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IntAna_IntQuadQuad.cxx

// File:    IntAna_IntQuadQuad.cxx
// Created: Mon Jun 29 11:59:35 1992
// Author:  Laurent BUCHARD
//          <lbr@topsn3>

#include <stdio.h>

#include <Standard_Stream.hxx>

#ifndef DEB
#define No_Standard_RangeError
#define No_Standard_OutOfRange
#endif


//======================================================================
//== I n t e r s e c t i o n   C O N E           Q U A D R I Q U E   
//==                           C Y L I N D R E   Q U A D R I Q U E
//======================================================================
#include <IntAna_IntQuadQuad.ixx>

#include <Standard_DomainError.hxx>
#include <Standard_OutOfRange.hxx>
#include <StdFail_NotDone.hxx>
#include <math_TrigonometricFunctionRoots.hxx>
      
#include <gp_Ax2.hxx>
#include <gp_Ax3.hxx>

//=======================================================================
//class : TrigonometricRoots
//purpose: Classe Interne (Donne des racines classees d un polynome trigo)
//======================================================================
class TrigonometricRoots {

 private:
  Standard_Real Roots[4];
  Standard_Boolean done;
  Standard_Integer NbRoots;
  Standard_Boolean infinite_roots;

 public:
  TrigonometricRoots(const Standard_Real CC,
                 const Standard_Real SC,
                 const Standard_Real C,
                 const Standard_Real S,
                 const Standard_Real Cte,
                 const Standard_Real Binf,
                 const Standard_Real Bsup);
  //IsDone
  Standard_Boolean IsDone() { 
    return done; 
  }
  //IsARoot
  Standard_Boolean IsARoot(Standard_Real u) {
    Standard_Integer i;
    Standard_Real aEps=RealEpsilon();
    Standard_Real PIpPI = Standard_PI+Standard_PI;
    //
    for(i=0 ; i<NbRoots; ++i) {
      if(Abs(u - Roots[i])<=aEps) {
      return Standard_True;
      }
      if(Abs(u - Roots[i]-PIpPI)<=aEps) {
      return Standard_True;
      }
    }
    return Standard_False;
  }
  //NbSolutions
  Standard_Integer NbSolutions() { 
    if(!done) {
      StdFail_NotDone::Raise();
    }
    return NbRoots; 
  }
  //InfiniteRoots
  Standard_Boolean InfiniteRoots() { 
    if(!done) {
      StdFail_NotDone::Raise();
    }    
    return infinite_roots; 
  }
  //Value
  Standard_Real Value(const Standard_Integer n) {
    if((!done)||(n>NbRoots)) {
      StdFail_NotDone::Raise();
    }    
    return Roots[n-1];
  }
}; 
//=======================================================================
//function : TrigonometricRoots::TrigonometricRoots
//purpose  : 
//=======================================================================
TrigonometricRoots::TrigonometricRoots(const Standard_Real CC,
                               const Standard_Real SC,
                               const Standard_Real C,
                               const Standard_Real S,
                               const Standard_Real Cte,
                               const Standard_Real Binf,
                               const Standard_Real Bsup) 
{
  Standard_Integer i, j, SvNbRoots;
  Standard_Boolean Triee;
  Standard_Real PIpPI = Standard_PI+Standard_PI;
  //
  done=Standard_False;
  //
  //-- F= AA*CN*CN+2*BB*CN*SN+CC*CN+DD*SN+EE;
  math_TrigonometricFunctionRoots MTFR(CC, SC, C, S, Cte, Binf, Bsup); 
  if(!MTFR.IsDone()) {
    return;
  }
  //
  done=Standard_True;
  if(MTFR.InfiniteRoots()) {
    infinite_roots=Standard_True;
    return;
  }
  //
  NbRoots=MTFR.NbSolutions();
  //
  for(i=0; i<NbRoots; ++i) {
    Roots[i]=MTFR.Value(i+1);
    if(Roots[i]<0.){
      Roots[i]+=PIpPI;
    }
    if(Roots[i]>PIpPI) {
      Roots[i]-=PIpPI;
    }
  }
  //
  //-- La recherche directe donne n importe quoi. 
  SvNbRoots=NbRoots;
  for(i=0; i<SvNbRoots; ++i) {
    Standard_Real y;
    Standard_Real co=cos(Roots[i]);
    Standard_Real si=sin(Roots[i]);
    y=co*(CC*co + (SC+SC)*si + C) + S*si + Cte;
    if(Abs(y)>1e-8) {
      done=Standard_False; 
      return; //-- le 1er avril 98 
    }
  }
  //
  do {
    Triee=Standard_True;
    for(i=1,j=0; i<SvNbRoots; ++i,++j) {
      if(Roots[i]<Roots[j]) {
      Triee=Standard_False;
      Standard_Real t=Roots[i]; 
      Roots[i]=Roots[j]; 
      Roots[j]=t;
      }
    }
  }
  while(!Triee);
  //
  infinite_roots=Standard_False;
  //
  if(!NbRoots) {//--!!!!! Detection du cas Pol = Cte ( 1e-50 ) !!!!
    if((Abs(CC) + Abs(SC) + Abs(C) + Abs(S)) < 1e-10) {
      if(Abs(Cte) < 1e-10)  {
      infinite_roots=Standard_True;
      }
    }
  }
}
//=======================================================================
//class    : MyTrigonometricFunction
//purpose  : 
//  Classe interne : Donne Value et Derivative sur un polynome 
//                   trigonometrique
//======================================================================
class MyTrigonometricFunction {
 
 private:
  Standard_Real CC,SS,SC,S,C,Cte;
 
 public:
  //
  MyTrigonometricFunction(const Standard_Real xCC,
                    const Standard_Real xSS,
                    const Standard_Real xSC,
                    const Standard_Real xC,
                    const Standard_Real xS,
                    const Standard_Real xCte) {
    CC=xCC; 
    SS=xSS; 
    SC=xSC; 
    S=xS; 
    C=xC; 
    Cte=xCte;
  }
 
  Standard_Real Value(const Standard_Real& U) {
    Standard_Real sinus, cosinus, aRet;
    //
    sinus=sin(U);
    cosinus=cos(U);
    aRet= CC*cosinus*cosinus + 
          SS*sinus*sinus +
        2.0*(sinus*(SC*cosinus+S)+cosinus*C)+
        Cte;
    //
    return aRet;
  }  
  //
  Standard_Real Derivative(const Standard_Real& U) {
    Standard_Real sinus, cosinus;
    //
    sinus=sin(U);
    cosinus=cos(U);
    //
    return(2.0*((sinus*cosinus)*(SS-CC)
            +S*cosinus
            -C*sinus
            +SC*(cosinus*cosinus-sinus*sinus)));
  }
};

//////////
//=======================================================================
//function : IntAna_IntQuadQuad::IntAna_IntQuadQuad
//purpose  : C o n s t r u c t e u r    v i d e   
//=======================================================================
00228 IntAna_IntQuadQuad::IntAna_IntQuadQuad(void) {
  done=Standard_False;
  identical = Standard_False;
  NbCurves=0;
  Nbpoints=0;
  myNbMaxCurves=12;
  myEpsilon=0.00000001;
  myEpsilonCoeffPolyNull=0.00000001;
}
//=======================================================================
//function : IntAna_IntQuadQuad::IntAna_IntQuadQuad
//purpose  : I n t e r s e c t i o n   C y l i n d r e   Q u a d r i q u e 
//=======================================================================
00241 IntAna_IntQuadQuad::IntAna_IntQuadQuad(const gp_Cylinder& Cyl,
                               const IntAna_Quadric& Quad,
                               const Standard_Real Tol) {
  myNbMaxCurves=12;
  myEpsilon=0.00000001;
  myEpsilonCoeffPolyNull=0.00000001;
  Perform(Cyl,Quad,Tol);
}
//=======================================================================
//function : Perform
//purpose  : I n t e r s e c t i o n   C y l i n d r e   Q u a d r i q u e 
//=======================================================================
00253 void IntAna_IntQuadQuad::Perform(const gp_Cylinder& Cyl,
                         const IntAna_Quadric& Quad,
                         const Standard_Real) 
{
  done = Standard_True;
  identical= Standard_False;
  NbCurves=0;
  Nbpoints=0;
  //
  Standard_Boolean UN_SEUL_Z_PAR_THETA, DEUX_Z_PAR_THETA, 
                   Z_POSITIF, Z_INDIFFERENT, Z_NEGATIF;
  //
  UN_SEUL_Z_PAR_THETA=Standard_False;
  DEUX_Z_PAR_THETA=Standard_True;
  Z_POSITIF=Standard_True;
  Z_INDIFFERENT=Standard_True;
  Z_NEGATIF=Standard_False;
  //
  Standard_Real Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1, aRealEpsilon, RCyl, R2;
  Standard_Real PIpPI = Standard_PI+Standard_PI;
  //
  for(Standard_Integer raz = 0 ; raz < myNbMaxCurves ; raz++) {
    previouscurve[raz] = nextcurve[raz] = 0;
  }
  //
  RCyl=Cyl.Radius();
  aRealEpsilon=RealEpsilon();
  //----------------------------------------------------------------------
  //-- Coefficients de la quadrique : 
  //--      2       2       2
  //-- Qxx x + Qyy y + Qzz z + 2 ( Qxy x y + Qxz x z + Qyz y z )
  //-- + 2 ( Qx x + Qy y + Qz z ) + Q1
  //----------------------------------------------------------------------
  Quad.Coefficients(Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1);
  
  //----------------------------------------------------------------------
  //-- Ecriture des Coefficients de la Quadrique dans le repere liee 
  //-- au Cylindre 
  //--                 Cyl.Position() -> Ax2
  //----------------------------------------------------------------------
  Quad.NewCoefficients(Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,Cyl.Position());
  
  //----------------------------------------------------------------------
  //-- Parametrage du Cylindre Cyl : 
  //--     X = Rcyl * Cos(Theta)
  //--     Y = Rcyl * Sin(Theta) 
  //--     Z = Z
  //----------------------------------------------------------------------
  //-- Donne une Equation de la forme :
  //--   F(Sin(Theta),Cos(Theta),ZCyl) = 0
  //--   (Equation du second degre en ZCyl) 
  //--    ZCyl**2  CoeffZ2(Theta) + ZCyl CoeffZ1(Theta) + CoeffZ0(Theta)
  //----------------------------------------------------------------------
  //-- CoeffZ0 = Q1 + 2*Qx*RCyl*Cos[Theta] + Qxx*RCyl^2*Cos[Theta]^2
  //--           2*RCyl*Sin[Theta]* ( Qy + Qxy*RCyl*Cos[Theta]);
  //--           Qyy*RCyl^2*Sin[Theta]^2;
  //-- CoeffZ1 =2.0 * ( Qz + RCyl*(Qxz*Cos[Theta] + Sin[Theta]*Qyz)) ;
  //-- CoeffZ2 =  Qzz;
  //-- !!!! Attention , si on norme sur Qzz pour detecter le cas 1er degre 
  //----------------------------------------------------------------------
  //-- On Cherche Les Racines en Theta du discriminant de cette equation :
  //-- DIS(Theta) = C_1 + C_SS Sin()**2 + C_CC Cos()**2 + 2 C_SC Sin() Cos()
  //--                  + 2 C_S  Sin()    + 2 C_C Cos()
  //--
  //-- Si Qzz = 0   Alors  On Resout Z=Fct(Theta)  sur le domaine de Theta
  //----------------------------------------------------------------------

  if(Abs(Qzz)<myEpsilonCoeffPolyNull) {
    done=Standard_False; //-- le 12 mars 98    
    return;
  }  
  else { //#0
    //----------------------------------------------------------------------
    //--- Cas  ou  F(Z,Theta) est du second degre en Z                  ----
    //----------------------------------------------------------------------
    R2   = RCyl*RCyl;
    
    Standard_Real C_1  = Qz*Qz - Qzz*Q1;
    Standard_Real C_SS = R2    * ( Qyz*Qyz - Qyy*Qzz );
    Standard_Real C_CC = R2    * ( Qxz*Qxz - Qxx*Qzz );
    Standard_Real C_S  = RCyl  * ( Qyz*Qz  - Qy*Qzz  );
    Standard_Real C_C  = RCyl  * ( Qxz*Qz  - Qx*Qzz  );
    Standard_Real C_SC = R2    * ( Qxz*Qyz - Qxy*Qzz );
    //
    MyTrigonometricFunction MTF(C_CC,C_SS,C_SC,C_C,C_S,C_1);
    TrigonometricRoots PolDIS(C_CC-C_SS,
                        C_SC,
                        C_C+C_C,
                        C_S+C_S,
                        C_1+C_SS, 0., PIpPI);
    //~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    if(PolDIS.IsDone()==Standard_False) {
      done=Standard_False;
      return;
    }
    //~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    if(PolDIS.InfiniteRoots()) {   
      TheCurve[0].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                              myEpsilon,0.,PIpPI,
                              UN_SEUL_Z_PAR_THETA,
                              Z_POSITIF);
      TheCurve[1].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                              myEpsilon,0.,PIpPI,
                              UN_SEUL_Z_PAR_THETA,
                              Z_NEGATIF);
      NbCurves=2;
    }

    else { //#1
      //---------------------------------------------------------------
      //-- La Recherche des Zero de DIS a reussi
      //---------------------------------------------------------------
      Standard_Integer nbsolDIS=PolDIS.NbSolutions();
      if(nbsolDIS==0) {
      //--------------------------------------------------------------
      //-- Le Discriminant a un signe constant : 
      //-- 
      //-- Si Positif  ---> 2 Courbes
      //-- Sinon       ---> Pas de solution
      //--------------------------------------------------------------
      if(MTF.Value(PI) >= -aRealEpsilon) {

        TheCurve[0].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                                  myEpsilon,0.0,PIpPI,
                                  UN_SEUL_Z_PAR_THETA,
                                  Z_POSITIF);
        TheCurve[1].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                                  myEpsilon,0.0,PIpPI,
                                  UN_SEUL_Z_PAR_THETA,
                                  Z_NEGATIF);
        
        NbCurves=2;
      }
      else {
        //------------------------------------------------------------
        //-- Le Discriminant est toujours Negatif
        //------------------------------------------------------------
        NbCurves=0;
      }
      }
      else { //#2
      //------------------------------------------------------------
      //-- Le Discriminant a des racines 
      //-- (Le Discriminant est une fonction periodique donc ... )
      //------------------------------------------------------------
      if( nbsolDIS==1 ) {
        //------------------------------------------------------------
        //-- Point de Tangence pour ce Theta Solution
        //-- Si Signe de Discriminant >=0 pour tout Theta
        //--     Alors  
        //--       Courbe Solution pour la partie Positive
        //--       entre les 2 racines ( Ici Tout le domaine )
        //-- Sinon Seulement un point Tangent
        //------------------------------------------------------------
        if(MTF.Value(PolDIS.Value(1)+PI) >= -aRealEpsilon ) {
          //------------------------------------------------------------
          //-- On a Un Point de Tangence + Une Courbe Solution
          //------------------------------------------------------------
          TheCurve[0].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                                    myEpsilon,0.0,PIpPI,
                                    UN_SEUL_Z_PAR_THETA,
                                    Z_POSITIF);
          TheCurve[1].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                                    myEpsilon,0.0,PIpPI,
                                    UN_SEUL_Z_PAR_THETA,
                                    Z_NEGATIF);
          
          NbCurves=2;
        }
        else {
          //------------------------------------------------------------
          //-- On a simplement un Point de tangence
          //------------------------------------------------------------
          //--Standard_Real Theta = PolDIS(1);
          //--Standard_Real SecPar= -0.5 * MTFZ1.Value(Theta) / MTFZ2.Value(Theta);
          //--Thepoints[Nbpoints] = ElSLib::CylinderValue(Theta,SecPar,Cyl.Position(),Cyl.Radius());
          //--Nbpoints++;
          NbCurves=0;
        }
      }
      else {  // #3
        //------------------------------------------------------------
        //-- On detecte   :  Les racines double   
        //--              :  Les Zones Positives [Modulo 2 PI]
        //-- Les Courbes solutions seront obtenues pour les valeurs
        //-- de Theta ou Discriminant(Theta) > 0  (>=0? en limite)
        //-- Par la resolution de l equation implicite avec Theta fixe
        //------------------------------------------------------------
        //-- Si tout est solution, Alors on est sur une iso 
        //-- Ce cas devrait etre traite en amont
        //------------------------------------------------------------
        //-- On construit la fonction permettant connaissant un Theta
        //-- de calculer les Z+ et Z- Correspondants.
        //-------------------------------------------------------------

        //-------------------------------------------------------------
        //-- Calcul des Intervalles ou Discriminant >=0
        //-- On a au plus nbsol intervalles ( en fait 2 )
        //--  (1,2) (2,3) .. (nbsol,1+2PI)
        //-------------------------------------------------------------
        Standard_Integer i;
        Standard_Real Theta1, Theta2, Theta3, autrepar, qwet;
        Standard_Boolean UnPtTg = Standard_False;
        //
        NbCurves=0;
        if(nbsolDIS == 2) { 
          for(i=1; i<=nbsolDIS ; i++) {
            Theta1=PolDIS.Value(i);
            Theta2=(i<nbsolDIS)? PolDIS.Value(i+1)  :  (PolDIS.Value(1)+PIpPI);
            //----------------------------------------------------------------
            //-- On detecte les racines doubles 
            //-- Si il n y a que 2 racines alors on prend tout l interval
            //----------------------------------------------------------------
            if(Abs(Theta2-Theta1)<=aRealEpsilon) {
            UnPtTg = Standard_True;
            autrepar=Theta1-0.1; 
            if(autrepar<0.) {
              autrepar=Theta1+0.1;
            }
            //
            qwet=MTF.Value(autrepar);
            if(qwet>=0.) { 
              TheCurve[NbCurves].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                                             myEpsilon,Theta1,Theta1+PIpPI,
                                             UN_SEUL_Z_PAR_THETA,
                                             Z_POSITIF);
              NbCurves++;
              TheCurve[NbCurves].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                                             myEpsilon,Theta1,Theta1+PIpPI,
                                             UN_SEUL_Z_PAR_THETA,
                                             Z_NEGATIF);
              NbCurves++;
            }
            }
          }
        }
        
        for(i=1; UnPtTg == (Standard_False) && (i<=nbsolDIS) ; i++) {
          Theta1=PolDIS.Value(i);
          Theta2=(i<nbsolDIS)? PolDIS.Value(i+1) : (PolDIS.Value(1)+PIpPI);
          //----------------------------------------------------------------
          //-- On detecte les racines doubles 
          //-- Si il n y a que 2 racines alors on prend tout l interval
          //----------------------------------------------------------------
          if(Abs(Theta2-Theta1)<=1e-12) {
            //-- cout<<"\n####### Un Point de Tangence en Theta = "<<Theta1<<endl;
            //-- i++;
          }
          else {  //-- On evite les pbs numeriques (Tout est du meme signe entre les racines) 
            qwet=MTF.Value(0.5*(Theta1+Theta2))
                +MTF.Value(0.4*Theta1+0.6*Theta2)
                  +MTF.Value(0.6*Theta1+0.4*Theta2);
            if(qwet >= 0.) {
            //------------------------------------------------------------
            //-- On est positif a partir de Theta1
            //-- L intervalle Theta1,Theta2 est retenu
            //------------------------------------------------------------
            
            //-- le 8 octobre 1997 : 
            //-- PB: Racine en 0    pi/2 pi/2  et 2pi
            //-- On cree 2 courbes : 0    -> pi/2    2zpartheta
            //--                     pi/2 -> 2pi     2zpartheta
            //-- 
            //-- la courbe 0 pi/2   passe par le double pt de tangence pi/2
            //-- il faut donc couper cette courbe en 2
            //--
            Theta3 = ((i+1)<nbsolDIS)? PolDIS.Value(i+2)  :  (PolDIS.Value(1)+PIpPI);
            //ft
            if((Theta3-Theta2)<5.e-8) { 
            //
              TheCurve[NbCurves].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                                             myEpsilon,Theta1,Theta2,
                                             UN_SEUL_Z_PAR_THETA,
                                             Z_POSITIF);
              NbCurves++;
              TheCurve[NbCurves].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                                             myEpsilon,Theta1,Theta2,
                                             UN_SEUL_Z_PAR_THETA,
                                             Z_NEGATIF);
              NbCurves++;
            }
            else { 
              TheCurve[NbCurves].SetCylinderQuadValues(Cyl,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                                             myEpsilon,Theta1,Theta2,
                                             DEUX_Z_PAR_THETA,
                                             Z_INDIFFERENT);
              NbCurves++;
            }
            }//if(qwet >= 0.)
          }//else {  //-- On evite les pbs numerique ...      
        } //for(i=1; UnPtTg == (Standard_False) && (i<=nbsolDIS) ; i++) {    
      }//else { // #3
      }//else { //#2
    }//else { //#1
    
  }//else { //#0
}

//=======================================================================
//function :IntAna_IntQuadQuad::IntAna_IntQuadQuad
//purpose  : 
//=======================================================================
00555 IntAna_IntQuadQuad::IntAna_IntQuadQuad(const gp_Cone& Cone,
                               const IntAna_Quadric& Quad,
                               const Standard_Real Tol) 
{ 
  myNbMaxCurves=12;
  myEpsilon=0.00000001;
  myEpsilonCoeffPolyNull=0.00000001;
  Perform(Cone,Quad,Tol);
}
//=======================================================================
//function :Perform
//purpose  : 
//=======================================================================
00568 void IntAna_IntQuadQuad::Perform(const gp_Cone& Cone,
                         const IntAna_Quadric& Quad,
                         const Standard_Real)  
{ 
  //
  Standard_Boolean UN_SEUL_Z_PAR_THETA, DEUX_Z_PAR_THETA, 
                   Z_POSITIF, Z_INDIFFERENT, Z_NEGATIF;
  //
  UN_SEUL_Z_PAR_THETA=Standard_False;
  DEUX_Z_PAR_THETA=Standard_True;
  Z_POSITIF=Standard_True;
  Z_INDIFFERENT=Standard_True;
  Z_NEGATIF=Standard_False;
  //
  Standard_Integer i;
  Standard_Real Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1;
  Standard_Real Theta1, Theta2, TgAngle;
  Standard_Real PIpPI = Standard_PI+Standard_PI;
  //
  done=Standard_True;
  identical = Standard_False;
  NbCurves=0;
  Nbpoints=0;
  //
  for(i=0 ; i<myNbMaxCurves ; ++i) {
    previouscurve[i]=0;
    nextcurve[i]=0;
  }
  //
  Quad.Coefficients(Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1);
  //
  gp_Ax3 tAx3(Cone.Position());
  tAx3.SetLocation(Cone.Apex());
  Quad.NewCoefficients(Qxx,Qyy,Qzz,
                   Qxy,Qxz,Qyz,
                   Qx,Qy,Qz,Q1,
                   tAx3);
  //
  TgAngle=1./(Tan(Cone.SemiAngle()));
  //
  // The parametrization of the Cone
  // 
  // x= z*tan(beta)*cos(t)
  // y= z*tan(beta)*sin(t)
  // z=z
  // 
  // The intersection curves are defined by the equation
  //
  //                        2
  //          f(z,t)= A(t)*z + B(t)*z + C(t)=0
  //
  //   
  // 1. Try to solve A(t)=0 -> PolZ2
  //
  Standard_Integer nbsol, nbsol1, nbsolZ2;
  Standard_Real Z2CC, Z2SS, Z2Cte, Z2SC, Z2C, Z2S;
  Standard_Real Z1CC, Z1SS, Z1Cte, Z1SC, Z1C, Z1S; 
  Standard_Real C_1,C_SS, C_CC, C_S, C_C, C_SC;
  //
  Z2CC = Qxx;
  Z2SS = Qyy;
  Z2Cte= Qzz * TgAngle*TgAngle;
  Z2SC = Qxy;
  Z2C  = (TgAngle)*Qxz;
  Z2S  = (TgAngle)*Qyz;
  //
  TrigonometricRoots PolZ2(Z2CC-Z2SS,Z2SC,Z2C+Z2C,Z2S+Z2S,Z2Cte+Z2SS,0.,PIpPI);
  if(!PolZ2.IsDone()) {
    done=!done;
    return;
  } 
  //
  //MyTrigonometricFunction MTF2(Z2CC,Z2SS,Z2SC,Z2C,Z2S,Z2Cte);
  nbsolZ2 = PolZ2.NbSolutions();
  //
  // 2. Try to solve B(t)=0  -> PolZ1
  //
  Z1Cte = 2.*(TgAngle)*Qz;
  Z1SS  = 0.;
  Z1CC  = 0.;
  Z1S   = Qy;
  Z1C   = Qx;
  Z1SC  = 0.;
  //
  TrigonometricRoots PolZ1(Z1CC-Z1SS,Z1SC, Z1C+Z1C,Z1S+Z1S, Z1Cte+Z1SS,0.,PIpPI);
  if(!PolZ1.IsDone()) {
    done=!done;
    return;
  }
  MyTrigonometricFunction MTFZ1(Z1CC,Z1SS,Z1SC,Z1C,Z1S,Z1Cte);
  //
  nbsol1=PolZ1.NbSolutions();
  if(PolZ2.InfiniteRoots()) { // i.e A(t)=0 for any t
    if(!PolZ1.InfiniteRoots()) {    
      if(!nbsol1) {
      //-- B(t)*z + C(t) = 0    avec C(t) != 0
      TheCurve[0].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,
                              myEpsilon,0.,PIpPI,
                              UN_SEUL_Z_PAR_THETA,
                              Z_POSITIF);
      NbCurves=1;
      }
      else {
      /*
      Standard_Integer ii;
      for(ii=1; ii<= nbsol1 ; ++ii) {
        Standard_Real Theta=PolZ1.Value(ii);
        if(Abs(MTFZ1.Value(Theta))<=myEpsilon) {
          //-- Une droite Solution  Z=  -INF -> +INF  pour Theta
          //-- cout<<"######## Droite Solution Pour Theta = "<<Theta<<endl;
        }
        else {
          //-- cout<<"\n#### _+_+_+_+_+ CAS A(t) Z + B = 0   avec A et B ==0 "<<endl;
        }
      }
      */
      }
    }
    else {
      if(Abs(Q1)<=myEpsilon) {
      done=!done;
      return;
      }
      else {
      //-- Pas de Solutions 
      }
    }
    return; 
  }
  //
  //else { //#5 
  //
  //                2
  //-- f(z,t)=A(t)*z + B(t)*z + C(t)=0   avec A n est pas toujours nul
  //
  //                                              2
  // 3. Try to explore s.c. Discriminant:  D(t)=B(t)-4*A(t)*C(t) => Pol
  //
  // The function D(t) is just a formula that has sense for quadratic
  // equation above.
  // For cases when A(t)=0 (say at t=ti, t=tj. etc) the equation
  // will be
  //
  //     f(z,t)=B(t)*z + C(t)=0, where B(t)!=0, 
  //
  // and the D(t) is nonsense for it.
  //
  C_1  = TgAngle*TgAngle * ( Qz * Qz - Qzz * Q1);
  C_SS = Qy * Qy - Qyy * Q1;
  C_CC = Qx * Qx - Qxx * Q1; 
  C_S  = TgAngle*( Qy * Qz - Qyz * Q1);
  C_C  = TgAngle*( Qx * Qz - Qxz * Q1);
  C_SC = Qx * Qy - Qxy * Q1;
  //
  TrigonometricRoots Pol(C_CC-C_SS, C_SC, C_C+C_C,C_S+C_S, C_1+C_SS,0.,PIpPI);
  if(!Pol.IsDone()) {
    done=!done;
    return;
  }
  //
  nbsol=Pol.NbSolutions();
  MyTrigonometricFunction MTF(C_CC,C_SS,C_SC,C_C,C_S,C_1);
  //
  if(Pol.InfiniteRoots()) { 
    //                             2
    //         f(z,t)= (z(t)-zo(t)) = 0       Discriminant(t)=0 pour tout t 
    //
    TheCurve[0].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1, myEpsilon,
                          0.,PIpPI,
                          UN_SEUL_Z_PAR_THETA, Z_POSITIF);
    TheCurve[1].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1, myEpsilon,
                          0.,PIpPI,
                          UN_SEUL_Z_PAR_THETA, Z_NEGATIF);
    NbCurves=2;
    return;
  }
  //else {//#4
  //                     2
  //        f(z,t)=A(t)*z + B(t)*z + C(t)      Discriminant(t) != 0 
  //
  if(!nbsol && (MTF.Value(PI)<0.) ) {
    //-- Discriminant signe constant negatif
    return;
  }
  //else {//# 3
  //
  //-- On Traite le cas : Discriminant(t) > 0 pour tout t en simulant 1 
  //   racine double en 0
  Standard_Boolean DiscriminantConstantPositif = Standard_False;
  if(!nbsol) { 
    nbsol=1;
    DiscriminantConstantPositif = Standard_True;
  }
  //----------------------------------------------------------------------
  //-- Le discriminant admet au moins une racine ( -> Point de Tangence )
  //-- ou est constant positif.
  //----------------------------------------------------------------------
  for(i=1; i<=nbsol; ++i) {
    if(DiscriminantConstantPositif) {
      Theta1 = 0.;
      Theta2 = PIpPI-myEpsilon; 
    }
    else {
      Theta1=Pol.Value(i);
      Theta2=(i<nbsol)? Pol.Value(i+1) : (Pol.Value(1)+PIpPI);
    }
    //
    if(Abs(Theta2-Theta1)<=myEpsilon){
      done=Standard_False; 
      return;// !!! pkv
    }
    //else {// #2
    Standard_Real qwet =MTF.Value(0.5*(Theta1+Theta2))+
                        MTF.Value(0.4*Theta1+0.6*Theta2)+
                      MTF.Value(0.6*Theta1+0.4*Theta2);
    if(qwet < 0.) {
      continue;
    }
    //---------------------------------------------------------------------
    //-- On a des Solutions entre Theta1 et Theta 2 
    //---------------------------------------------------------------------
    
    //---------------------------------------------------------------------
    //-- On Subdivise si necessaire Theta1-->Theta2
    //-- en Theta1-->t1   t1--->t2   ...  tn--->Theta2
    //--  ou t1,t2,...tn sont des racines de A(t)
    //--
    //-- Seule la courbe Z_NEGATIF est affectee
    //----------------------------------------------------------------------
    Standard_Boolean RacinesdePolZ2DansTheta1Theta2;
    Standard_Integer i2;
    Standard_Real r;
    //
    //nbsolZ2=PolZ2.NbSolutions();
    RacinesdePolZ2DansTheta1Theta2=Standard_False;
    for(i2=1; i2<=nbsolZ2 && !RacinesdePolZ2DansTheta1Theta2; ++i2) {
      r=PolZ2.Value(i2);
      if(r>Theta1 && r<Theta2) {
      RacinesdePolZ2DansTheta1Theta2=Standard_True;
      }
      else { 
      r+=PIpPI;
      if(r>Theta1 && r<Theta2){ 
        RacinesdePolZ2DansTheta1Theta2=Standard_True;
      }
      }
    }
    //
    if(!RacinesdePolZ2DansTheta1Theta2) {
      //--------------------------------------------------------------------
      //-- Pas de Branches Infinies 
      TheCurve[NbCurves].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,myEpsilon,
                                 Theta1,Theta2,
                                 UN_SEUL_Z_PAR_THETA,Z_POSITIF);
      NbCurves++;
      TheCurve[NbCurves].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,myEpsilon,
                                 Theta1,Theta2,
                                 UN_SEUL_Z_PAR_THETA,
                                 Z_NEGATIF);
      NbCurves++;
    }
    
    else { //#1
      Standard_Boolean NoChanges;
      Standard_Real NewMin, NewMax, to;
      //
      NewMin=Theta1;
      NewMax=Theta2;
      NoChanges=Standard_True;
      //
      for(i2=1; i2<= (nbsolZ2+nbsolZ2) ; ++i2) {
      if(i2>nbsolZ2) {
        to=PolZ2.Value(i2-nbsolZ2) + PIpPI; 
      }
      else {
        to=PolZ2.Value(i2);
      }
      //
      // to is value of the parameter t where A(t)=0, i.e. 
      // f(z,to)=B(to)*z + C(to)=0, B(to)!=0. 
      //                   
      // z=-C(to)/B(to) is one and only
      // solution inspite of the fact that D(t)>0 for any value of t.
      //
      if(to<NewMax && to>NewMin) {
        //-----------------------------------------------------------------
        //-- On coupe au moins une fois le domaine Theta1 Theta2
        //-----------------------------------------------------------------
        NoChanges=Standard_False;
        TheCurve[NbCurves].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1, myEpsilon,
                                     NewMin,to,
                                     UN_SEUL_Z_PAR_THETA, Z_NEGATIF);
        //
        NbCurves++;
        TheCurve[NbCurves].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1, myEpsilon,
                                     NewMin,to,
                                     UN_SEUL_Z_PAR_THETA, Z_POSITIF);
        //------------------------------------------------------------
        //-- A == 0    
        //-- Si B < 0    Alors Infini sur   Z+  
        //-- Sinon             Infini sur   Z-
        //------------------------------------------------------------
        if(PolZ2.IsARoot(NewMin)) {
          if(MTFZ1.Value(NewMin) < 0.) {
            TheCurve[NbCurves].SetIsFirstOpen(Standard_True);
          }
          else {
            TheCurve[NbCurves-1].SetIsFirstOpen(Standard_True);
          }
        }
        if(MTFZ1.Value(to) < 0.) {
          TheCurve[NbCurves].SetIsLastOpen(Standard_True);
        }
        else {
          TheCurve[NbCurves-1].SetIsLastOpen(Standard_True);
        }
        //------------------------------------------------------------
        NbCurves++;
        NewMin=to;
      }//if(to<NewMax && to>NewMin)
      }// for(i2=1; i2<= (nbsolZ2+nbsolZ2) ; ++i2)
      //
      if(NoChanges) {
      TheCurve[NbCurves].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1,  myEpsilon,
                                   Theta1,Theta2,
                                   UN_SEUL_Z_PAR_THETA, Z_NEGATIF);
      NbCurves++;
      TheCurve[NbCurves].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1, myEpsilon,
                                   Theta1,Theta2,
                                   UN_SEUL_Z_PAR_THETA, Z_POSITIF);
      //------------------------------------------------------------
      //-- A == 0    
      //-- Si B < 0    Alors Infini sur   Z+  
      //-- Sinon             Infini sur   Z-
      //------------------------------------------------------------
      if(PolZ2.IsARoot(Theta1)) {
        if(MTFZ1.Value(Theta1) < 0.) {
          TheCurve[NbCurves].SetIsFirstOpen(Standard_True);
        }
        else {
          TheCurve[NbCurves-1].SetIsFirstOpen(Standard_True);
        }
      }
      if(PolZ2.IsARoot(Theta2)) {
        if(MTFZ1.Value(Theta2) < 0.) {
          TheCurve[NbCurves].SetIsLastOpen(Standard_True);
        }
        else {
          TheCurve[NbCurves-1].SetIsLastOpen(Standard_True);
        }
      }
      //------------------------------------------------------------
      NbCurves++;
      }//if(NoChanges) {
      else {// #0
      TheCurve[NbCurves].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1, myEpsilon,
                                   NewMin,Theta2,
                                   UN_SEUL_Z_PAR_THETA, Z_NEGATIF);
      NbCurves++;
      TheCurve[NbCurves].SetConeQuadValues(Cone,Qxx,Qyy,Qzz,Qxy,Qxz,Qyz,Qx,Qy,Qz,Q1, myEpsilon,
                                   NewMin,Theta2,
                                   UN_SEUL_Z_PAR_THETA, Z_POSITIF);
      //------------------------------------------------------------
      //-- A == 0    
      //-- Si B < 0    Alors Infini sur   Z+  
      //-- Sinon             Infini sur   Z-
      //------------------------------------------------------------
      if(PolZ2.IsARoot(NewMin)) {
        if(MTFZ1.Value(NewMin) < 0.) {
          TheCurve[NbCurves].SetIsFirstOpen(Standard_True);
        }
        else {
          TheCurve[NbCurves-1].SetIsFirstOpen(Standard_True);
        }
      }
      if(PolZ2.IsARoot(Theta2)) {
        if(MTFZ1.Value(Theta2) < 0.) {
          TheCurve[NbCurves].SetIsLastOpen(Standard_True);
        }
        else {
          TheCurve[NbCurves-1].SetIsLastOpen(Standard_True);
        }
      }
      //------------------------------------------------------------
      
      NbCurves++;
      }//else {// #0
    }//else { //#1
  }//for(i=1; i<=nbsol ; i++) {
  //}//else { //#5
  InternalSetNextAndPrevious();
}
//=======================================================================
//function :InternalSetNextAndPrevious
//purpose  : 
//-- Raccordement des courbes bout a bout 
//--    - Utilisation du champ : IsFirstOpen 
//--    -                        IsLastOpen
//-- Pas de verification si ces champs retournent Vrai.
//--
//--
//-- 1: Test de      Fin(C1)    = Debut(C2)   ->N(C1) et P(C2)
//-- 2:              Debut(C1)  = Fin(C2)     ->P(C1) et N(C2)
//=======================================================================
00972 void IntAna_IntQuadQuad::InternalSetNextAndPrevious() 
{
  Standard_Boolean NotLastOpenC2, NotFirstOpenC2;
  Standard_Integer c1,c2;
  Standard_Real aEps, aEPSILON_DISTANCE;
  Standard_Real DInfC1,DSupC1, DInfC2,DSupC2;
  //
  aEps=0.0000001;
  aEPSILON_DISTANCE=0.0000000001;
  //
  for(c1=0; c1<NbCurves; c1++) {
    nextcurve[c1] =0;
    previouscurve[c1] = 0;
  }
  //
  for(c1=0; c1 < NbCurves; c1++) {
    TheCurve[c1].Domain(DInfC1,DSupC1);
    
    for(c2 = 0; (c2 < NbCurves) && (c2!=c1) ; c2++) {
      
      NotLastOpenC2  = ! TheCurve[c2].IsLastOpen();
      NotFirstOpenC2 = ! TheCurve[c2].IsFirstOpen();
      TheCurve[c2].Domain(DInfC2,DSupC2);
      if(TheCurve[c1].IsFirstOpen() == Standard_False) {
      if(NotLastOpenC2) {
        if(Abs(DInfC1-DSupC2)<=aEps && 
           (TheCurve[c1].Value(DInfC1)
            .Distance(TheCurve[c2].Value(DSupC2))<aEPSILON_DISTANCE)) {
          previouscurve[c1] = c2+1;
          nextcurve[c2]     = c1+1;
        }
      }
      if(NotFirstOpenC2) {
        if(Abs(DInfC1-DInfC2)<=aEps
           && (TheCurve[c1].Value(DInfC1)
             .Distance(TheCurve[c2].Value(DInfC2))<aEPSILON_DISTANCE)) {
          previouscurve[c1] = -(c2+1);
          previouscurve[c2] = -(c1+1);
        }
      }
      }
      if(TheCurve[c1].IsLastOpen() == Standard_False) {
      if(NotLastOpenC2) {
        if(Abs(DSupC1-DSupC2)<=aEps
           && (TheCurve[c1].Value(DSupC1)
             .Distance(TheCurve[c2].Value(DSupC2))<aEPSILON_DISTANCE)) {
          
          nextcurve[c1] = -(c2+1);
          nextcurve[c2] = -(c1+1);
        }
      }
      if(NotFirstOpenC2) {
        if(Abs(DSupC1-DInfC2)<=aEps 
           && (TheCurve[c1].Value(DSupC1) 
             .Distance(TheCurve[c2].Value(DInfC2))<aEPSILON_DISTANCE)) { 
          nextcurve[c1]     = c2+1;
          previouscurve[c2] = c1+1;
        }
      }
      }
    }
  }
}
//=======================================================================
//function :HasPreviousCurve
//purpose  : 
//=======================================================================    
01039 Standard_Boolean IntAna_IntQuadQuad::HasPreviousCurve(const Standard_Integer I) const
{
  if(!done) {
    StdFail_NotDone::Raise("IntQuadQuad Not done");
  }  
  if (identical) {
    Standard_DomainError::Raise("IntQuadQuad identical");
  }
  if((I>NbCurves)||(I<=0)) {
    Standard_OutOfRange::Raise("Incorrect Curve Number 'HasPrevious Curve'");
  }
  if(previouscurve[I-1]) {
    return Standard_True;
  }
  return Standard_False;
}
//=======================================================================
//function :HasNextCurve
//purpose  : 
//=======================================================================    
01059 Standard_Boolean IntAna_IntQuadQuad::HasNextCurve(const Standard_Integer I) const
{
  if(!done) {
    StdFail_NotDone::Raise("IntQuadQuad Not done");
  }  
  if (identical) {
    Standard_DomainError::Raise("IntQuadQuad identical");
  }
  if((I>NbCurves)||(I<=0)) {
    Standard_OutOfRange::Raise("Incorrect Curve Number 'HasNextCurve'");
  }
  if(nextcurve[I-1]) {
    return Standard_True;
  }
  return(Standard_False);
}
//=======================================================================
//function :PreviousCurve
//purpose  : 
//=======================================================================     
01079 Standard_Integer IntAna_IntQuadQuad::PreviousCurve  (const Standard_Integer I,
                                         Standard_Boolean& Opposite) const
{
  if(HasPreviousCurve(I)) {
    if(previouscurve[I-1]>0) {
      Opposite = Standard_False;
      return(previouscurve[I-1]);
    }
    else {
      Opposite = Standard_True;
      return( - previouscurve[I-1]);
    }
  }
  else {
    Standard_DomainError::Raise("Incorrect Curve Number 'PreviousCurve'"); return(0);
  }
}
//=======================================================================
//function :NextCurve
//purpose  : 
//=======================================================================    
01100 Standard_Integer IntAna_IntQuadQuad::NextCurve (const Standard_Integer I,
                                    Standard_Boolean& Opposite) const
{
  if(HasNextCurve(I)) {
    if(nextcurve[I]>0) {
      Opposite = Standard_False;
      return(nextcurve[I-1]);
    }
    else {
      Opposite = Standard_True;
      return( - nextcurve[I-1]);
    }
  }
  else {
    Standard_DomainError::Raise("Incorrect Curve Number 'NextCurve'"); 
    return(0);
  }
}
//=======================================================================
//function :Curve
//purpose  : 
//=======================================================================       
01122 const IntAna_Curve& IntAna_IntQuadQuad::Curve(const Standard_Integer i) const
{
  if(!done) {
    StdFail_NotDone::Raise("IntQuadQuad Not done");
  }
  if (identical) {
    Standard_DomainError::Raise("IntQuadQuad identical");
  }
  if((i <= 0) || (i>NbCurves)) {
    Standard_OutOfRange::Raise("Incorrect Curve Number");
  }
  return TheCurve[i-1];
}
//=======================================================================
//function :Point
//purpose  : 
//=======================================================================    
01139 const gp_Pnt& IntAna_IntQuadQuad::Point (const Standard_Integer i) const
{
  if(!done) {
    StdFail_NotDone::Raise("IntQuadQuad Not done");
  }  
  if (identical) {
    Standard_DomainError::Raise("IntQuadQuad identical");
  }
  if((i <= 0) || (i>Nbpoints)) {
    Standard_OutOfRange::Raise("Incorrect Point Number");
  }
  return Thepoints[i-1];
}
//=======================================================================
//function :Parameters
//purpose  : 
//=======================================================================    
01156 void IntAna_IntQuadQuad::Parameters (const Standard_Integer, //i, 
                             Standard_Real& , 
                             Standard_Real& ) const
{
  cout << "IntAna_IntQuadQuad::Parameters(...) is not yet implemented" << endl;
}

/*********************************************************************************

Mathematica Pour Cone Quadrique 
In[6]:= y[r_,t_]:=r*Sin[t]

In[7]:= x[r_,t_]:=r*Cos[t]

In[8]:= z[r_,t_]:=r*d

In[14]:= Quad[x_,y_,z_]:=Qxx x x + Qyy y y + Qzz z z + 2 (Qxy x y + Qxz x z + Qyz y z + Qx x + Qy y + Qz z ) + Q1

In[15]:= Quad[x[r,t],y[r,t],z[r,t]]

               2      2        2       2        2       2
Out[15]= Q1 + d  Qzz r  + Qxx r  Cos[t]  + Qyy r  Sin[t]  + 
 
                                      2
>    2 (d Qz r + Qx r Cos[t] + d Qxz r  Cos[t] + Qy r Sin[t] + 
 
               2               2
>       d Qyz r  Sin[t] + Qxy r  Cos[t] Sin[t])

In[16]:= QQ=%



In[17]:= Collect[QQ,r]
Collect[QQ,r]

Out[17]= Q1 + r (2 d Qz + 2 Qx Cos[t] + 2 Qy Sin[t]) + 
 
      2   2                                  2
>    r  (d  Qzz + 2 d Qxz Cos[t] + Qxx Cos[t]  + 2 d Qyz Sin[t] + 
 
                                        2
>       2 Qxy Cos[t] Sin[t] + Qyy Sin[t] )
********************************************************************************/


  //**********************************************************************
  //***                   C O N E   - Q U A D R I Q U E                ***
  //***   2    2                                  2                    ***
  //***  R  ( d  Qzz + 2 d Qxz Cos[t] + Qxx Cos[t]  + 2 d Qyz Sin[t] + ***
  //***                                                                ***
  //***       2 Qxy Cos[t] Sin[t] + Qyy Sin[t] )                       ***
  //***                                                                ***
  //*** + R  ( 2 d Qz + 2 Qx Cos[t] + 2 Qy Sin[t] )                    ***
  //***                                                                ***
  //*** + Q1                                                           ***
  //**********************************************************************
  //FortranForm= ( DIS = QQ1 QQ1 - 4 QQ0 QQ2  ) / 4 
  //   -  d**2*Qz**2 - d**2*Qzz*Q1 + (Qx**2 - Qxx*Q1)*Cos(t)**2 + 
  //   -   (2*d*Qy*Qz - 2*d*Qyz*Q1)*Sin(t) + (Qy**2 - Qyy*Q1)*Sin(t)**2 + 
  //   -   Cos(t)*(2*d*Qx*Qz - 2*d*Qxz*Q1 + (2*Qx*Qy - 2*Qxy*Q1)*Sin(t))
  //**********************************************************************
//modified by NIZNHY-PKV Fri Dec  2 10:56:03 2005f
/*
static
  void DumpCurve(const Standard_Integer aIndex,
             IntAna_Curve& aC);
//=======================================================================
//function : DumpCurve
//purpose  : 
//=======================================================================
void DumpCurve(const Standard_Integer aIndex,
             IntAna_Curve& aC)
{
  Standard_Boolean bIsOpen, bIsConstant, bIsFirstOpen, bIsLastOpen;
  Standard_Integer i, aNb;
  Standard_Real aT1, aT2, aT, dT;
  gp_Pnt aP;
  //
  aC.Domain(aT1, aT2);
  bIsOpen=aC.IsOpen();
  bIsConstant=aC.IsConstant();
  bIsFirstOpen=aC.IsFirstOpen();
  bIsLastOpen=aC.IsLastOpen();
  //
  printf("\n");
  printf(" * IntAna_Curve #%d*\n", aIndex);
  printf(" Domain: [ %lf, %lf ]\n", aT1, aT2);
  printf(" IsOpen=%d\n", bIsOpen);
  printf(" IsConstant=%d\n", bIsConstant);
  printf(" IsFirstOpen =%d\n", bIsFirstOpen);
  printf(" IsLastOpen =%d\n", bIsLastOpen);
  //
  aNb=11;
  dT=(aT2-aT1)/(aNb-1);
  for (i=0; i<aNb; ++i) {
    aT=aT1+i*dT;
    if (i==(aNb-1)){
      aT=aT2;
    }
    //
    aP=aC.Value(aT);
    printf("point p%d_%d %lf %lf %lf\n", 
         aIndex, i, aP.X(), aP.Y(), aP.Z());
  }
  printf(" ---- end of curve ----\n");
}
*/
//modified by NIZNHY-PKV Fri Dec  2 10:42:16 2005t

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