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BSplCLib_2.cxx

// File:    BSplCLib_2.cxx
// Created: Tue Jan 17 14:17:48 1995
// Author:  Laurent BOURESCHE
//          <lbo@phylox>
// xab : modified 15-Mar-95 : added BuildBSplMatrix,FactorBandedMatrix,
//                            EvalBsplineBasis,
//                            EvalPolynomial : Horners method

#define No_Standard_RangeError
#define No_Standard_OutOfRange

#include <Standard_Stream.hxx>

#include <BSplCLib.hxx>
#include <gp_Mat2d.hxx>
#include <PLib.hxx>
#include <Standard_ConstructionError.hxx>
#include <TColStd_Array1OfReal.hxx>
#include <TColStd_Array1OfInteger.hxx>
#include <TColStd_HArray1OfReal.hxx>
#include <TColStd_HArray1OfInteger.hxx>

#include <math_Matrix.hxx>

// static arrays for temporary computations

static Standard_Integer locpolessize = 0;
static Standard_Real *locpoles;
static Standard_Integer locknotssize = 0;
static Standard_Real *locknots;
static Standard_Integer locderssize = 0;
static Standard_Real *locders;

static void LocalArray(const Standard_Integer newsize,
                   Standard_Integer& size,
                   Standard_Real** arr)
{
  // update a local array
  if (newsize > size) {
    if (*arr) delete [] *arr;
    size = newsize;
    *arr = new Standard_Real [size];
  }
}

// methods for 1 dimensional BSplines

//=======================================================================
//function : BuildEval
//purpose  : builds the local array for evaluation
//=======================================================================

void  BSplCLib::BuildEval(const Standard_Integer         Degree,
                    const Standard_Integer         Index,
                    const TColStd_Array1OfReal&    Poles, 
                    const TColStd_Array1OfReal&    Weights,
                    Standard_Real&                 LP)
{
  Standard_Integer PLower = Poles.Lower();
  Standard_Integer PUpper = Poles.Upper();
  Standard_Integer i;
  Standard_Integer ip = PLower + Index - 1;
  Standard_Real w, *pole = &LP;
  if (&Weights == NULL) {
    
    for (i = 0; i <= Degree; i++) {
      ip++;
      if (ip > PUpper) ip = PLower;
      pole[0] = Poles(ip);
      pole += 1;
    }
  }
  else {
    
    for (i = 0; i <= Degree; i++) {
      ip++;
      if (ip > PUpper) ip = PLower;
      pole[1] = w = Weights(ip);
      pole[0] = Poles(ip) * w;
      pole += 2;
    }
  }
}

//=======================================================================
//function : PrepareEval
//purpose  : stores data for Eval in the local arrays
//           locpoles and locknots
//=======================================================================

static void PrepareEval
(Standard_Real&                 u,                  
 Standard_Integer&              index, 
 Standard_Integer&              dim,
 Standard_Boolean&              rational,
 const Standard_Integer         Degree,     
 const Standard_Boolean         Periodic,
 const TColStd_Array1OfReal&    Poles,  
 const TColStd_Array1OfReal&    Weights,
 const TColStd_Array1OfReal&    Knots,  
 const TColStd_Array1OfInteger& Mults) 
{                    
  // Set the Index
  BSplCLib::LocateParameter(Degree,Knots,Mults,u,Periodic,index,u);
  
  // make the knots
  LocalArray(Degree << 1, locknotssize, &locknots);
  BSplCLib::BuildKnots(Degree,index,Periodic,Knots,Mults,*locknots);
  if (&Mults == NULL)
    index -= Knots.Lower() + Degree;
  else
    index = BSplCLib::PoleIndex(Degree,index,Periodic,Mults);
  
  // check truly rational
  rational = (&Weights != NULL);
  if (rational) {
    Standard_Integer WLower = Weights.Lower() + index;
    rational = BSplCLib::IsRational(Weights, WLower, WLower + Degree);
  }
  
  // make the poles
  if(rational) {
    dim = 2;
    LocalArray((Degree + 1) << 1, locpolessize, &locpoles);
    BSplCLib::BuildEval(Degree, index, Poles, Weights              , *locpoles);
  }
  else {
    dim = 1;
    LocalArray((Degree + 1)     , locpolessize, &locpoles);
    BSplCLib::BuildEval(Degree, index, Poles, BSplCLib::NoWeights(), *locpoles);
  }
}

//=======================================================================
//function : D0
//purpose  : 
//=======================================================================

void BSplCLib::D0 
(const Standard_Real            U,                  
 const Standard_Integer         Index,          
 const Standard_Integer         Degree,     
 const Standard_Boolean         Periodic,
 const TColStd_Array1OfReal&    Poles,  
 const TColStd_Array1OfReal&    Weights,
 const TColStd_Array1OfReal&    Knots,  
 const TColStd_Array1OfInteger& Mults,  
 Standard_Real&                 P) 
{                    
  Standard_Integer dim,index = Index;
  Standard_Real    u = U;
  Standard_Boolean rational;
  PrepareEval(u,index,dim,rational,Degree,Periodic,Poles,Weights,Knots,Mults);
  BSplCLib::Eval(u,Degree,*locknots,dim,*locpoles);
  if (rational) P = locpoles[0] / locpoles[1];
  else          P = locpoles[0];
}

//=======================================================================
//function : D1
//purpose  : 
//=======================================================================

void BSplCLib::D1
(const Standard_Real            U,                  
 const Standard_Integer         Index,          
 const Standard_Integer         Degree,     
 const Standard_Boolean         Periodic,
 const TColStd_Array1OfReal&    Poles,  
 const TColStd_Array1OfReal&    Weights,
 const TColStd_Array1OfReal&    Knots,  
 const TColStd_Array1OfInteger& Mults,  
 Standard_Real&                 P,
 Standard_Real&                 V) 
{                    
  Standard_Integer dim,index = Index;
  Standard_Real    u = U;
  Standard_Boolean rational;
  PrepareEval(u,index,dim,rational,Degree,Periodic,Poles,Weights,Knots,Mults);
  BSplCLib::Bohm(u,Degree,1,*locknots,dim,*locpoles);
  Standard_Real *result = locpoles;
  if (rational) {
    LocalArray(2,locderssize,&locders);
    PLib::RationalDerivative(Degree,1,1,*locpoles,*locders);
    result = locders;
  }
  P = result[0];
  V = result[1];
}

//=======================================================================
//function : D2
//purpose  : 
//=======================================================================

void BSplCLib::D2
(const Standard_Real            U,                  
 const Standard_Integer         Index,          
 const Standard_Integer         Degree,     
 const Standard_Boolean         Periodic,
 const TColStd_Array1OfReal&    Poles,  
 const TColStd_Array1OfReal&    Weights,
 const TColStd_Array1OfReal&    Knots,  
 const TColStd_Array1OfInteger& Mults,  
 Standard_Real&                 P,
 Standard_Real&                 V1,
 Standard_Real&                 V2) 
{                    
  Standard_Integer dim,index = Index;
  Standard_Real    u = U;
  Standard_Boolean rational;
  PrepareEval(u,index,dim,rational,Degree,Periodic,Poles,Weights,Knots,Mults);
  BSplCLib::Bohm(u,Degree,2,*locknots,dim,*locpoles);
  Standard_Real *result = locpoles;
  if (rational) {
    LocalArray(3,locderssize,&locders);
    PLib::RationalDerivative(Degree,2,1,*locpoles,*locders);
    result = locders;
  }
  P  = result[0];
  V1 = result[1];
  if (!rational && (Degree < 2)) V2 = 0.;
  else                           V2 = result[2];
}

//=======================================================================
//function : D3
//purpose  : 
//=======================================================================

void BSplCLib::D3
(const Standard_Real            U,                  
 const Standard_Integer         Index,          
 const Standard_Integer         Degree,     
 const Standard_Boolean         Periodic,
 const TColStd_Array1OfReal&    Poles,  
 const TColStd_Array1OfReal&    Weights,
 const TColStd_Array1OfReal&    Knots,  
 const TColStd_Array1OfInteger& Mults,  
 Standard_Real&                 P,
 Standard_Real&                 V1,
 Standard_Real&                 V2,
 Standard_Real&                 V3) 
{                    
  Standard_Integer dim,index = Index;
  Standard_Real    u = U;
  Standard_Boolean rational;
  PrepareEval(u,index,dim,rational,Degree,Periodic,Poles,Weights,Knots,Mults);
  BSplCLib::Bohm(u,Degree,3,*locknots,dim,*locpoles);
  Standard_Real *result = locpoles;
  if (rational) {
    LocalArray(4,locderssize,&locders);
    PLib::RationalDerivative(Degree,3,1,*locpoles,*locders);
    result = locders;
  }
  P  = result[0];
  V1 = result[1];
  if (!rational && (Degree < 2)) V2 = 0.;
  else                           V2 = result[2];
  if (!rational && (Degree < 3)) V3 = 0.;
  else                           V3 = result[3];
}

//=======================================================================
//function : DN
//purpose  : 
//=======================================================================

void BSplCLib::DN
(const Standard_Real            U,                  
 const Standard_Integer         N,          
 const Standard_Integer         Index,          
 const Standard_Integer         Degree,     
 const Standard_Boolean         Periodic,
 const TColStd_Array1OfReal&    Poles,  
 const TColStd_Array1OfReal&    Weights,
 const TColStd_Array1OfReal&    Knots,  
 const TColStd_Array1OfInteger& Mults,  
 Standard_Real&                 VN) 
{                    
  Standard_Integer dim,index = Index;
  Standard_Real    u = U;
  Standard_Boolean rational;
  PrepareEval(u,index,dim,rational,Degree,Periodic,Poles,Weights,Knots,Mults);
  BSplCLib::Bohm(u,Degree,N,*locknots,dim,*locpoles);
  if (rational) {
    Standard_Real v;
    PLib::RationalDerivative(Degree,N,1,*locpoles,v,Standard_False);
    VN = v;
  }
  else {
    if (N > Degree) VN = 0.;
    else            VN = locpoles[N];
  }
}

//=======================================================================
//function : Build BSpline Matrix
//purpose  : Builds the Bspline Matrix
//=======================================================================

Standard_Integer 
00303 BSplCLib::BuildBSpMatrix(const  TColStd_Array1OfReal&     Parameters,
                   const  TColStd_Array1OfInteger&  ContactOrderArray,
                   const  TColStd_Array1OfReal&     FlatKnots,
                   const  Standard_Integer          Degree,   
                   math_Matrix&                     Matrix,
                   Standard_Integer&                UpperBandWidth,
                   Standard_Integer&                LowerBandWidth) 
{
  Standard_Integer ii,
  jj,
  Index,
  ErrorCode,
  ReturnCode = 0,
  FirstNonZeroBsplineIndex,
  BandWidth,
  MaxOrder = 21,
  Order ;
  
  math_Matrix   BSplineBasis(1, MaxOrder,
                             1, MaxOrder) ;
  
  Order = Degree + 1 ;
  UpperBandWidth = Degree ;
  LowerBandWidth = Degree ;
  BandWidth = UpperBandWidth + LowerBandWidth + 1 ;
  if (Matrix.LowerRow() != Parameters.Lower()  || 
      Matrix.UpperRow() != Parameters.Upper()  ||
      Matrix.LowerCol() != 1  || 
      Matrix.UpperCol() != BandWidth) {
    ReturnCode = 1;
    goto FINISH ;
  }
  
  for (ii = Parameters.Lower() ; ii <= Parameters.Upper() ; ii++) {
    ErrorCode =
      BSplCLib::EvalBsplineBasis(1,
                         ContactOrderArray(ii),
                         Order,
                         FlatKnots,
                         Parameters(ii),
                         
                         FirstNonZeroBsplineIndex,
                         BSplineBasis) ;
    if (ErrorCode != 0) {
      ReturnCode = 2 ;
      goto FINISH ;
    }
    Index = LowerBandWidth + 1 + FirstNonZeroBsplineIndex - ii ;

    for (jj = 1 ; jj < Index ; jj++) {
      Matrix.Value(ii,jj) = 0.0e0 ;
    }

    for (jj = 1 ; jj <= Order ; jj++) {
      Matrix.Value(ii,Index) = BSplineBasis(ContactOrderArray(ii) + 1, jj) ;
      Index += 1 ;
    }
    
    for (jj = Index ; jj <= BandWidth ; jj++) {
      Matrix.Value(ii,jj) = 0.0e0 ;
    }
  } 
  FINISH : ;
  return (ReturnCode) ;
}

//=======================================================================
//function : Makes LU decompositiomn without Pivoting
//purpose  : Builds the Bspline Matrix
//=======================================================================

Standard_Integer 
00375 BSplCLib::FactorBandedMatrix(math_Matrix&   Matrix,
                       const Standard_Integer UpperBandWidth,
                       const Standard_Integer LowerBandWidth,
                       Standard_Integer&  PivotIndexProblem) 
{
  Standard_Integer ii,
  jj,
  kk,
  Index,
  MinIndex,
  MaxIndex,
  ReturnCode = 0,
  BandWidth = UpperBandWidth + LowerBandWidth + 1 ;
  
  Standard_Real Inverse ;
  PivotIndexProblem = 0 ;

  for (ii = Matrix.LowerRow() + 1 ; ii <= Matrix.UpperRow() ; ii++) {
    MinIndex = ( LowerBandWidth - ii + 2 >= 1 ? LowerBandWidth - ii + 2 : 1) ;

    for (jj = MinIndex ; jj <= LowerBandWidth  ; jj++) {
      Index = ii - LowerBandWidth + jj - 1 ; 
      Inverse = Matrix(Index ,LowerBandWidth + 1) ;
      if (Abs(Inverse) > RealSmall()) {
      Inverse = -1.0e0/Inverse ;
      }
      else {
      ReturnCode = 1 ;
      PivotIndexProblem = Index ;
      goto FINISH ;
      }
      Matrix(ii,jj) = Matrix(ii,jj) * Inverse ;
      MaxIndex = BandWidth + Index - ii ;

      for (kk = jj + 1 ; kk <= MaxIndex ; kk++) {
      Matrix(ii,kk) += Matrix(ii,jj) * Matrix(Index, kk + ii - Index) ;
      }
    }                 
  }
  FINISH :
    return (ReturnCode) ;
}

//=======================================================================
//function : Build BSpline Matrix
//purpose  : Builds the Bspline Matrix
//=======================================================================

Standard_Integer 
BSplCLib::EvalBsplineBasis
//(const Standard_Integer              Side, // = 1 rigth side, -1 left side 
00426 (const Standard_Integer              , // = 1 rigth side, -1 left side 
 const  Standard_Integer              DerivativeRequest,
 const  Standard_Integer              Order,
 const  TColStd_Array1OfReal&         FlatKnots,
 const  Standard_Real                 Parameter,
 Standard_Integer&             FirstNonZeroBsplineIndex,
 math_Matrix&                  BsplineBasis)
{
  // the matrix must have at least DerivativeRequest + 1
  //   row and Order columns
  // the result are stored in the following way in
  // the Bspline matrix 
  // Let i be the FirstNonZeroBsplineIndex and 
  // t be the parameter value, k the order of the 
  // knot vector, r the DerivativeRequest :
  //   
  //   B (t)   B (t)                     B (t)
  //    i       i+1                       i+k-1
  //   
  //    (1)     (1)                       (1) 
  //   B (t)   B (t)                     B (t)
  //    i       i+1                       i+k-1
  //  
  //
  //
  //
  //    (r)     (r)                       (r) 
  //   B (t)   B (t)                     B (t)
  //    i       i+1                       i+k-1
  //
  Standard_Integer  
    ReturnCode,
  ii,
  pp,
  qq,
  ss,
  NumPoles,
  LocalRequest ;
//  ,Index ;
  
  Standard_Real NewParameter,
  Inverse,
  Factor,
  LocalInverse,
  Saved ;
// , *FlatKnotsArray ;
  
  ReturnCode = 0 ;
  FirstNonZeroBsplineIndex = 0 ;
  LocalRequest = DerivativeRequest ;
  if (DerivativeRequest >= Order) {
    LocalRequest = Order - 1 ;
  }
  
  if (BsplineBasis.LowerCol() != 1    ||
      BsplineBasis.UpperCol() < Order ||
      BsplineBasis.LowerRow() != 1    ||
      BsplineBasis.UpperRow() <= LocalRequest) {
    ReturnCode = 1;
    goto FINISH ;
  }
  NumPoles = FlatKnots.Upper() - FlatKnots.Lower() + 1 - Order ;
  BSplCLib::LocateParameter(Order - 1, 
                      FlatKnots,
                      Parameter,
                      Standard_False, 
                      Order, 
                      NumPoles+1, 
                      ii,
                            NewParameter) ;
  
  FirstNonZeroBsplineIndex = ii - Order + 1 ;
  
  BsplineBasis(1,1) = 1.0e0 ;
  LocalRequest = DerivativeRequest ;
  if (DerivativeRequest >= Order) {
    LocalRequest = Order - 1 ;
  }

  for (qq = 2 ; qq <= Order - LocalRequest ; qq++) {
    BsplineBasis(1,qq) = 0.0e0 ;
    
    for (pp = 1 ; pp <= qq - 1 ; pp++) {
      //
      // this should be always invertible if ii is correctly computed 
      //
      Factor = (Parameter - FlatKnots(ii - qq + pp + 1)) 
      / (FlatKnots(ii + pp)   - FlatKnots(ii - qq + pp + 1)) ; 
      Saved = Factor *    BsplineBasis(1,pp) ;
      BsplineBasis(1,pp) *= (1.0e0 - Factor) ;
      BsplineBasis(1,pp) += BsplineBasis(1,qq) ;
      BsplineBasis(1,qq) = Saved ;
    }
  }
  
  for (qq = Order - LocalRequest + 1 ; qq <= Order ; qq++) {
    
    for (pp = 1 ; pp <= qq - 1 ; pp++) {
      BsplineBasis(Order - qq + 2,pp) = BsplineBasis(1,pp) ;
    }
    BsplineBasis(1,qq) = 0.0e0 ;

    for (ss = Order - LocalRequest + 1 ; ss <= qq ; ss++) {
      BsplineBasis(Order - ss + 2,qq) = 0.0e0 ;
    }

    for (pp = 1 ; pp <= qq - 1 ; pp++) {
      Inverse = 1.0e0 / (FlatKnots(ii + pp)  - FlatKnots(ii - qq + pp + 1)) ;
      Factor  =  (Parameter - FlatKnots(ii - qq + pp + 1)) * Inverse ;
      Saved = Factor *                 BsplineBasis(1,pp) ;
      BsplineBasis(1,pp) *= (1.0e0 - Factor) ;
      BsplineBasis(1,pp) += BsplineBasis(1,qq) ;
      BsplineBasis(1,qq) = Saved ;
      LocalInverse = (Standard_Real) (qq - 1) * Inverse ;

      for (ss = Order - LocalRequest + 1 ; ss <= qq ; ss++) {
      Saved = LocalInverse * BsplineBasis(Order - ss + 2, pp) ;
      BsplineBasis(Order - ss + 2, pp) *= - LocalInverse  ;
      BsplineBasis(Order - ss + 2, pp) +=   BsplineBasis(Order - ss + 2,qq) ;
      BsplineBasis(Order - ss + 2,qq) = Saved ;
      }
    }
  }
  FINISH :
    return (ReturnCode) ;
}

//=======================================================================
//function : MovePointAndTangent
//purpose  : 
//=======================================================================

00558 void BSplCLib::MovePointAndTangent(const Standard_Real    U,
                           const Standard_Integer ArrayDimension,
                           Standard_Real    &Delta,
                           Standard_Real    &DeltaDerivatives,
                           const Standard_Real    Tolerance,
                           const Standard_Integer Degree,
                           const Standard_Boolean Rational,
                           const Standard_Integer StartingCondition,
                           const Standard_Integer EndingCondition,
                           Standard_Real&         Poles,
                           const TColStd_Array1OfReal&   Weights,
                           const TColStd_Array1OfReal&   FlatKnots,
                           Standard_Real&        NewPoles,
                           Standard_Integer&     ErrorStatus) 
{
  Standard_Integer num_poles,
  num_knots,
  ii,
  jj,
  conditions,
  start_num_poles,
  end_num_poles,
  index,
  start_index,
  end_index,
  other_index,
  type,
  order ;
  
  Standard_Real    new_parameter,
  value,
  divide,
  end_value,
  start_value,
  *poles_array,
  *new_poles_array,
  *delta_array,
  *derivatives_array,
  *weights_array ;
  
  ErrorStatus = 0 ;
  weights_array = NULL ;
  if (Rational) {
    weights_array = (Standard_Real *) &Weights(Weights.Lower()) ;
  }
  
  poles_array = &Poles ;
  new_poles_array = &NewPoles ;
  delta_array = &Delta ;
  derivatives_array = &DeltaDerivatives ;
  order = Degree + 1 ;
  num_knots = FlatKnots.Length() ;
  num_poles = num_knots - order ;
  conditions = StartingCondition + EndingCondition + 4 ;
  //
  // check validity of input data
  //
  if (StartingCondition >= -1 &&  
      StartingCondition <= Degree &&
      EndingCondition >= -1 &&
      EndingCondition <= Degree &&
      conditions <= num_poles) {
    //
    // check the parameter is within bounds 
    //
    start_index = FlatKnots.Lower() + Degree ;
    end_index = FlatKnots.Upper() - Degree ;
    //
    //  check if there is enough room to move the poles
    //
    conditions = 1 ;
    if (StartingCondition == -1) {
      conditions = conditions && (FlatKnots(start_index) <=  U) ;
    }
    else {
      conditions = conditions && (FlatKnots(start_index) + Tolerance < U) ;
    }
    if (EndingCondition == -1) {
      conditions = conditions && (FlatKnots(end_index) >= U) ;
    }
    else {
      conditions = conditions && (FlatKnots(end_index) - Tolerance > U) ; 
    }
    
    if (conditions) {
      //
      // build 2 auxialiary functions 
      // 
      TColStd_Array1OfReal schoenberg_points(1,num_poles) ;
      TColStd_Array1OfReal first_function  (1,num_poles) ;
      TColStd_Array1OfReal second_function (1,num_poles) ;
      
      BuildSchoenbergPoints(Degree,
                      FlatKnots,
                      schoenberg_points) ;
      start_index = StartingCondition + 2 ;
      end_index = num_poles - EndingCondition - 1 ;
      LocateParameter(schoenberg_points,
                  U,
                  Standard_False,
                  start_index,
                  end_index,
                  index,
                  new_parameter, 
                  0, 1) ;
      
      if (index == start_index) {
      other_index = index + 1 ;
      } 
      else if (index == end_index) {
      other_index = index -1  ;
      }
      else if (U - FlatKnots(index) < FlatKnots(index + 1) - U ) {
      other_index = index - 1 ;
      }
      else {
      other_index = index + 1 ;
      }
      type = 3 ;
      
      start_num_poles = StartingCondition + 2 ;
      end_num_poles   = num_poles - EndingCondition - 1 ;
      if (start_num_poles == 1) {
      start_value = schoenberg_points(num_poles) - schoenberg_points(1) ;
      start_value = schoenberg_points(1) - start_value ;
      }
      else {
      start_value = schoenberg_points(start_num_poles - 1) ;
      }
      if (end_num_poles == num_poles) {
      end_value = schoenberg_points(num_poles) - schoenberg_points(1) ;
      end_value = schoenberg_points(num_poles) + end_value ;
      }
      else {
      end_value = schoenberg_points(end_num_poles + 1) ;
      }
      
      for (ii = 1 ; ii < start_num_poles ; ii++) {
      first_function(ii) = 0.e0 ;
      second_function(ii) = 0.0e0 ;
      }

      for (ii = end_num_poles + 1 ; ii <= num_poles ; ii++) {
      first_function(ii) = 0.e0 ;
      second_function(ii) = 0.0e0 ;
      }
      divide = 1.0e0 / (schoenberg_points(index) - start_value) ;

      for (ii = start_num_poles  ; ii <= index ; ii++) {
      value = schoenberg_points(ii) - start_value ;
      value *= divide ;
      first_function(ii) = 1.0e0 ;
      
      for (jj = 0 ; jj < type ; jj++) {
        first_function(ii) *= value ;
      }
      }
      divide = 1.0e0 /(end_value - schoenberg_points(index)) ;

      for (ii = index ; ii <= end_num_poles ; ii++) {
      value = end_value - schoenberg_points(ii) ;
        value *= divide ;
      first_function(ii) = 1.0e0 ;

      for (jj = 0 ; jj < type ; jj++) {
        first_function(ii) *= value ;
      }
      }
      
      divide = 1.0e0 / (schoenberg_points(other_index) - start_value) ;

      for (ii = start_num_poles  ; ii <= other_index ; ii++) {
      value = schoenberg_points(ii) - start_value ;
        value *= divide ;
      second_function(ii) = 1.0e0 ;

      for (jj = 0 ; jj < type ; jj++) {
        second_function(ii) *= value ;
      }
      }
      divide = 1.0e0/( end_value - schoenberg_points(other_index)) ;

      for (ii = other_index ; ii <= end_num_poles ; ii++) {
      value = end_value - schoenberg_points(ii) ;
      value *= divide ;
      second_function(ii) = 1.0e0 ;

      for (jj = 0 ; jj < type ; jj++) {
        second_function(ii) *= value ;
      }
      }

      //
      //  compute the point and derivatives of both functions
      //    
      Standard_Real results[2][2],
      weights_results[2][2];
      Standard_Integer extrap_mode[2],
      derivative_request = 1,
      dimension = 1 ;
      Standard_Boolean periodic_flag = Standard_False ;
      
      extrap_mode[0] = Degree ;
      extrap_mode[1] = Degree ;
      if (Rational) {
      //
      // evaluate in homogenised form
      //
      Eval(U,
           periodic_flag,
           derivative_request,
           extrap_mode[0],
           Degree,
           FlatKnots,
           dimension,
           first_function(1),
           weights_array[0],
           results[0][0],
           weights_results[0][0]) ;
      
      Eval(U,
           periodic_flag,
           derivative_request,
           extrap_mode[0],
           Degree,
           FlatKnots,
           dimension,
           second_function(1),
           weights_array[0],
           results[1][0],
           weights_results[1][0]) ;
      //
      //  compute the rational derivatives values
      //       

      for (ii = 0 ; ii < 2 ; ii++) {
        PLib::RationalDerivatives(1,
                            1,
                            results[ii][0],
                            weights_results[ii][0],
                            results[ii][0]) ;
      }
      }
      else {
      Eval(U,
           Standard_False,
           1,
           extrap_mode[0],
           Degree,
           FlatKnots,
           1,
           first_function(1),
           results[0][0]) ;
      
      Eval(U,
           Standard_False,
           1,
           extrap_mode[0],
           Degree,
           FlatKnots,
           1,
           second_function(1),
           results[1][0]) ;
      }
      gp_Mat2d  a_matrix ;

      for (ii = 0 ; ii < 2 ; ii++) {

      for (jj = 0 ; jj < 2 ; jj++) {
        a_matrix.SetValue(ii+1,jj+1,results[ii][jj]) ;
      }
      }
      a_matrix.Invert() ;
      TColStd_Array1OfReal the_a_vector(0,ArrayDimension-1) ;
      TColStd_Array1OfReal the_b_vector(0,ArrayDimension-1) ;

      for( ii = 0 ; ii < ArrayDimension ; ii++) {
      the_a_vector(ii) = 
        a_matrix.Value(1,1) * delta_array[ii] +
          a_matrix.Value(2,1) * derivatives_array[ii] ;
      the_b_vector(ii) =
        a_matrix.Value(1,2) * delta_array[ii] +
          a_matrix.Value(2,2) * derivatives_array[ii] ;
      }
      index = 0 ;
      
      for (ii = 0 ; ii < num_poles ; ii++) {
      
      for (jj = 0 ; jj < ArrayDimension ; jj++) {
        new_poles_array[index] = poles_array[index] ; 
        new_poles_array[index] += 
          first_function(ii+1) * the_a_vector(jj) ;
        new_poles_array[index] += 
          second_function(ii+1) * the_b_vector(jj) ;
        index += 1 ;
      }
      }
    }
    else {
      ErrorStatus = 1 ;
    }
  }
  else {
    ErrorStatus = 2 ;
  }
}

//=======================================================================
//function : FunctionMultiply
//purpose  : 
//=======================================================================

void BSplCLib::FunctionMultiply
00871 (const BSplCLib_EvaluatorFunction & FunctionPtr,
 const Standard_Integer             BSplineDegree,
 const TColStd_Array1OfReal &       BSplineFlatKnots,
 const Standard_Integer             PolesDimension,
 Standard_Real &                    Poles,
 const TColStd_Array1OfReal &       FlatKnots,
 const Standard_Integer             NewDegree,
 Standard_Real &                    NewPoles,
 Standard_Integer &                 Status) 
{
  Standard_Integer ii,
  jj,
  index ;
  Standard_Integer extrap_mode[2],
  error_code,
  num_new_poles,
  derivative_request = 0 ;
  Standard_Boolean  periodic_flag = Standard_False ;
  Standard_Real  result,
  start_end[2],
  *array_of_poles,
  *array_of_new_poles ;
  
  array_of_poles = (Standard_Real *) &NewPoles ;
  extrap_mode[0] = 
    extrap_mode[1] = BSplineDegree ;
  num_new_poles =
    FlatKnots.Length() - NewDegree - 1 ;
  start_end[0] = FlatKnots(NewDegree+1) ;
  start_end[1] = FlatKnots(num_new_poles+1) ;
  TColStd_Array1OfReal  parameters(1,num_new_poles) ;
  TColStd_Array1OfInteger contact_order_array(1,num_new_poles) ;
  TColStd_Array1OfReal  new_poles_array(1, num_new_poles * PolesDimension) ;
  
  array_of_new_poles = 
    (Standard_Real *) &new_poles_array(1) ;
  BuildSchoenbergPoints(NewDegree,
                  FlatKnots,
                  parameters) ;
  //
  // on recadre sur les bornes
  // 
  if (parameters(1) < start_end[0]) {
    parameters(1) = start_end[0] ;
  }
  if (parameters(num_new_poles) > start_end[1]) {
    parameters(num_new_poles) = start_end[1] ;
  }
  index = 0 ; 

  for (ii = 1 ; ii <= num_new_poles ; ii++) {
    contact_order_array(ii) = 0 ;
    (*FunctionPtr)(contact_order_array(ii),
               start_end,
               parameters(ii),
               result,
               error_code) ;
    if (error_code) {
      Status = 1 ;
      goto FINISH ;
    }     
    
    Eval(parameters(ii),
       periodic_flag,
       derivative_request,
       extrap_mode[0],
       BSplineDegree,
       BSplineFlatKnots,
       PolesDimension,
       Poles,
       array_of_new_poles[index]) ;
    
    for (jj = 0 ; jj < PolesDimension ; jj++) {
      array_of_new_poles[index] *= result ;
      index += 1 ;
    }
  }
  Interpolate(NewDegree,
            FlatKnots,
            parameters,
            contact_order_array,
            PolesDimension,
            array_of_new_poles[0],
            Status) ;
  
  for (ii = 0 ; ii < num_new_poles * PolesDimension ; ii++) {
    array_of_poles[ii] = array_of_new_poles[ii] ;
    
  }
  FINISH :
    ;
}

//=======================================================================
// function : FunctionMultiply
//purpose  : 
//=======================================================================

void BSplCLib::FunctionReparameterise
00970 (const BSplCLib_EvaluatorFunction & FunctionPtr,
 const Standard_Integer             BSplineDegree,
 const TColStd_Array1OfReal &       BSplineFlatKnots,
 const Standard_Integer             PolesDimension,
 Standard_Real &                    Poles,
 const TColStd_Array1OfReal &       FlatKnots,
 const Standard_Integer             NewDegree,
 Standard_Real &                    NewPoles,
 Standard_Integer &                 Status) 
{
  Standard_Integer ii,
//  jj,
  index ;
  Standard_Integer extrap_mode[2],
  error_code,
  num_new_poles,
  derivative_request = 0 ;
  Standard_Boolean  periodic_flag = Standard_False ;
  Standard_Real  result,
  start_end[2],
  *array_of_poles,
  *array_of_new_poles ;
  
  array_of_poles = (Standard_Real *) &NewPoles ;
  extrap_mode[0] = 
    extrap_mode[1] = BSplineDegree ;
  num_new_poles =
    FlatKnots.Length() - NewDegree - 1 ;
  start_end[0] = FlatKnots(NewDegree+1) ;
  start_end[1] = FlatKnots(num_new_poles+1) ;
  TColStd_Array1OfReal  parameters(1,num_new_poles) ;
  TColStd_Array1OfInteger contact_order_array(1,num_new_poles) ;
  TColStd_Array1OfReal  new_poles_array(1, num_new_poles * PolesDimension) ;
  
  array_of_new_poles = 
    (Standard_Real *) &new_poles_array(1) ;
  BuildSchoenbergPoints(NewDegree,
                  FlatKnots,
                  parameters) ;
  index = 0 ; 

  for (ii = 1 ; ii <= num_new_poles ; ii++) {
    contact_order_array(ii) = 0 ;
    (*FunctionPtr)(contact_order_array(ii),
               start_end,
               parameters(ii),
               result,
               error_code) ;
    if (error_code) {
      Status = 1 ;
      goto FINISH ;
    }     
    
    Eval(result,
       periodic_flag,
       derivative_request,
       extrap_mode[0],
       BSplineDegree,
       BSplineFlatKnots,
       PolesDimension,
       Poles,
       array_of_new_poles[index]) ;
    index += PolesDimension ;
  }
  Interpolate(NewDegree,
            FlatKnots,
            parameters,
            contact_order_array,
            PolesDimension,
            array_of_new_poles[0],
            Status) ;
  
  for (ii = 0 ; ii < num_new_poles * PolesDimension ; ii++) {
    array_of_poles[ii] = array_of_new_poles[ii] ;
    
  }
  FINISH :
    ;
}

//=======================================================================
//function : FunctionMultiply
//purpose  : 
//=======================================================================

void BSplCLib::FunctionMultiply
01056 (const BSplCLib_EvaluatorFunction & FunctionPtr,
 const Standard_Integer              BSplineDegree,
 const TColStd_Array1OfReal &        BSplineFlatKnots,
 const TColStd_Array1OfReal &        Poles,
 const TColStd_Array1OfReal &        FlatKnots,
 const Standard_Integer              NewDegree,
 TColStd_Array1OfReal &              NewPoles,
 Standard_Integer &                  Status) 
{ 
  Standard_Integer num_bspline_poles =
    BSplineFlatKnots.Length() - BSplineDegree - 1 ;
  Standard_Integer num_new_poles =
    FlatKnots.Length() - NewDegree - 1 ;
  
  if (Poles.Length() != num_bspline_poles ||
      NewPoles.Length() != num_new_poles) {
    Standard_ConstructionError::Raise();  
  }
  Standard_Real  * array_of_poles =
    (Standard_Real *) &Poles(Poles.Lower()) ;
  Standard_Real  * array_of_new_poles =
    (Standard_Real *) &NewPoles(NewPoles.Lower()) ;
  BSplCLib::FunctionMultiply(FunctionPtr,
                       BSplineDegree,
                       BSplineFlatKnots,
                       1,
                       array_of_poles[0],
                       FlatKnots,
                       NewDegree,
                       array_of_new_poles[0],
                       Status) ;
}

//=======================================================================
//function : FunctionReparameterise
//purpose  : 
//=======================================================================

void BSplCLib::FunctionReparameterise
01095 (const BSplCLib_EvaluatorFunction & FunctionPtr,
 const Standard_Integer              BSplineDegree,
 const TColStd_Array1OfReal &        BSplineFlatKnots,
 const TColStd_Array1OfReal &        Poles,
 const TColStd_Array1OfReal &        FlatKnots,
 const Standard_Integer              NewDegree,
 TColStd_Array1OfReal &              NewPoles,
 Standard_Integer &                  Status) 
{ 
  Standard_Integer num_bspline_poles =
    BSplineFlatKnots.Length() - BSplineDegree - 1 ;
  Standard_Integer num_new_poles =
    FlatKnots.Length() - NewDegree - 1 ;
  
  if (Poles.Length() != num_bspline_poles ||
      NewPoles.Length() != num_new_poles) {
    Standard_ConstructionError::Raise();  
  }
  Standard_Real  * array_of_poles =
    (Standard_Real *) &Poles(Poles.Lower()) ;
  Standard_Real  * array_of_new_poles =
    (Standard_Real *) &NewPoles(NewPoles.Lower()) ;
  BSplCLib::FunctionReparameterise(
                           FunctionPtr,
                           BSplineDegree,
                           BSplineFlatKnots,
                           1,
                           array_of_poles[0],
                           FlatKnots,
                           NewDegree,
                           array_of_new_poles[0],
                           Status) ;
}

//=======================================================================
//function : MergeBSplineKnots
//purpose  : 
//=======================================================================
void BSplCLib::MergeBSplineKnots
01134 (const Standard_Real                 Tolerance,
 const Standard_Real                 StartValue,
 const Standard_Real                 EndValue,
 const Standard_Integer              Degree1,
 const TColStd_Array1OfReal          & Knots1,
 const TColStd_Array1OfInteger       & Mults1,
 const Standard_Integer              Degree2,
 const TColStd_Array1OfReal          & Knots2,
 const TColStd_Array1OfInteger       & Mults2,
 Standard_Integer &                  NumPoles,
 Handle(TColStd_HArray1OfReal) &     NewKnots,
 Handle(TColStd_HArray1OfInteger) &  NewMults) 
{
  Standard_Integer ii,
  jj,
  continuity,
  set_mults_flag,
  degree,
  index,
  num_knots ;
  if (StartValue < EndValue - Tolerance) {
    TColStd_Array1OfReal knots1(1,Knots1.Length()) ;
    TColStd_Array1OfReal knots2(1,Knots2.Length()) ;
    degree = Degree1 + Degree2 ;
    index = 1 ;

    for (ii = Knots1.Lower() ; ii <= Knots1.Upper() ; ii++) {
      knots1(index) = Knots1(ii) ;
      index += 1 ;
    }
    index = 1 ;

    for (ii = Knots2.Lower() ; ii <= Knots2.Upper() ; ii++) {
      knots2(index) = Knots2(ii) ;
      index += 1 ;
    }
    BSplCLib::Reparametrize(StartValue,
                      EndValue,
                      knots1) ;
    
    BSplCLib::Reparametrize(StartValue,
                      EndValue,
                      knots2) ;
    num_knots = 0 ;
    jj =  1 ;

    for (ii = 1 ; ii <= knots1.Length() ; ii++) {

      while (jj <= knots2.Length() && knots2(jj) <= knots1(ii) - Tolerance) {
      jj += 1 ;
      num_knots += 1 ;
      }

      while (jj <= knots2.Length() && knots2(jj) <= knots1(ii) + Tolerance) {
      jj += 1 ;
      }
      num_knots += 1 ;
    }
    NewKnots = 
      new TColStd_HArray1OfReal(1,num_knots) ;
    NewMults =
      new TColStd_HArray1OfInteger(1,num_knots) ;
    num_knots = 1 ;
    jj = 1 ;

    for (ii = 1 ; ii <= knots1.Length() ; ii++) {

      while (jj <= knots2.Length() && knots2(jj) <= knots1(ii) - Tolerance) {
      NewKnots->ChangeArray1()(num_knots) = knots2(jj) ;
        NewMults->ChangeArray1()(num_knots) = Mults2(jj) + Degree1 ;
      jj += 1 ;
      num_knots += 1 ;
      }
      set_mults_flag = 0 ;

      while (jj <= knots2.Length() && knots2(jj) <= knots1(ii) + Tolerance) {
      continuity = Min(Degree1 - Mults1(ii), Degree2 - Mults2(jj)) ;
        set_mults_flag = 1 ;
      NewMults->ChangeArray1()(num_knots) = degree - continuity ;
      jj += 1 ;
      }

      NewKnots->ChangeArray1()(num_knots) = knots1(ii) ;
      if (! set_mults_flag) {
      NewMults->ChangeArray1()(num_knots) = Mults1(ii) + Degree2 ;
      }
      num_knots += 1 ;
    }
    num_knots -= 1 ;
    NewMults->ChangeArray1()(1) = degree + 1 ;
    NewMults->ChangeArray1()(num_knots) = degree + 1 ;
    index = 0 ;

    for (ii = 1 ; ii <= num_knots ; ii++) {
      index += NewMults->Value(ii) ;
    }
    NumPoles = index  - degree - 1  ;
  }
}
      

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