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Extrema_ExtElC.cxx

#include <stdio.h>

#include <Extrema_ExtElC.ixx>
#include <StdFail_InfiniteSolutions.hxx>
#include <StdFail_NotDone.hxx>
#include <ElCLib.hxx>
#include <math_TrigonometricFunctionRoots.hxx>
#include <math_DirectPolynomialRoots.hxx>
#include <Standard_OutOfRange.hxx>
#include <Standard_NotImplemented.hxx>
#include <Precision.hxx>
#include <Extrema_ExtPElC.hxx>
#include <IntAna_QuadQuadGeo.hxx>
#include <Extrema_ExtPElC.hxx>
#include <gp_Pln.hxx>
#include <gp_Ax3.hxx>
#include <gp_Ax2.hxx>





//======================================================================
//==  Classe Interne (Donne des racines classees d un polynome trigo)
//==  Code duplique avec IntAna_IntQuadQuad.cxx (lbr le 26 mars 98)
//==  Solution fiable aux problemes de coefficients proches de 0 
//==  avec essai de rattrapage si coeff<1.e-10 (jct le 27 avril 98) 
//======================================================================
static const Standard_Real PIpPI=Standard_PI+Standard_PI;

class ExtremaExtElC_TrigonometricRoots {
 private:
  Standard_Real Roots[4];
  Standard_Boolean done;
  Standard_Integer NbRoots;
  Standard_Boolean infinite_roots;
 public:
  ExtremaExtElC_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_Boolean IsDone() { return(done); }

  Standard_Boolean IsARoot(Standard_Real u) {
    for(Standard_Integer i=0 ; i<NbRoots; i++) {
      if(Abs(u - Roots[i])<=RealEpsilon()) return(Standard_True);
      if(Abs(u - Roots[i]-PIpPI)<=RealEpsilon()) return(Standard_True);
    }
    return(Standard_False);
  }

  Standard_Integer NbSolutions() { 
    if(!done) StdFail_NotDone::Raise();
    return(NbRoots); 
  }
  Standard_Boolean InfiniteRoots() { 
    if(!done) StdFail_NotDone::Raise();
    return(infinite_roots); 
  }
  Standard_Real    Value(const Standard_Integer& n) {
    if((!done)||(n>NbRoots)) StdFail_NotDone::Raise();
    return(Roots[n-1]);
  }
}; 
//----------------------------------------------------------------------
ExtremaExtElC_TrigonometricRoots::ExtremaExtElC_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 ;
  Standard_Integer nbessai = 1;
  Standard_Real cc = CC, sc = SC, c = C, s = S, cte = Cte;
  done=Standard_False;
  while (nbessai<=2 && !done) {
    //-- 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()) {
      done=Standard_True;
      if(MTFR.InfiniteRoots()) {
      infinite_roots=Standard_True;
      }
      else {
      NbRoots=MTFR.NbSolutions();
      for( i=0;i<NbRoots;i++) {
        Roots[i]=MTFR.Value(i+1);
        if(Roots[i]<0.0) Roots[i]+=PI+PI;
        if(Roots[i]>(PI+PI)) Roots[i]-=PI+PI;
      }
      Standard_Boolean Triee;
      Standard_Integer j;
      
      //-- La recherche directe donne n importe quoi. 
      // modified by OCC  Tue Oct  3 18:41:27 2006.BEGIN
      Standard_Real aMaxCoef = Max(CC,SC);
      aMaxCoef = Max(aMaxCoef,C);
      aMaxCoef = Max(aMaxCoef,S);
      aMaxCoef = Max(aMaxCoef,Cte);
      const Standard_Real aPrecision = Max(1.e-8,1.e-12*aMaxCoef);
      // modified by OCC  Tue Oct  3 18:41:33 2006.END

      Standard_Integer 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;
        // modified by OCC  Tue Oct  3 18:43:00 2006
        if(Abs(y)>aPrecision) {
#ifdef DEB
          printf("\n**IntAna_IntQuadQuad** Solution : %g ( %g cos2 + 2  %g cos sin + %g cos + %g sin  + %g) = %g\n",
               Roots[i],CC,SC,C,S,Cte,y);
#endif
          NbRoots--;
          Roots[i]=1000.0;
        }
      }
      
      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==0) { //--!!!!! 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;
          }
        }
      }
      }
    }
    else {
      // on essaie en mettant les tres petits coeff. a ZERO
      if (Abs(CC)<1e-10) cc = 0.0;
      if (Abs(SC)<1e-10) sc = 0.0;
      if (Abs(C)<1e-10) c = 0.0;
      if (Abs(S)<1e-10) s = 0.0;
      if (Abs(Cte)<1e-10) cte = 0.0;
      nbessai++;
    }
  }
}

//=============================================================================

Extrema_ExtElC::Extrema_ExtElC () { myDone = Standard_False; }
//=============================================================================

00164 Extrema_ExtElC::Extrema_ExtElC (const gp_Lin& C1, const gp_Lin& C2,
                        const Standard_Real)
// Fonction:
//    Recherche de la distance minimale entre 2 droites.

// Methode:
//   Soit D1 et D2, les 2 directions des droites C1 et C2.
//   2 cas sont consideres:
//   1- si Angle(D1,D2) < AngTol, les droites sont paralleles.
//      La distance est la distance entre un point quelconque de C1 et la droite
//      C2.
//   2- si Angle(D1,D2) > AngTol:
//      Soit P1=C1(u1) et P2=C2(u2) les 2 points solutions:
//      Alors, ( P1P2.D1 = 0. (1)
//             ( P1P2.D2 = 0. (2)
//   Soit O1 et O2 les origines de C1 et C2;
//   Alors, (1) <=> (O1P2-u1*D1).D1 = 0.       car O1P1 = u1*D1
//         <=> u1 = O1P2.D1               car D1.D1 = 1.
//          (2) <=> (P1O2+u2*D2).D2 = 0.       car O2P2 = u2*D2
//         <=> u2 = O2P1.D2               car D2.D2 = 1.
//         <=> u2 = (O2O1+O1P1).D2
//         <=> u2 = O2O1.D2+((O1P2.T1)T1).T2) car O1P1 = u1*T1 = (O1P2.T1)T1
//         <=> u2 = O2O1.D2+(((O1O2+O2P2).D1)D1).D2)
//         <=> u2 = O2O1.D2+((O1O2.D1)D1).D2)+(O2P2.D1)(D1.D2)
//         <=> u2 = ((O1O2.D1)D1-O1O2).D2 + u2*(D2.D1)(D1.D2)
//         <=> u2 = (((O1O2.D1)D1-O1O2).D2) / 1.-(D1.D2)**2
{
  myDone = Standard_False;
  myNbExt = 0;

  gp_Dir D1 = C1.Position().Direction();
  gp_Dir D2 = C2.Position().Direction();
  // MSV 16/01/2000: avoid "divide by zero"
  Standard_Real D1DotD2 = D1.Dot(D2);
  Standard_Real aSin = 1.-D1DotD2*D1DotD2;
  if (aSin < gp::Resolution() ||
      D1.IsParallel(D2, Precision::Angular())) {
    myIsPar = Standard_True;
    myValue[0] = C2.Distance(C1.Location());
    myValue[1] = myValue[0];
  }
  else {
    myIsPar = Standard_False;
    gp_Pnt O1 = C1.Location();
    gp_Pnt O2 = C2.Location();
    gp_Vec O1O2 (O1,O2);
    Standard_Real U2 = (D1.XYZ()*(O1O2.Dot(D1))-(O1O2.XYZ())).Dot(D2.XYZ());
    if( Precision::IsInfinite(U2) ) {
      myIsPar = Standard_True;
      myValue[0] = C2.Distance(C1.Location());
      myValue[1] = myValue[0];
    }
    else {
      U2 /= aSin;
      if( Precision::IsInfinite(U2) ) {
      myIsPar = Standard_True;
      myValue[0] = C2.Distance(C1.Location());
      myValue[1] = myValue[0];
      }
      else {
      gp_Pnt P2(ElCLib::Value(U2,C2));
      Standard_Real U1 = (gp_Vec(O1,P2)).Dot(D1);
      if( Precision::IsInfinite(U1) ) {
        myIsPar = Standard_True;
        myValue[0] = C2.Distance(C1.Location());
        myValue[1] = myValue[0];
      }
      else {
        gp_Pnt P1(ElCLib::Value(U1,C1));
        myValue[myNbExt] = P1.Distance(P2);
        myPoint[myNbExt][0] = Extrema_POnCurv(U1,P1);
        myPoint[myNbExt][1] = Extrema_POnCurv(U2,P2);
        myNbExt = 1;
      }
      }
    }
  }
  myDone = Standard_True;
}
//=============================================================================

00245 Extrema_ExtElC::Extrema_ExtElC (const gp_Lin& C1, const gp_Circ& C2,
                        const Standard_Real)
/*-----------------------------------------------------------------------------
Fonction:
   Recherche des distances extremales entre la droite C1 et le cercle C2.

Methode:
   Soit P1=C1(u1) et P2=C2(u2) deux points solutions
        D la direction de la droite C1
      T la tangente au point P2;
  Alors, ( P1P2.D = 0. (1)
         ( P1P2.T = 0. (2)
  Soit O1 et O2 les origines de C1 et C2;
  Alors, (1) <=> (O1P2-u1*D).D = 0.         car O1P1 = u1*D
           <=> u1 = O1P2.D                car D.D = 1.
         (2) <=> P1O2.T = 0.                car O2P2.T = 0.
             <=> ((P2O1.D)D+O1O2).T = 0.    car P1O1 = -u1*D = (P2O1.D)D
           <=> (((P2O2+O2O1).D)D+O1O2).T = 0.
           <=> ((P2O2.D)(D.T)+((O2O1.D)D-O2O1).T = 0.
  On se place dans le repere du cercle; soit:
         Cos = Cos(u2) et Sin = Sin(u2),
         P2 (R*Cos,R*Sin,0.),
         T (-R*Sin,R*Cos,0.),
       D (Dx,Dy,Dz),
       V (Vx,Vy,Vz) = (O2O1.D)D-O2O1;
  Alors, on obtient l'equation en Cos et Sin suivante:
    -(2*R*R*Dx*Dy)   * Cos**2  +       A1
   R*R*(Dx**2-Dy**2) * Cos*Sin +    2* A2
         R*Vy        * Cos     +       A3
      -R*Vx        * Sin     +       A4
      R*R*Dx*Dy                = 0.    A5
     On utilise l'algorithme math_TrigonometricFunctionRoots pour resoudre
    cette equation.
-----------------------------------------------------------------------------*/
{
  myIsPar = Standard_False;
  myDone = Standard_False;
  myNbExt = 0;

// Calcul de T1 dans le repere du cercle ...
  gp_Dir D = C1.Direction();
  gp_Dir D1 = D;
  gp_Dir x2, y2, z2;
  x2 = C2.XAxis().Direction();
  y2 = C2.YAxis().Direction();
  z2 = C2.Axis().Direction();
  Standard_Real Dx = D.Dot(x2);
  Standard_Real Dy = D.Dot(y2);
  Standard_Real Dz = D.Dot(z2);
  D.SetCoord(Dx,Dy,Dz);

// Calcul de V dans le repere du cercle:
  gp_Pnt O1 = C1.Location();
  gp_Pnt O2 = C2.Location();
  gp_Vec O2O1 (O2,O1);
  O2O1.SetCoord(O2O1.Dot(x2), O2O1.Dot(y2), O2O1.Dot(z2));
  gp_XYZ Vxyz = (D.XYZ()*(O2O1.Dot(D)))-O2O1.XYZ();

// Calcul des coefficients de l equation en Cos et Sin ...
  Standard_Real R = C2.Radius();
  Standard_Real A5 = R*R*Dx*Dy;
  Standard_Real A1 = -2.*A5;
  Standard_Real A2 = R*R*(Dx*Dx-Dy*Dy)/2.0;
  Standard_Real A3 = R*Vxyz.Y();
  Standard_Real A4 = -R*Vxyz.X();
 // Standard_Real TolU = Tol * R;

  
  if(fabs(A5) <= 1.e-12) A5 = 0.;
  if(fabs(A1) <= 1.e-12) A1 = 0.;
  if(fabs(A2) <= 1.e-12) A2 = 0.;
  if(fabs(A3) <= 1.e-12) A3 = 0.;
  if(fabs(A4) <= 1.e-12) A4 = 0.;
  

  ExtremaExtElC_TrigonometricRoots Sol(A1,A2,A3,A4,A5,0.,PI+PI);
  if (!Sol.IsDone()) { return; }
  if (Sol.InfiniteRoots()) { 
    myIsPar = Standard_True;
    myValue[0] = R;
    myDone = Standard_True;
    return; 
  }
// Stockage des solutions ...
  gp_Pnt P1,P2;
  Standard_Real U1,U2;
  Standard_Integer NbSol = Sol.NbSolutions();
  for (Standard_Integer NoSol = 1; NoSol <= NbSol; NoSol++) {
    U2 = Sol.Value(NoSol);
    P2 = ElCLib::Value(U2,C2);
    U1 = (gp_Vec(O1,P2)).Dot(D1);
    P1 = ElCLib::Value(U1,C1);
    myValue[myNbExt] = P1.Distance(P2);
    myPoint[myNbExt][0] = Extrema_POnCurv(U1,P1);
    myPoint[myNbExt][1] = Extrema_POnCurv(U2,P2);
    myNbExt++;
  }
  myDone = Standard_True;
}



00347 Extrema_ExtElC::Extrema_ExtElC (const gp_Lin& C1, const gp_Elips& C2)
{
/*-----------------------------------------------------------------------------
Fonction:
   Recherche des distances extremales entre la droite C1 et l ellipse C2.

Methode:
   Soit P1=C1(u1) et P2=C2(u2) deux points solutions
        D la direction de la droite C1
      T la tangente au point P2;
  Alors, ( P1P2.D = 0. (1)
         ( P1P2.T = 0. (2)
  Soit O1 et O2 les origines de C1 et C2;
  Alors, (1) <=> (O1P2-u1*D).D = 0.         car O1P1 = u1*D
           <=> u1 = O1P2.D                car D.D = 1.
         (2) <=> P1O2.T = 0.                car O2P2.T = 0.
             <=> ((P2O1.D)D+O1O2).T = 0.    car P1O1 = -u1*D = (P2O1.D)D
           <=> (((P2O2+O2O1).D)D+O1O2).T = 0.
           <=> ((P2O2.D)(D.T)+((O2O1.D)D-O2O1).T = 0.
  On se place dans le repere de l ellipse; soit:
         Cos = Cos(u2) et Sin = Sin(u2),
         P2 (MajR*Cos,MinR*Sin,0.),
         T (-MajR*Sin,MinR*Cos,0.),
       D (Dx,Dy,Dz),
       V (Vx,Vy,Vz) = (O2O1.D)D-O2O1;
  Alors, on obtient l'equation en Cos et Sin suivante:
    -(2*MajR*MinR*Dx*Dy)             * Cos**2  +
   (MajR*MajR*Dx**2-MinR*MinR*Dy**2) * Cos*Sin +
         MinR*Vy                     * Cos     +
       - MajR*Vx                     * Sin     +
      MinR*MajR*Dx*Dy                = 0.
     On utilise l'algorithme math_TrigonometricFunctionRoots pour resoudre
    cette equation.
-----------------------------------------------------------------------------*/
  myIsPar = Standard_False;
  myDone = Standard_False;
  myNbExt = 0;

// Calcul de T1 dans le repere de l'ellipse ...
  gp_Dir D = C1.Direction();
  gp_Dir D1 = D;
  gp_Dir x2, y2, z2;
  x2 = C2.XAxis().Direction();
  y2 = C2.YAxis().Direction();
  z2 = C2.Axis().Direction();
  Standard_Real Dx = D.Dot(x2);
  Standard_Real Dy = D.Dot(y2);
  Standard_Real Dz = D.Dot(z2);
  D.SetCoord(Dx,Dy,Dz);

// Calcul de V ...
  gp_Pnt O1 = C1.Location();
  gp_Pnt O2 = C2.Location();
  gp_Vec O2O1 (O2,O1);
  O2O1.SetCoord(O2O1.Dot(x2), O2O1.Dot(y2), O2O1.Dot(z2));
  gp_XYZ Vxyz = (D.XYZ()*(O2O1.Dot(D)))-O2O1.XYZ();

// Calcul des coefficients de l equation en Cos et Sin ...
  Standard_Real MajR = C2.MajorRadius();
  Standard_Real MinR = C2.MinorRadius();
  Standard_Real A5 = MajR*MinR*Dx*Dy;
  Standard_Real A1 = -2.*A5;
  Standard_Real R2 = MajR*MajR;
  Standard_Real r2 = MinR*MinR;
  Standard_Real A2 =(R2*Dx*Dx -r2*Dy*Dy -R2 +r2)/2.0;
  Standard_Real A3 = MinR*Vxyz.Y();
  Standard_Real A4 = -MajR*Vxyz.X();

  ExtremaExtElC_TrigonometricRoots Sol(A1,A2,A3,A4,A5,0.,PI+PI);
  if (!Sol.IsDone()) { return; }

// Stockage des solutions ...
  gp_Pnt P1,P2;
  Standard_Real U1,U2;
  Standard_Integer NbSol = Sol.NbSolutions();
  for (Standard_Integer NoSol = 1; NoSol <= NbSol; NoSol++) {
    U2 = Sol.Value(NoSol);
    P2 = ElCLib::Value(U2,C2);
    U1 = (gp_Vec(O1,P2)).Dot(D1);
    P1 = ElCLib::Value(U1,C1);
    myValue[myNbExt] = P1.Distance(P2);
    myPoint[myNbExt][0] = Extrema_POnCurv(U1,P1);
    myPoint[myNbExt][1] = Extrema_POnCurv(U2,P2);
    myNbExt++;
  }
  myDone = Standard_True;
}
//=============================================================================

00436 Extrema_ExtElC::Extrema_ExtElC (const gp_Lin& C1, const gp_Hypr& C2)
{
/*-----------------------------------------------------------------------------
Fonction:
   Recherche des distances extremales entre la droite C1 et l'hyperbole C2.

Methode:
   Soit P1=C1(u1) et P2=C2(u2) deux points solutions
        D la direction de la droite C1
      T la tangente au point P2;
  Alors, ( P1P2.D = 0. (1)
         ( P1P2.T = 0. (2)
  Soit O1 et O2 les origines de C1 et C2;
  Alors, (1) <=> (O1P2-u1*D).D = 0.         car O1P1 = u1*D
           <=> u1 = O1P2.D                car D.D = 1.
         (2) <=> (P1O2 + O2P2).T= 0.
             <=> ((P2O1.D)D+O1O2 + O2P2).T = 0.  car P1O1 = -u1*D = (P2O1.D)D
           <=> (((P2O2+O2O1).D)D+O1O2 + O2P2).T = 0.
           <=> (P2O2.D)(D.T)+((O2O1.D)D-O2O1).T + O2P2.T= 0.
  On se place dans le repere de l'hyperbole; soit:
         en ecrivant P (R* Chu, r* Shu, 0.0)
       et Chu = (v**2 + 1)/(2*v) ,
          Shu = (V**2 - 1)/(2*v)

       T(R*Shu, r*Chu)
       D (Dx,Dy,Dz),
       V (Vx,Vy,Vz) = (O2O1.D)D-O2O1;

  Alors, on obtient l'equation en v suivante:
         (-2*R*r*Dx*Dy - R*R*Dx*Dx-r*r*Dy*Dy + R*R + r*r)     * v**4  +
       (2*R*Vx + 2*r*Vy)                                    * v**3  +
       (-2*R*Vx + 2*r*Vy)                                   * v     +
       (-2*R*r*Dx*Dy - (R*R*Dx*Dx-r*r*Dy*Dy + R*R + r*r))  = 0


     On utilise l'algorithme math_DirectPolynomialRoots pour resoudre
    cette equation.
-----------------------------------------------------------------------------*/
  myIsPar = Standard_False;
  myDone = Standard_False;
  myNbExt = 0;

// Calcul de T1 dans le repere de l'hyperbole ...
  gp_Dir D = C1.Direction();
  gp_Dir D1 = D;
  gp_Dir x2, y2, z2;
  x2 = C2.XAxis().Direction();
  y2 = C2.YAxis().Direction();
  z2 = C2.Axis().Direction();
  Standard_Real Dx = D.Dot(x2);
  Standard_Real Dy = D.Dot(y2);
  Standard_Real Dz = D.Dot(z2);
  D.SetCoord(Dx,Dy,Dz);

// Calcul de V ...
  gp_Pnt O1 = C1.Location();
  gp_Pnt O2 = C2.Location();
  gp_Vec O2O1 (O2,O1);
  O2O1.SetCoord(O2O1.Dot(x2), O2O1.Dot(y2), O2O1.Dot(z2));
  gp_XYZ Vxyz = (D.XYZ()*(O2O1.Dot(D)))-O2O1.XYZ();
  Standard_Real Vx = Vxyz.X();
  Standard_Real Vy = Vxyz.Y();

// Calcul des coefficients de l equation en v
  Standard_Real R = C2.MajorRadius();
  Standard_Real r = C2.MinorRadius();
  Standard_Real a = -2*R*r*Dx*Dy;
  Standard_Real b = -R*R*Dx*Dx - r*r*Dy*Dy + R*R + r*r;
  Standard_Real A1 = a + b;
  Standard_Real A2 = 2*R*Vx + 2*r*Vy;
  Standard_Real A4 = -2*R*Vx + 2*r*Vy;
  Standard_Real A5 = a - b;

  math_DirectPolynomialRoots Sol(A1,A2,0.0,A4, A5);
  if (!Sol.IsDone()) { return; }

// Stockage des solutions ...
  gp_Pnt P1,P2;
  Standard_Real U1,U2, v;
  Standard_Integer NbSol = Sol.NbSolutions();
  for (Standard_Integer NoSol = 1; NoSol <= NbSol; NoSol++) {
    v = Sol.Value(NoSol);
    if (v > 0.0) {
      U2 = Log(v);
      P2 = ElCLib::Value(U2,C2);
      U1 = (gp_Vec(O1,P2)).Dot(D1);
      P1 = ElCLib::Value(U1,C1);
      myValue[myNbExt] = P1.Distance(P2);
      myPoint[myNbExt][0] = Extrema_POnCurv(U1,P1);
      myPoint[myNbExt][1] = Extrema_POnCurv(U2,P2);
      myNbExt++;
    }
  }
  myDone = Standard_True;
}
//=============================================================================

00533 Extrema_ExtElC::Extrema_ExtElC (const gp_Lin& C1, const gp_Parab& C2)
{
/*-----------------------------------------------------------------------------
Fonction:
   Recherche des distances extremales entre la droite C1 et la parabole C2.

Methode:
   Soit P1=C1(u1) et P2=C2(u2) deux points solutions
        D la direction de la droite C1
      T la tangente au point P2;
  Alors, ( P1P2.D = 0. (1)
         ( P1P2.T = 0. (2)
  Soit O1 et O2 les origines de C1 et C2;
  Alors, (1) <=> (O1P2-u1*D).D = 0.         car O1P1 = u1*D
           <=> u1 = O1P2.D                car D.D = 1.
         (2) <=> (P1O2 + O2P2).T= 0.
             <=> ((P2O1.D)D+O1O2 + O2P2).T = 0.  car P1O1 = -u1*D = (P2O1.D)D
           <=> (((P2O2+O2O1).D)D+O1O2 + O2P2).T = 0.
           <=> (P2O2.D)(D.T)+((O2O1.D)D-O2O1).T + O2P2.T = 0.
  On se place dans le repere de la parabole; soit:
         P2 (y*y/(2*p), y, 0)
         T (y/p, 1, 0)
       D (Dx,Dy,Dz),
       V (Vx,Vy,Vz) = (O2O1.D)D-O2O1;

  Alors, on obtient l'equation en y suivante:
     ((1-Dx*Dx)/(2*p*p))            *  y*y*y  +        A1
     (-3*Dx*Dy/(2*p))               *  y*y    +        A2
     (1-Dy*Dy + Vx/p)               *  y      +        A3 
        Vy                          = 0.               A4

     On utilise l'algorithme math_DirectPolynomialRoots pour resoudre
    cette equation.
-----------------------------------------------------------------------------*/
  myIsPar = Standard_False;
  myDone = Standard_False;
  myNbExt = 0;

// Calcul de T1 dans le repere de la parabole ...
  gp_Dir D = C1.Direction();
  gp_Dir D1 = D;
  gp_Dir x2, y2, z2;
  x2 = C2.XAxis().Direction();
  y2 = C2.YAxis().Direction();
  z2 = C2.Axis().Direction();
  Standard_Real Dx = D.Dot(x2);
  Standard_Real Dy = D.Dot(y2);
  Standard_Real Dz = D.Dot(z2);
  D.SetCoord(Dx,Dy,Dz);

// Calcul de V ...
  gp_Pnt O1 = C1.Location();
  gp_Pnt O2 = C2.Location();
  gp_Vec O2O1 (O2,O1);
  O2O1.SetCoord(O2O1.Dot(x2), O2O1.Dot(y2), O2O1.Dot(z2));
  gp_XYZ Vxyz = (D.XYZ()*(O2O1.Dot(D)))-O2O1.XYZ();

// Calcul des coefficients de l equation en y
  Standard_Real P = C2.Parameter();
  Standard_Real A1 = (1-Dx*Dx)/(2.0*P*P);
  Standard_Real A2 = (-3.0*Dx*Dy/(2.0*P));
  Standard_Real A3 = (1 - Dy*Dy + Vxyz.X()/P);
  Standard_Real A4 = Vxyz.Y();

  math_DirectPolynomialRoots Sol(A1,A2,A3,A4);
  if (!Sol.IsDone()) { return; }

// Stockage des solutions ...
  gp_Pnt P1,P2;
  Standard_Real U1,U2;
  Standard_Integer NbSol = Sol.NbSolutions();
  for (Standard_Integer NoSol = 1; NoSol <= NbSol; NoSol++) {
    U2 = Sol.Value(NoSol);
    P2 = ElCLib::Value(U2,C2);
    U1 = (gp_Vec(O1,P2)).Dot(D1);
    P1 = ElCLib::Value(U1,C1);
    myValue[myNbExt] = P1.Distance(P2);
    myPoint[myNbExt][0] = Extrema_POnCurv(U1,P1);
    myPoint[myNbExt][1] = Extrema_POnCurv(U2,P2);
    myNbExt++;
  }
  myDone = Standard_True;
}
//=============================================================================

00618 Extrema_ExtElC::Extrema_ExtElC (const gp_Circ& C1, const gp_Circ& C2)
{

  myIsPar = Standard_False;
  myDone = Standard_False;
  myNbExt = 0;

  gp_Ax1 ax1 = C1.Axis();
  gp_Ax1 ax2 = C2.Axis();

  if(ax1.IsCoaxial(ax2, Precision::Angular(), Precision::Confusion())) { 
    myIsPar = Standard_True;
    Standard_Real dR = C1.Radius() - C2.Radius();
    Standard_Real dC = C1.Location().Distance(C2.Location());
    myValue[0] = Sqrt(dR*dR + dC*dC);
    dR = C1.Radius() + C2.Radius();
    myValue[1] = Sqrt(dR*dR + dC*dC);
  }

  myDone = Standard_True;   

}
//=============================================================================

00642 Extrema_ExtElC::Extrema_ExtElC (const gp_Circ&, const gp_Elips&)
{
  Standard_NotImplemented::Raise();
}
//=============================================================================

00648 Extrema_ExtElC::Extrema_ExtElC (const gp_Circ&, const gp_Hypr&)
{
  Standard_NotImplemented::Raise();
}
//=============================================================================

00654 Extrema_ExtElC::Extrema_ExtElC (const gp_Circ&, const gp_Parab&)
{
  Standard_NotImplemented::Raise();
}
//=============================================================================

00660 Extrema_ExtElC::Extrema_ExtElC (const gp_Elips&, const gp_Elips&)
{
  Standard_NotImplemented::Raise();
}
//=============================================================================

00666 Extrema_ExtElC::Extrema_ExtElC (const gp_Elips&, const gp_Hypr&)
{
  Standard_NotImplemented::Raise();
}
//=============================================================================

00672 Extrema_ExtElC::Extrema_ExtElC (const gp_Elips&, const gp_Parab&)
{
  Standard_NotImplemented::Raise();
}
//=============================================================================

00678 Extrema_ExtElC::Extrema_ExtElC (const gp_Hypr&, const gp_Hypr&)
{
  Standard_NotImplemented::Raise();
}
//=============================================================================

00684 Extrema_ExtElC::Extrema_ExtElC (const gp_Hypr&, const gp_Parab&)
{
  Standard_NotImplemented::Raise();
}
//=============================================================================

00690 Extrema_ExtElC::Extrema_ExtElC (const gp_Parab&, const gp_Parab&)
{
  Standard_NotImplemented::Raise();
}
//=============================================================================

00696 Standard_Boolean Extrema_ExtElC::IsDone () const { return myDone; }
//=============================================================================

00699 Standard_Boolean Extrema_ExtElC::IsParallel () const
{
  if (!IsDone()) { StdFail_NotDone::Raise(); }
  return myIsPar;
}
//=============================================================================

00706 Standard_Integer Extrema_ExtElC::NbExt () const
{
  if (IsParallel()) { StdFail_InfiniteSolutions::Raise(); }
  return myNbExt;
}
//=============================================================================

00713 Standard_Real Extrema_ExtElC::Value (const Standard_Integer N) const
{
  if (!myDone) { StdFail_NotDone::Raise(); }
  if (myIsPar) {
    if (N < 1 || N > 2) { Standard_OutOfRange::Raise(); }
  }
  else {  
    if (N < 1 || N > NbExt()) { Standard_OutOfRange::Raise(); }
  }
  return myValue[N-1];
}
//=============================================================================

00726 void Extrema_ExtElC::Points (const Standard_Integer N,
                       Extrema_POnCurv& P1, Extrema_POnCurv& P2) const
{
  if (N < 1 || N > NbExt()) { Standard_OutOfRange::Raise(); }
  P1 = myPoint[N-1][0];
  P2 = myPoint[N-1][1];
}
//=============================================================================

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