marks whether we went once through all baselines without finding any without two triangles
        bool failflag = false;

        if (LCList == NULL) {
                LCList = new LinkedCell(mol, 2.*RADIUS);
                freeLC = true;
        }

        Tess->Find_starting_triangle(out, mol, RADIUS, LCList);

        baseline = Tess->LinesOnBoundary.begin();
        while ((baseline != Tess->LinesOnBoundary.end()) || (flag)) {
                if (baseline->second->TrianglesCount == 1) {
                        failflag = Tess->Find_next_suitable_triangle(out, mol, *(baseline->second), *(((baseline->second->triangles.begin()))->second), RADIUS, N, filename, LCList); //the line is there, so there is a triangle, but only one.
                        flag = flag || failflag;
                        if (!failflag)
                                cerr << "WARNING: Find_next_suitable_triangle failed." << endl;
                } else {
                        cout << Verbose(1) << "Line " << *baseline->second << " has " << baseline->second->TrianglesCount << " triangles adjacent" << endl;
                }
                N++;
                baseline++;
                if ((baseline == Tess->LinesOnBoundary.end()) && (flag)) {
                        baseline = Tess->LinesOnBoundary.begin();        // restart if we reach end due to newly inserted lines
                        flag = false;
                }
        }
        if (1) { //failflag) {
                *out << Verbose(1) << "Writing final tecplot file\n";
                if (DoTecplotOutput) {
                        string OutputName(filename);
                        OutputName.append(TecplotSuffix);
                        ofstream *tecplot = new ofstream(OutputName.c_str());
                        write_tecplot_file(out, tecplot, Tess, mol, -1);
                        tecplot->close();
                        delete(tecplot);
                }
                if (DoRaster3DOutput) {
                        string OutputName(filename);
                        OutputName.append(Raster3DSuffix);
                        ofstream *raster = new ofstream(OutputName.c_str());
                        write_raster3d_file(out, raster, Tess, mol);
                        raster->close();
                        delete(raster);
                }
        } else {
                cerr << "ERROR: Could definately not find all necessary triangles!" << endl;
        }
        if (freeTess)
                delete(Tess);
        if (freeLC)
                delete(LCList);
        *out << Verbose(0) << "End of Find_non_convex_border\n";
};