text: /*=========================================================================
  
    Program:   Visualization Toolkit
    Module:    $RCSfile: vtkPyramid.cxx,v $
  
    Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen
    All rights reserved.
    See Copyright.txt or http://www.kitware.com/Copyright.htm for details.
  
       This software is distributed WITHOUT ANY WARRANTY; without even
       the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
       PURPOSE.  See the above copyright notice for more information.
  
  =========================================================================*/
  #include "vtkPyramid.h"
  
  #include "vtkCellArray.h"
  #include "vtkCellData.h"
  #include "vtkLine.h"
  #include "vtkMath.h"
  #include "vtkObjectFactory.h"
  #include "vtkPointData.h"
  #include "vtkPointLocator.h"
  #include "vtkQuad.h"
  #include "vtkTriangle.h"
  #include "vtkUnstructuredGrid.h"
  
  vtkCxxRevisionMacro(vtkPyramid, "$Revision: 1.3 $");
  vtkStandardNewMacro(vtkPyramid);
  
  static const double VTK_DIVERGED = 1.e6;
  //----------------------------------------------------------------------------
  //
  // Construct the pyramid with five points.
  //
  vtkPyramid::vtkPyramid()
  {
    this->Points->SetNumberOfPoints(5);
    this->PointIds->SetNumberOfIds(5);
    for (int i = 0; i < 5; i++)
      {
      this->Points->SetPoint(i, 0.0, 0.0, 0.0);
      this->PointIds->SetId(i,0);
      }
    this->Line = vtkLine::New();
    this->Triangle = vtkTriangle::New();
    this->Quad = vtkQuad::New();
  }
  
  //----------------------------------------------------------------------------
  vtkPyramid::~vtkPyramid()
  {
    this->Line->Delete();
    this->Triangle->Delete();
    this->Quad->Delete();
  }
  
  static const int VTK_MAX_ITERATION=10;
  static const double VTK_CONVERGED=1.e-03;
  //----------------------------------------------------------------------------
  int vtkPyramid::EvaluatePosition(double x[3], double closestPoint[3],
                                   int& subId, double pcoords[3], 
                                   double& dist2, double *weights)
  {
    int iteration, converged;
    double  params[3];
    double  fcol[3], rcol[3], scol[3], tcol[3];
    int i, j;
    double  d, pt[3];
    double derivs[15];
  
    //  set initial position for Newton's method
    subId = 0;
    pcoords[0] = pcoords[1] = pcoords[2] = 0.5;
    params[0] = params[1] = params[2] = 0.3333333;
  
    //  enter iteration loop
    for (iteration=converged=0; !converged && (iteration < VTK_MAX_ITERATION);
    iteration++) 
      {
      //  calculate element interpolation functions and derivatives
      this->InterpolationFunctions(pcoords, weights);
      this->InterpolationDerivs(pcoords, derivs);
  
      //  calculate newton functions
      for (i=0; i<3; i++) 
        {
        fcol[i] = rcol[i] = scol[i] = tcol[i] = 0.0;
        }
      for (i=0; i<5; i++)
        {
        this->Points->GetPoint(i, pt);
        for (j=0; j<3; j++)
          {
          fcol[j] += pt[j] * weights[i];
          rcol[j] += pt[j] * derivs[i];
          scol[j] += pt[j] * derivs[i+5];
          tcol[j] += pt[j] * derivs[i+10];
          }
        }
  
      for (i=0; i<3; i++) 
        {
        fcol[i] -= x[i];
        }
  
      //  compute determinants and generate improvements
      d=vtkMath::Determinant3x3(rcol,scol,tcol);
      if ( fabs(d) < 1.e-20) 
        {
        return -1;
        }
  
      pcoords[0] = params[0] - vtkMath::Determinant3x3 (fcol,scol,tcol) / d;
      pcoords[1] = params[1] - vtkMath::Determinant3x3 (rcol,fcol,tcol) / d;
      pcoords[2] = params[2] - vtkMath::Determinant3x3 (rcol,scol,fcol) / d;
  
      //  check for convergence
      if ( ((fabs(pcoords[0]-params[0])) < VTK_CONVERGED) &&
           ((fabs(pcoords[1]-params[1])) < VTK_CONVERGED) &&
           ((fabs(pcoords[2]-params[2])) < VTK_CONVERGED) )
        {
        converged = 1;
        }
      // Test for bad divergence (S.Hirschberg 11.12.2001)
      else if ((fabs(pcoords[0]) > VTK_DIVERGED) || 
               (fabs(pcoords[1]) > VTK_DIVERGED) || 
               (fabs(pcoords[2]) > VTK_DIVERGED))
        {
        return -1;
        }
      //  if not converged, repeat
      else 
        {
        params[0] = pcoords[0];
        params[1] = pcoords[1];
        params[2] = pcoords[2];
        }
      }
  
    //  if not converged, set the parametric coordinates to arbitrary values
    //  outside of element
    if ( !converged ) 
      {
      return -1;
      }
  
    this->InterpolationFunctions(pcoords, weights);
  
    if ( pcoords[0] >= -0.001 && pcoords[0] <= 1.001 &&
    pcoords[1] >= -0.001 && pcoords[1] <= 1.001 &&
    pcoords[2] >= -0.001 && pcoords[2] <= 1.001 )
      {
      if (closestPoint)
        {
        closestPoint[0] = x[0]; closestPoint[1] = x[1]; closestPoint[2] = x[2];
        dist2 = 0.0; //inside pyramid
        }
      return 1;
      }
    else
      {
      double pc[3], w[5];
      if (closestPoint)
        {
        for (i=0; i<3; i++) //only approximate, not really true for warped hexa
          {
          if (pcoords[i] < 0.0) 
            {
            pc[i] = 0.0;
            }
          else if (pcoords[i] > 1.0) 
            {
            pc[i] = 1.0;
            }
          else 
            {
            pc[i] = pcoords[i];
            }
          }
        this->EvaluateLocation(subId, pc, closestPoint, (double *)w);
        dist2 = vtkMath::Distance2BetweenPoints(closestPoint,x);
        }
      return 0;
      }
  }
  
  //----------------------------------------------------------------------------
  void vtkPyramid::EvaluateLocation(int& vtkNotUsed(subId), double pcoords[3], 
                                  double x[3], double *weights)
  {
    int i, j;
    double pt[3];
  
    this->InterpolationFunctions(pcoords, weights);
  
    x[0] = x[1] = x[2] = 0.0;
    for (i=0; i<5; i++)
      {
      this->Points->GetPoint(i, pt);
      for (j=0; j<3; j++)
        {
        x[j] += pt[j] * weights[i];
        }
      }
  }
  
  //----------------------------------------------------------------------------
  // Returns the closest face to the point specified. Closeness is measured
  // parametrically.
  int vtkPyramid::CellBoundary(int vtkNotUsed(subId), double pcoords[3], 
                             vtkIdList *pts)
  {
    int i;
  
    // define 6 planes that separate regions
    static double normals[6][3] = { 
      {0.0,-0.5547002,0.8320503}, {0.5547002,0.0,0.8320503}, {0.0,0.5547002,0.8320503},
      {-0.5547002,0.0,0.8320503}, {0.70710670,-0.70710670,0.0}, {0.70710670,0.70710670,0.0} };
    static double point[3] = {0.5,0.5,0.3333333};
    double vals[6];
  
    // evaluate 6 plane equations
    for (i=0; i<6; i++)
      {
      vals[i] = normals[i][0]*(pcoords[0]-point[0]) + 
        normals[i][1]*(pcoords[1]-point[1]) + normals[i][2]*(pcoords[2]-point[2]);
      }
  
    // compare against six planes in parametric space that divide element
    // into five pieces (each corresponding to a face).
    if ( vals[4] >= 0.0 && vals[5] <= 0.0 && vals[0] >= 0.0 )
      {
      pts->SetNumberOfIds(3); //triangle face
      pts->SetId(0,this->PointIds->GetId(0));
      pts->SetId(1,this->PointIds->GetId(1));
      pts->SetId(2,this->PointIds->GetId(4));
      }
  
    else if ( vals[4] >= 0.0 && vals[5] >= 0.0 && vals[1] >= 0.0 )
      {
      pts->SetNumberOfIds(3); //triangle face
      pts->SetId(0,this->PointIds->GetId(1));
      pts->SetId(1,this->PointIds->GetId(2));
      pts->SetId(2,this->PointIds->GetId(4));
      }
  
    else if ( vals[4] <= 0.0 && vals[5] >= 0.0 && vals[2] >= 0.0 )
      {
      pts->SetNumberOfIds(3); //triangle face
      pts->SetId(0,this->PointIds->GetId(2));
      pts->SetId(1,this->PointIds->GetId(3));
      pts->SetId(2,this->PointIds->GetId(4));
      }
  
    else if ( vals[4] <= 0.0 && vals[5] <= 0.0 && vals[3] >= 0.0 )
      {
      pts->SetNumberOfIds(3); //triangle face
      pts->SetId(0,this->PointIds->GetId(3));
      pts->SetId(1,this->PointIds->GetId(0));
      pts->SetId(2,this->PointIds->GetId(4));
      }
  
    else 
      {
      pts->SetNumberOfIds(4); //quad face
      pts->SetId(0,this->PointIds->GetId(0));
      pts->SetId(1,this->PointIds->GetId(1));
      pts->SetId(2,this->PointIds->GetId(2));
      pts->SetId(3,this->PointIds->GetId(3));
      }
  
    if ( pcoords[0] < 0.0 || pcoords[0] > 1.0 ||
         pcoords[1] < 0.0 || pcoords[1] > 1.0 || 
         pcoords[2] < 0.0 || pcoords[2] > 1.0 )
      {
      return 0;
      }
    else
      {
      return 1;
      }
  }
  
  //----------------------------------------------------------------------------
  // Marching pyramids (contouring)
  //
  static int edges[8][2] = { {0,1}, {1,2}, {2,3}, 
                             {3,0}, {0,4}, {1,4},
                             {2,4}, {3,4} };
  static int faces[5][4] = { {0,3,2,1}, {0,1,4,-1}, 
                             {1,2,4,-1}, {2,3,4,-1}, {3,0,4,-1} };
  
  typedef int EDGE_LIST;
  typedef struct {
         EDGE_LIST edges[13];
  } TRIANGLE_CASES;
  
  static TRIANGLE_CASES triCases[] = { 
    {{-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}}, //0
    {{ 3,  4,  0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}}, //1
    {{ 5,  1,  0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}}, //2
    {{ 5,  1,  4,  1,  3,  4, -1, -1, -1, -1, -1, -1, -1}}, //3
    {{ 6,  2,  1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}}, //4
    {{ 3,  4,  0,  6,  2,  1, -1, -1, -1, -1, -1, -1, -1}}, //5
    {{ 5,  2,  0,  6,  2,  5, -1, -1, -1, -1, -1, -1, -1}}, //6
    {{ 2,  3,  4,  2,  4,  6,  4,  5,  6, -1, -1, -1, -1}}, //7
    {{ 2,  7,  3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}}, //8
    {{ 2,  7,  4,  4,  0,  2, -1, -1, -1, -1, -1, -1, -1}}, //9
    {{ 5,  1,  0,  2,  7,  3, -1, -1, -1, -1, -1, -1, -1}}, //10
    {{ 5,  7,  4,  1,  7,  5,  2,  7,  1, -1, -1, -1, -1}}, //11
    {{ 6,  3,  1,  7,  3,  6, -1, -1, -1, -1, -1, -1, -1}}, //12
    {{ 4,  6,  7,  0,  6,  4,  1,  6,  0, -1, -1, -1, -1}}, //13
    {{ 7,  5,  6,  3,  5,  7,  0,  5,  3, -1, -1, -1, -1}}, //14
    {{ 7,  4,  5,  7,  5,  6, -1, -1, -1, -1, -1, -1, -1}}, //15
    {{ 5,  7,  4,  6,  7,  5, -1, -1, -1, -1, -1, -1, -1}}, //16
    {{ 0,  5,  3,  5,  6,  3,  6,  7,  3, -1, -1, -1, -1}}, //17
    {{ 0,  1,  4,  1,  7,  4,  1,  6,  7, -1, -1, -1, -1}}, //18
    {{ 1,  6,  3,  6,  7,  3, -1, -1, -1, -1, -1, -1, -1}}, //19
    {{ 7,  5,  4,  7,  1,  5,  7,  2,  1, -1, -1, -1, -1}}, //20
    {{ 3,  7,  0,  7,  5,  0,  7,  2,  5,  2,  1,  5, -1}}, //21
    {{ 4,  2,  0,  7,  2,  4, -1, -1, -1, -1, -1, -1, -1}}, //22
    {{ 7,  2,  3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}}, //23
    {{ 2,  4,  3,  5,  4,  2,  6,  5,  2, -1, -1, -1, -1}}, //24
    {{ 2,  5,  0,  2,  6,  5, -1, -1, -1, -1, -1, -1, -1}}, //25
    {{ 6,  1,  0,  4,  6,  0,  3,  6,  4,  3,  2,  6, -1}}, //26
    {{ 2,  6,  1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}}, //27
    {{ 1,  4,  3,  1,  5,  4, -1, -1, -1, -1, -1, -1, -1}}, //28
    {{ 1,  5,  0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}}, //29
    {{ 4,  3,  0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}}, //30
    {{-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}}  //31
  };
  
  //----------------------------------------------------------------------------
  void vtkPyramid::Contour(double value, vtkDataArray *cellScalars, 
                           vtkPointLocator *locator,
                           vtkCellArray *verts, 
                           vtkCellArray *lines, 
                           vtkCellArray *polys,
                           vtkPointData *inPd, vtkPointData *outPd,
                           vtkCellData *inCd, vtkIdType cellId,
                           vtkCellData *outCd)
  {
    static int CASE_MASK[5] = {1,2,4,8,16};
    TRIANGLE_CASES *triCase;
    EDGE_LIST  *edge;
    int i, j, index, *vert, v1, v2, newCellId;
    vtkIdType pts[3];
    double t, x1[3], x2[3], x[3], deltaScalar;
    vtkIdType offset = verts->GetNumberOfCells() + lines->GetNumberOfCells();
  
    // Build the case table
    for ( i=0, index = 0; i < 5; i++)
      {
      if (cellScalars->GetComponent(i,0) >= value)
        {
        index |= CASE_MASK[i];
        }
      }
  
    triCase = triCases + index;
    edge = triCase->edges;
  
    for ( ; edge[0] > -1; edge += 3 )
      {
      for (i=0; i<3; i++) // insert triangle
        {
        vert = edges[edge[i]];
  
        // calculate a preferred interpolation direction
        deltaScalar = (cellScalars->GetComponent(vert[1],0) 
                       - cellScalars->GetComponent(vert[0],0));
        if (deltaScalar > 0)
          {
          v1 = vert[0]; v2 = vert[1];
          }
        else
          {
          v1 = vert[1]; v2 = vert[0];
          deltaScalar = -deltaScalar;
          }
  
        // linear interpolation
        t = ( deltaScalar == 0.0 ? 0.0 :
              (value - cellScalars->GetComponent(v1,0)) / deltaScalar );
  
        this->Points->GetPoint(v1, x1);
        this->Points->GetPoint(v2, x2);
  
        for (j=0; j<3; j++)
          {
          x[j] = x1[j] + t * (x2[j] - x1[j]);
          }
        if ( locator->InsertUniquePoint(x, pts[i]) )
          {
          if ( outPd ) 
            {
            vtkIdType p1 = this->PointIds->GetId(v1);
            vtkIdType p2 = this->PointIds->GetId(v2);
            outPd->InterpolateEdge(inPd,pts[i],p1,p2,t);
            }
          }
        }
  
      // check for degenerate triangle
      if ( pts[0] != pts[1] && pts[0] != pts[2] && pts[1] != pts[2] )
        {
        newCellId = offset + polys->InsertNextCell(3,pts);
        outCd->CopyData(inCd,cellId,newCellId);
        }
      }
  }
  
  //----------------------------------------------------------------------------
  int *vtkPyramid::GetEdgeArray(int edgeId)
  {
    return edges[edgeId];
  }
  
  //----------------------------------------------------------------------------
  vtkCell *vtkPyramid::GetEdge(int edgeId)
  {
    int *verts;
  
    verts = edges[edgeId];
  
    // load point id's
    this->Line->PointIds->SetId(0,this->PointIds->GetId(verts[0]));
    this->Line->PointIds->SetId(1,this->PointIds->GetId(verts[1]));
  
    // load coordinates
    this->Line->Points->SetPoint(0,this->Points->GetPoint(verts[0]));
    this->Line->Points->SetPoint(1,this->Points->GetPoint(verts[1]));
  
    return this->Line;
  }
  
  //----------------------------------------------------------------------------
  int *vtkPyramid::GetFaceArray(int faceId)
  {
    return faces[faceId];
  }
  
  //----------------------------------------------------------------------------
  vtkCell *vtkPyramid::GetFace(int faceId)
  {
    int *verts;
  
    verts = faces[faceId];
  
    if ( verts[3] != -1 ) // quad cell
      {
      // load point id's
      this->Quad->PointIds->SetId(0,this->PointIds->GetId(verts[0]));
      this->Quad->PointIds->SetId(1,this->PointIds->GetId(verts[1]));
      this->Quad->PointIds->SetId(2,this->PointIds->GetId(verts[2]));
      this->Quad->PointIds->SetId(3,this->PointIds->GetId(verts[3]));
  
      // load coordinates
      this->Quad->Points->SetPoint(0,this->Points->GetPoint(verts[0]));
      this->Quad->Points->SetPoint(1,this->Points->GetPoint(verts[1]));
      this->Quad->Points->SetPoint(2,this->Points->GetPoint(verts[2]));
      this->Quad->Points->SetPoint(3,this->Points->GetPoint(verts[3]));
  
      return this->Quad;
      }
    else
      {
      // load point id's
      this->Triangle->PointIds->SetId(0,this->PointIds->GetId(verts[0]));
      this->Triangle->PointIds->SetId(1,this->PointIds->GetId(verts[1]));
      this->Triangle->PointIds->SetId(2,this->PointIds->GetId(verts[2]));
  
      // load coordinates
      this->Triangle->Points->SetPoint(0,this->Points->GetPoint(verts[0]));
      this->Triangle->Points->SetPoint(1,this->Points->GetPoint(verts[1]));
      this->Triangle->Points->SetPoint(2,this->Points->GetPoint(verts[2]));
  
      return this->Triangle;
      }
  }
  
  //----------------------------------------------------------------------------
  // Intersect faces against line.
  //
  int vtkPyramid::IntersectWithLine(double p1[3], double p2[3], double tol, double& t,
                                 double x[3], double pcoords[3], int& subId)
  {
    int intersection=0;
    double pt1[3], pt2[3], pt3[3], pt4[3];
    double tTemp;
    double pc[3], xTemp[3], dist2, weights[5];
  
    int faceNum;
  
    t = VTK_DOUBLE_MAX;
  
    //first intersect the triangle faces
    for (faceNum=1; faceNum<5; faceNum++)
      {
      this->Points->GetPoint(faces[faceNum][0], pt1);
      this->Points->GetPoint(faces[faceNum][1], pt2);
      this->Points->GetPoint(faces[faceNum][2], pt3);
  
      this->Triangle->Points->SetPoint(0,pt1);
      this->Triangle->Points->SetPoint(1,pt2);
      this->Triangle->Points->SetPoint(2,pt3);
  
      if ( this->Triangle->IntersectWithLine(p1, p2, tol, tTemp, xTemp, pc, subId) )
        {
        intersection = 1;
        if ( tTemp < t )
          {
          t = tTemp;
          x[0] = xTemp[0]; x[1] = xTemp[1]; x[2] = xTemp[2]; 
          this->EvaluatePosition(x, xTemp, subId, pcoords, dist2, weights);
          }
        }
      }
  
    //now intersect the quad face
    this->Points->GetPoint(faces[0][0], pt1);
    this->Points->GetPoint(faces[0][1], pt2);
    this->Points->GetPoint(faces[0][2], pt3);
    this->Points->GetPoint(faces[0][3], pt4);
  
    this->Quad->Points->SetPoint(0,pt1);
    this->Quad->Points->SetPoint(1,pt2);
    this->Quad->Points->SetPoint(2,pt3);
    this->Quad->Points->SetPoint(3,pt4);
  
    if ( this->Quad->IntersectWithLine(p1, p2, tol, tTemp, xTemp, pc, subId) )
      {
      intersection = 1;
      if ( tTemp < t )
        {
        t = tTemp;
        x[0] = xTemp[0]; x[1] = xTemp[1]; x[2] = xTemp[2]; 
        pcoords[0] = pc[0]; pcoords[1] = pc[1]; pcoords[2] = 0.0;
        }
      }
  
    return intersection;
  }
  
  //----------------------------------------------------------------------------
  int vtkPyramid::Triangulate(int vtkNotUsed(index), vtkIdList *ptIds, vtkPoints *pts)
  {
    int p[4], i;
    ptIds->Reset();
    pts->Reset();
  
    // The base of the pyramid must be split into two triangles.  There are two
    // ways to do this (across either diagonal).  Pick the shorter diagonal.
    double base_points[4][3];
    for (i = 0; i < 4; i++)
      {
      this->Points->GetPoint(i, base_points[i]);
      }
    double diagonal1, diagonal2;
    diagonal1 = vtkMath::Distance2BetweenPoints(base_points[0], base_points[2]);
    diagonal2 = vtkMath::Distance2BetweenPoints(base_points[1], base_points[3]);
  
    if (diagonal1 < diagonal2)
      {
      for (i=0; i < 4; i++)
        {
        p[0] = 0; p[1] = 1; p[2] = 2; p[3] = 4;
        ptIds->InsertNextId(this->PointIds->GetId(p[i]));
        pts->InsertNextPoint(this->Points->GetPoint(p[i]));
        }
      for (i=0; i < 4; i++)
        {
        p[0] = 0; p[1] = 2; p[2] = 3; p[3] = 4;
        ptIds->InsertNextId(this->PointIds->GetId(p[i]));
        pts->InsertNextPoint(this->Points->GetPoint(p[i]));
        }
      }
    else
      {
      for (i=0; i < 4; i++)
        {
        p[0] = 0; p[1] = 1; p[2] = 3; p[3] = 4;
        ptIds->InsertNextId(this->PointIds->GetId(p[i]));
        pts->InsertNextPoint(this->Points->GetPoint(p[i]));
        }
      for (i=0; i < 4; i++)
        {
        p[0] = 1; p[1] = 2; p[2] = 3; p[3] = 4;
        ptIds->InsertNextId(this->PointIds->GetId(p[i]));
        pts->InsertNextPoint(this->Points->GetPoint(p[i]));
        }
      }
  
    return !(diagonal1 == diagonal2);
  }
  
  //----------------------------------------------------------------------------
  void vtkPyramid::Derivatives(int vtkNotUsed(subId), double pcoords[3],
                               double *values, int dim, double *derivs)
  {
    double *jI[3], j0[3], j1[3], j2[3];
    double functionDerivs[15], sum[3], value;
    int i, j, k;
  
    // compute inverse Jacobian and interpolation function derivatives
    jI[0] = j0; jI[1] = j1; jI[2] = j2;
    this->JacobianInverse(pcoords, jI, functionDerivs);
  
    // now compute derivates of values provided
    for (k=0; k < dim; k++) //loop over values per vertex
      {
      sum[0] = sum[1] = sum[2] = 0.0;
      for ( i=0; i < 5; i++) //loop over interp. function derivatives
        {
        value = values[dim*i + k];
        sum[0] += functionDerivs[i] * value;
        sum[1] += functionDerivs[5 + i] * value;
        sum[2] += functionDerivs[10 + i] * value;
        }
  
      for (j=0; j < 3; j++) //loop over derivative directions
        {
        derivs[3*k + j] = sum[0]*jI[0][j] + sum[1]*jI[1][j] + sum[2]*jI[2][j];
        }
      }
  }
  
  //----------------------------------------------------------------------------
  // Compute iso-parametric interpolation functions for pyramid
  //
  void vtkPyramid::InterpolationFunctions(double pcoords[3], double sf[5])
  {
    double rm, sm, tm;
  
    rm = 1. - pcoords[0];
    sm = 1. - pcoords[1];
    tm = 1. - pcoords[2];
  
    sf[0] = rm*sm*tm;
    sf[1] = pcoords[0]*sm*tm;
    sf[2] = pcoords[0]*pcoords[1]*tm;
    sf[3] = rm*pcoords[1]*tm;
    sf[4] = pcoords[2];
  }
  
  //----------------------------------------------------------------------------
  void vtkPyramid::InterpolationDerivs(double pcoords[3], double derivs[15])
  { 
    double rm, sm, tm;
  
    rm = 1. - pcoords[0];
    sm = 1. - pcoords[1];
    tm = 1. - pcoords[2];
  
    // r-derivatives
    derivs[0] = -sm*tm;
    derivs[1] = sm*tm;
    derivs[2] = pcoords[1]*tm;
    derivs[3] = -pcoords[1]*tm;
    derivs[4] = 0.0;
  
    // s-derivatives
    derivs[5] = -rm*tm;
    derivs[6] = -pcoords[0]*tm;
    derivs[7] = pcoords[0]*tm;
    derivs[8] = rm*tm;
    derivs[9] = 0.0;
  
    // t-derivatives
    derivs[10] = -rm*sm;
    derivs[11] = -pcoords[0]*sm;
    derivs[12] = -pcoords[0]*pcoords[1];
    derivs[13] = -rm*pcoords[1];
    derivs[14] = 1.0;
  }
  
  //----------------------------------------------------------------------------
  // Given parametric coordinates compute inverse Jacobian transformation
  // matrix. Returns 9 elements of 3x3 inverse Jacobian plus interpolation
  // function derivatives. Returns 0 if no inverse exists.
  // Note for pyramid: the inverse Jacobian is undefined at the apex.
  int vtkPyramid::JacobianInverse(double pcoords[3], double **inverse, double derivs[15])
  {
    int i, j;
    double *m[3], m0[3], m1[3], m2[3];
    double x[3];
  
    // compute interpolation function derivatives
    this->InterpolationDerivs(pcoords,derivs);
  
    // create Jacobian matrix
    m[0] = m0; m[1] = m1; m[2] = m2;
    for (i=0; i < 3; i++) //initialize matrix
      {
      m0[i] = m1[i] = m2[i] = 0.0;
      }
  
    for ( j=0; j < 5; j++ )
      {
      this->Points->GetPoint(j, x);
      for ( i=0; i < 3; i++ )
        {
        m0[i] += x[i] * derivs[j];
        m1[i] += x[i] * derivs[5 + j];
        m2[i] += x[i] * derivs[10 + j];
        }
      }
  
    // now find the inverse
    if ( vtkMath::InvertMatrix(m,inverse,3) == 0 )
      {
  #define VTK_MAX_WARNS 3    
      static int numWarns=0;
      if ( numWarns++ < VTK_MAX_WARNS )
        {
        vtkErrorMacro(<<"Jacobian inverse not found");
        vtkErrorMacro(<<"Matrix:" << m[0][0] << " " << m[0][1] << " " << m[0][2]
        << m[1][0] << " " << m[1][1] << " " << m[1][2] 
        << m[2][0] << " " << m[2][1] << " " << m[2][2] );
        return 0;
        }
      }
  
    return 1;
  }
  
  //----------------------------------------------------------------------------
  void vtkPyramid::GetEdgePoints(int edgeId, int* &pts)
  {
    pts = this->GetEdgeArray(edgeId);
  }
  
  //----------------------------------------------------------------------------
  void vtkPyramid::GetFacePoints(int faceId, int* &pts)
  {
    pts = this->GetFaceArray(faceId);
  }
  
  static double vtkPyramidCellPCoords[15] = {0.0,0.0,0.0, 1.0,0.0,0.0,
                                             1.0,1.0,0.0, 0.0,1.0,0.0, 
                                             0.0,0.0,1.0};
  
  //----------------------------------------------------------------------------
  double *vtkPyramid::GetParametricCoords()
  {
    return vtkPyramidCellPCoords;
  }
  
  //----------------------------------------------------------------------------
  void vtkPyramid::PrintSelf(ostream& os, vtkIndent indent)
  {
    this->Superclass::PrintSelf(os,indent);
    
    os << indent << "Line:\n";
    this->Line->PrintSelf(os,indent.GetNextIndent());
    os << indent << "Triangle:\n";
    this->Triangle->PrintSelf(os,indent.GetNextIndent());
    os << indent << "Quad:\n";
    this->Quad->PrintSelf(os,indent.GetNextIndent());
  }
  