occt/src/Geom/Geom_BSplineCurve.hxx

// Created on: 1993-03-09
// Created by: JCV
// Copyright (c) 1993-1999 Matra Datavision
// Copyright (c) 1999-2014 OPEN CASCADE SAS
//
// This file is part of Open CASCADE Technology software library.
//
// This library is free software; you can redistribute it and/or modify it under
// the terms of the GNU Lesser General Public License version 2.1 as published
// by the Free Software Foundation, with special exception defined in the file
// OCCT_LGPL_EXCEPTION.txt. Consult the file LICENSE_LGPL_21.txt included in OCCT
// distribution for complete text of the license and disclaimer of any warranty.
//
// Alternatively, this file may be used under the terms of Open CASCADE
// commercial license or contractual agreement.

#ifndef _Geom_BSplineCurve_HeaderFile
#define _Geom_BSplineCurve_HeaderFile

#include <Standard.hxx>
#include <Standard_Type.hxx>

#include <Standard_Boolean.hxx>
#include <GeomAbs_BSplKnotDistribution.hxx>
#include <GeomAbs_Shape.hxx>
#include <Standard_Integer.hxx>
#include <TColgp_HArray1OfPnt.hxx>
#include <TColStd_HArray1OfReal.hxx>
#include <TColStd_HArray1OfInteger.hxx>
#include <Standard_Real.hxx>
#include <Geom_BoundedCurve.hxx>
#include <TColgp_Array1OfPnt.hxx>
#include <TColStd_Array1OfReal.hxx>
#include <TColStd_Array1OfInteger.hxx>
class Standard_ConstructionError;
class Standard_DimensionError;
class Standard_DomainError;
class Standard_OutOfRange;
class Standard_RangeError;
class Standard_NoSuchObject;
class Geom_UndefinedDerivative;
class gp_Pnt;
class gp_Vec;
class gp_Trsf;
class Geom_Geometry;


class Geom_BSplineCurve;
DEFINE_STANDARD_HANDLE(Geom_BSplineCurve, Geom_BoundedCurve)

//! Definition of the B_spline curve.
//! A B-spline curve can be
//! Uniform  or non-uniform
//! Rational or non-rational
//! Periodic or non-periodic
//!
//! a b-spline curve is defined by :
//! its degree; the degree for a
//! Geom_BSplineCurve is limited to a value (25)
//! which is defined and controlled by the system.
//! This value is returned by the function MaxDegree;
//! - its periodic or non-periodic nature;
//! - a table of poles (also called control points), with
//! their associated weights if the BSpline curve is
//! rational. The poles of the curve are "control
//! points" used to deform the curve. If the curve is
//! non-periodic, the first pole is the start point of
//! the curve, and the last pole is the end point of
//! the curve. The segment which joins the first pole
//! to the second pole is the tangent to the curve at
//! its start point, and the segment which joins the
//! last pole to the second-from-last pole is the
//! tangent to the curve at its end point. If the curve
//! is periodic, these geometric properties are not
//! verified. It is more difficult to give a geometric
//! signification to the weights but are useful for
//! providing exact representations of the arcs of a
//! circle or ellipse. Moreover, if the weights of all the
//! poles are equal, the curve has a polynomial
//! equation; it is therefore a non-rational curve.
//! - a table of knots with their multiplicities. For a
//! Geom_BSplineCurve, the table of knots is an
//! increasing sequence of reals without repetition;
//! the multiplicities define the repetition of the knots.
//! A BSpline curve is a piecewise polynomial or
//! rational curve. The knots are the parameters of
//! junction points between two pieces. The
//! multiplicity Mult(i) of the knot Knot(i) of
//! the BSpline curve is related to the degree of
//! continuity of the curve at the knot Knot(i),
//! which is equal to Degree - Mult(i)
//! where Degree is the degree of the BSpline curve.
//! If the knots are regularly spaced (i.e. the difference
//! between two consecutive knots is a constant), three
//! specific and frequently used cases of knot
//! distribution can be identified:
//! - "uniform" if all multiplicities are equal to 1,
//! - "quasi-uniform" if all multiplicities are equal to 1,
//! except the first and the last knot which have a
//! multiplicity of Degree + 1, where Degree is
//! the degree of the BSpline curve,
//! - "Piecewise Bezier" if all multiplicities are equal to
//! Degree except the first and last knot which
//! have a multiplicity of Degree + 1, where
//! Degree is the degree of the BSpline curve. A
//! curve of this type is a concatenation of arcs of Bezier curves.
//! If the BSpline curve is not periodic:
//! - the bounds of the Poles and Weights tables are 1
//! and NbPoles, where NbPoles is the number
//! of poles of the BSpline curve,
//! - the bounds of the Knots and Multiplicities tables
//! are 1 and NbKnots, where NbKnots is the
//! number of knots of the BSpline curve.
//! If the BSpline curve is periodic, and if there are k
//! periodic knots and p periodic poles, the period is:
//! period = Knot(k + 1) - Knot(1)
//! and the poles and knots tables can be considered
//! as infinite tables, verifying:
//! - Knot(i+k) = Knot(i) + period
//! - Pole(i+p) = Pole(i)
//! Note: data structures of a periodic BSpline curve
//! are more complex than those of a non-periodic one.
//! Warning
//! In this class, weight value is considered to be zero if
//! the weight is less than or equal to gp::Resolution().
//!
//! References :
//! . A survey of curve and surface methods in CADG Wolfgang BOHM
//! CAGD 1 (1984)
//! . On de Boor-like algorithms and blossoming Wolfgang BOEHM
//! cagd 5 (1988)
//! . Blossoming and knot insertion algorithms for B-spline curves
//! Ronald N. GOLDMAN
//! . Modelisation des surfaces en CAO, Henri GIAUME Peugeot SA
//! . Curves and Surfaces for Computer Aided Geometric Design,
//! a practical guide Gerald Farin
class Geom_BSplineCurve : public Geom_BoundedCurve
{

public:

  
  //! Creates a  non-rational B_spline curve   on  the
  //! basis <Knots, Multiplicities> of degree <Degree>.
  Standard_EXPORT Geom_BSplineCurve(const TColgp_Array1OfPnt& Poles, const TColStd_Array1OfReal& Knots, const TColStd_Array1OfInteger& Multiplicities, const Standard_Integer Degree, const Standard_Boolean Periodic = Standard_False);
  
  //! Creates  a rational B_spline  curve  on the basis
  //! <Knots, Multiplicities> of degree <Degree>.
  //! Raises ConstructionError subject to the following conditions
  //! 0 < Degree <= MaxDegree.
  //!
  //! Weights.Length() == Poles.Length()
  //!
  //! Knots.Length() == Mults.Length() >= 2
  //!
  //! Knots(i) < Knots(i+1) (Knots are increasing)
  //!
  //! 1 <= Mults(i) <= Degree
  //!
  //! On a non periodic curve the first and last multiplicities
  //! may be Degree+1 (this is even recommanded if you want the
  //! curve to start and finish on the first and last pole).
  //!
  //! On a periodic  curve the first  and  the last multicities
  //! must be the same.
  //!
  //! on non-periodic curves
  //!
  //! Poles.Length() == Sum(Mults(i)) - Degree - 1 >= 2
  //!
  //! on periodic curves
  //!
  //! Poles.Length() == Sum(Mults(i)) except the first or last
  Standard_EXPORT Geom_BSplineCurve(const TColgp_Array1OfPnt& Poles, const TColStd_Array1OfReal& Weights, const TColStd_Array1OfReal& Knots, const TColStd_Array1OfInteger& Multiplicities, const Standard_Integer Degree, const Standard_Boolean Periodic = Standard_False, const Standard_Boolean CheckRational = Standard_True);
  
  //! Increases the degree of this BSpline curve to
  //! Degree. As a result, the poles, weights and
  //! multiplicities tables are modified; the knots table is
  //! not changed. Nothing is done if Degree is less than
  //! or equal to the current degree.
  //! Exceptions
  //! Standard_ConstructionError if Degree is greater than
  //! Geom_BSplineCurve::MaxDegree().
  Standard_EXPORT void IncreaseDegree (const Standard_Integer Degree);
  
  //! Increases the multiplicity  of the knot <Index> to
  //! <M>.
  //!
  //! If   <M>   is   lower   or  equal   to  the current
  //! multiplicity nothing is done. If <M> is higher than
  //! the degree the degree is used.
  //! If <Index> is not in [FirstUKnotIndex, LastUKnotIndex]
  Standard_EXPORT void IncreaseMultiplicity (const Standard_Integer Index, const Standard_Integer M);
  
  //! Increases  the  multiplicities   of  the knots  in
  //! [I1,I2] to <M>.
  //!
  //! For each knot if  <M>  is  lower  or equal  to  the
  //! current multiplicity  nothing  is  done. If <M>  is
  //! higher than the degree the degree is used.
  //! If <I1,I2> are not in [FirstUKnotIndex, LastUKnotIndex]
  Standard_EXPORT void IncreaseMultiplicity (const Standard_Integer I1, const Standard_Integer I2, const Standard_Integer M);
  
  //! Increment  the  multiplicities   of  the knots  in
  //! [I1,I2] by <M>.
  //!
  //! If <M> is not positive nithing is done.
  //!
  //! For   each  knot   the resulting   multiplicity  is
  //! limited to the Degree.
  //! If <I1,I2> are not in [FirstUKnotIndex, LastUKnotIndex]
  Standard_EXPORT void IncrementMultiplicity (const Standard_Integer I1, const Standard_Integer I2, const Standard_Integer M);
  
  //! Inserts a knot value in the sequence of knots.  If
  //! <U>  is an  existing knot     the multiplicity  is
  //! increased by <M>.
  //!
  //! If U  is  not  on the parameter  range  nothing is
  //! done.
  //!
  //! If the multiplicity is negative or null nothing is
  //! done. The  new   multiplicity  is limited  to  the
  //! degree.
  //!
  //! The  tolerance criterion  for  knots  equality  is
  //! the max of Epsilon(U) and ParametricTolerance.
  Standard_EXPORT void InsertKnot (const Standard_Real U, const Standard_Integer M = 1, const Standard_Real ParametricTolerance = 0.0, const Standard_Boolean Add = Standard_True);
  
  //! Inserts a set of knots  values in  the sequence of
  //! knots.
  //!
  //! For each U = Knots(i), M = Mults(i)
  //!
  //! If <U>  is an existing  knot  the  multiplicity is
  //! increased by  <M> if  <Add>  is True, increased to
  //! <M> if <Add> is False.
  //!
  //! If U  is  not  on the parameter  range  nothing is
  //! done.
  //!
  //! If the multiplicity is negative or null nothing is
  //! done. The  new   multiplicity  is limited  to  the
  //! degree.
  //!
  //! The  tolerance criterion  for  knots  equality  is
  //! the max of Epsilon(U) and ParametricTolerance.
  Standard_EXPORT void InsertKnots (const TColStd_Array1OfReal& Knots, const TColStd_Array1OfInteger& Mults, const Standard_Real ParametricTolerance = 0.0, const Standard_Boolean Add = Standard_False);
  
  //! Reduces the multiplicity of the knot of index Index
  //! to M. If M is equal to 0, the knot is removed.
  //! With a modification of this type, the array of poles is also modified.
  //! Two different algorithms are systematically used to
  //! compute the new poles of the curve. If, for each
  //! pole, the distance between the pole calculated
  //! using the first algorithm and the same pole
  //! calculated using the second algorithm, is less than
  //! Tolerance, this ensures that the curve is not
  //! modified by more than Tolerance. Under these
  //! conditions, true is returned; otherwise, false is returned.
  //! A low tolerance is used to prevent modification of
  //! the curve. A high tolerance is used to "smooth" the curve.
  //! Exceptions
  //! Standard_OutOfRange if Index is outside the
  //! bounds of the knots table.
  //! pole insertion and pole removing
  //! this operation is limited to the Uniform or QuasiUniform
  //! BSplineCurve. The knot values are modified . If the BSpline is
  //! NonUniform or Piecewise Bezier an exception Construction error
  //! is raised.
  Standard_EXPORT Standard_Boolean RemoveKnot (const Standard_Integer Index, const Standard_Integer M, const Standard_Real Tolerance);
  

  //! Changes the direction of parametrization of <me>. The Knot
  //! sequence is modified, the FirstParameter and the
  //! LastParameter are not modified. The StartPoint of the
  //! initial curve becomes the EndPoint of the reversed curve
  //! and the EndPoint of the initial curve becomes the StartPoint
  //! of the reversed curve.
  Standard_EXPORT void Reverse() Standard_OVERRIDE;
  
  //! Returns the  parameter on the  reversed  curve for
  //! the point of parameter U on <me>.
  //!
  //! returns UFirst + ULast - U
  Standard_EXPORT Standard_Real ReversedParameter (const Standard_Real U) const Standard_OVERRIDE;
  
  //! Modifies this BSpline curve by segmenting it between
  //! U1 and U2. Either of these values can be outside the
  //! bounds of the curve, but U2 must be greater than U1.
  //! All data structure tables of this BSpline curve are
  //! modified, but the knots located between U1 and U2
  //! are retained. The degree of the curve is not modified.
  //! Warnings :
  //! Even if <me> is not closed it can become closed after the
  //! segmentation for example if U1 or U2 are out of the bounds
  //! of the curve <me> or if the curve makes loop.
  //! After the segmentation the length of a curve can be null.
  //! raises if U2 < U1.
  //! Standard_DomainError if U2 - U1 exceeds the period for periodic curves.
  //! i.e. ((U2 - U1) - Period) > Precision::PConfusion().
  Standard_EXPORT void Segment (const Standard_Real U1, const Standard_Real U2);
  
  //! Modifies this BSpline curve by assigning the value K
  //! to the knot of index Index in the knots table. This is a
  //! relatively local modification because K must be such that:
  //! Knots(Index - 1) < K < Knots(Index + 1)
  //! The second syntax allows you also to increase the
  //! multiplicity of the knot to M (but it is not possible to
  //! decrease the multiplicity of the knot with this function).
  //! Standard_ConstructionError if:
  //! - K is not such that:
  //! Knots(Index - 1) < K < Knots(Index + 1)
  //! - M is greater than the degree of this BSpline curve
  //! or lower than the previous multiplicity of knot of
  //! index Index in the knots table.
  //! Standard_OutOfRange if Index is outside the bounds of the knots table.
  Standard_EXPORT void SetKnot (const Standard_Integer Index, const Standard_Real K);
  
  //! Modifies this BSpline curve by assigning the array
  //! K to its knots table. The multiplicity of the knots is not modified.
  //! Exceptions
  //! Standard_ConstructionError if the values in the
  //! array K are not in ascending order.
  //! Standard_OutOfRange if the bounds of the array
  //! K are not respectively 1 and the number of knots of this BSpline curve.
  Standard_EXPORT void SetKnots (const TColStd_Array1OfReal& K);
  

  //! Changes the knot of range Index with its multiplicity.
  //! You can increase the multiplicity of a knot but it is
  //! not allowed to decrease the multiplicity of an existing knot.
  //!
  //! Raised if K >= Knots(Index+1) or K <= Knots(Index-1).
  //! Raised if M is greater than Degree or lower than the previous
  //! multiplicity of knot of range Index.
  //! Raised if Index < 1 || Index > NbKnots
  Standard_EXPORT void SetKnot (const Standard_Integer Index, const Standard_Real K, const Standard_Integer M);
  
  //! returns the parameter normalized within
  //! the period if the curve is periodic : otherwise
  //! does not do anything
  Standard_EXPORT void PeriodicNormalization (Standard_Real& U) const;
  
  //! Changes this BSpline curve into a periodic curve.
  //! To become periodic, the curve must first be closed.
  //! Next, the knot sequence must be periodic. For this,
  //! FirstUKnotIndex and LastUKnotIndex are used
  //! to compute I1 and I2, the indexes in the knots
  //! array of the knots corresponding to the first and
  //! last parameters of this BSpline curve.
  //! The period is therefore: Knots(I2) - Knots(I1).
  //! Consequently, the knots and poles tables are modified.
  //! Exceptions
  //! Standard_ConstructionError if this BSpline curve is not closed.
  Standard_EXPORT void SetPeriodic();
  
  //! Assigns the knot of index Index in the knots table as
  //! the origin of this periodic BSpline curve. As a
  //! consequence, the knots and poles tables are modified.
  //! Exceptions
  //! Standard_NoSuchObject if this curve is not periodic.
  //! Standard_DomainError if Index is outside the bounds of the knots table.
  Standard_EXPORT void SetOrigin (const Standard_Integer Index);
  
  //! Set the origin of a periodic curve at Knot U. If U
  //! is  not a  knot  of  the  BSpline  a  new knot  is
  //! inseted. KnotVector and poles are modified.
  //! Raised if the curve is not periodic
  Standard_EXPORT void SetOrigin (const Standard_Real U, const Standard_Real Tol);
  
  //! Changes this BSpline curve into a non-periodic
  //! curve. If this curve is already non-periodic, it is not modified.
  //! Note: the poles and knots tables are modified.
  //! Warning
  //! If this curve is periodic, as the multiplicity of the first
  //! and last knots is not modified, and is not equal to
  //! Degree + 1, where Degree is the degree of
  //! this BSpline curve, the start and end points of the
  //! curve are not its first and last poles.
  Standard_EXPORT void SetNotPeriodic();
  
  //! Modifies this BSpline curve by assigning P to the pole
  //! of index Index in the poles table.
  //! Exceptions
  //! Standard_OutOfRange if Index is outside the
  //! bounds of the poles table.
  //! Standard_ConstructionError if Weight is negative or null.
  Standard_EXPORT void SetPole (const Standard_Integer Index, const gp_Pnt& P);
  
  //! Modifies this BSpline curve by assigning P to the pole
  //! of index Index in the poles table.
  //! This syntax also allows you to modify the
  //! weight of the modified pole, which becomes Weight.
  //! In this case, if this BSpline curve is non-rational, it
  //! can become rational and vice versa.
  //! Exceptions
  //! Standard_OutOfRange if Index is outside the
  //! bounds of the poles table.
  //! Standard_ConstructionError if Weight is negative or null.
  Standard_EXPORT void SetPole (const Standard_Integer Index, const gp_Pnt& P, const Standard_Real Weight);
  

  //! Changes the weight for the pole of range Index.
  //! If the curve was non rational it can become rational.
  //! If the curve was rational it can become non rational.
  //!
  //! Raised if Index < 1 || Index > NbPoles
  //! Raised if Weight <= 0.0
  Standard_EXPORT void SetWeight (const Standard_Integer Index, const Standard_Real Weight);
  
  //! Moves the point of parameter U of this BSpline curve
  //! to P. Index1 and Index2 are the indexes in the table
  //! of poles of this BSpline curve of the first and last
  //! poles designated to be moved.
  //! FirstModifiedPole and LastModifiedPole are the
  //! indexes of the first and last poles which are effectively modified.
  //! In the event of incompatibility between Index1, Index2 and the value U:
  //! - no change is made to this BSpline curve, and
  //! - the FirstModifiedPole and LastModifiedPole are returned null.
  //! Exceptions
  //! Standard_OutOfRange if:
  //! - Index1 is greater than or equal to Index2, or
  //! - Index1 or Index2 is less than 1 or greater than the
  //! number of poles of this BSpline curve.
  Standard_EXPORT void MovePoint (const Standard_Real U, const gp_Pnt& P, const Standard_Integer Index1, const Standard_Integer Index2, Standard_Integer& FirstModifiedPole, Standard_Integer& LastModifiedPole);
  

  //! Move a point with parameter U to P.
  //! and makes it tangent at U be Tangent.
  //! StartingCondition = -1 means first can move
  //! EndingCondition   = -1 means last point can move
  //! StartingCondition = 0 means the first point cannot move
  //! EndingCondition   = 0 means the last point cannot move
  //! StartingCondition = 1 means the first point and tangent cannot move
  //! EndingCondition   = 1 means the last point and tangent cannot move
  //! and so forth
  //! ErrorStatus != 0 means that there are not enought degree of freedom
  //! with the constrain to deform the curve accordingly
  Standard_EXPORT void MovePointAndTangent (const Standard_Real U, const gp_Pnt& P, const gp_Vec& Tangent, const Standard_Real Tolerance, const Standard_Integer StartingCondition, const Standard_Integer EndingCondition, Standard_Integer& ErrorStatus);
  

  //! Returns the continuity of the curve, the curve is at least C0.
  //! Raised if N < 0.
  Standard_EXPORT Standard_Boolean IsCN (const Standard_Integer N) const Standard_OVERRIDE;
  

  //! Check if curve has at least G1 continuity in interval [theTf, theTl]
  //! Returns true if IsCN(1)
  //! or
  //! angle betweem "left" and "right" first derivatives at
  //! knots with C0 continuity is less then theAngTol
  //! only knots in interval [theTf, theTl] is checked
  Standard_EXPORT Standard_Boolean IsG1 (const Standard_Real theTf, const Standard_Real theTl, const Standard_Real theAngTol) const;
  

  //! Returns true if the distance between the first point and the
  //! last point of the curve is lower or equal to Resolution
  //! from package gp.
  //! Warnings :
  //! The first and the last point can be different from the first
  //! pole and the last pole of the curve.
  Standard_EXPORT Standard_Boolean IsClosed() const Standard_OVERRIDE;
  
  //! Returns True if the curve is periodic.
  Standard_EXPORT Standard_Boolean IsPeriodic() const Standard_OVERRIDE;
  

  //! Returns True if the weights are not identical.
  //! The tolerance criterion is Epsilon of the class Real.
  Standard_EXPORT Standard_Boolean IsRational() const;
  

  //! Returns the global continuity of the curve :
  //! C0 : only geometric continuity,
  //! C1 : continuity of the first derivative all along the Curve,
  //! C2 : continuity of the second derivative all along the Curve,
  //! C3 : continuity of the third derivative all along the Curve,
  //! CN : the order of continuity is infinite.
  //! For a B-spline curve of degree d if a knot Ui has a
  //! multiplicity p the B-spline curve is only Cd-p continuous
  //! at Ui. So the global continuity of the curve can't be greater
  //! than Cd-p where p is the maximum multiplicity of the interior
  //! Knots. In the interior of a knot span the curve is infinitely
  //! continuously differentiable.
  Standard_EXPORT GeomAbs_Shape Continuity() const Standard_OVERRIDE;
  
  //! Returns the degree of this BSpline curve.
  //! The degree of a Geom_BSplineCurve curve cannot
  //! be greater than Geom_BSplineCurve::MaxDegree().
  //! Computation of value and derivatives
  Standard_EXPORT Standard_Integer Degree() const;
  
  //! Returns in P the point of parameter U.
  Standard_EXPORT void D0 (const Standard_Real U, gp_Pnt& P) const Standard_OVERRIDE;
  
  //! Raised if the continuity of the curve is not C1.
  Standard_EXPORT void D1 (const Standard_Real U, gp_Pnt& P, gp_Vec& V1) const Standard_OVERRIDE;
  
  //! Raised if the continuity of the curve is not C2.
  Standard_EXPORT void D2 (const Standard_Real U, gp_Pnt& P, gp_Vec& V1, gp_Vec& V2) const Standard_OVERRIDE;
  
  //! Raised if the continuity of the curve is not C3.
  Standard_EXPORT void D3 (const Standard_Real U, gp_Pnt& P, gp_Vec& V1, gp_Vec& V2, gp_Vec& V3) const Standard_OVERRIDE;
  
  //! For the point of parameter U of this BSpline curve,
  //! computes the vector corresponding to the Nth derivative.
  //! Warning
  //! On a point where the continuity of the curve is not the
  //! one requested, this function impacts the part defined
  //! by the parameter with a value greater than U, i.e. the
  //! part of the curve to the "right" of the singularity.
  //! Exceptions
  //! Standard_RangeError if N is less than 1.
  //!
  //! The following functions compute the point of parameter U
  //! and the derivatives at this point on the B-spline curve
  //! arc defined between the knot FromK1 and the knot ToK2.
  //! U can be out of bounds [Knot (FromK1),  Knot (ToK2)] but
  //! for the computation we only use the definition of the curve
  //! between these two knots. This method is useful to compute
  //! local derivative, if the order of continuity of the whole
  //! curve is not greater enough.    Inside the parametric
  //! domain Knot (FromK1), Knot (ToK2) the evaluations are
  //! the same as if we consider the whole definition of the
  //! curve. Of course the evaluations are different outside
  //! this parametric domain.
  Standard_EXPORT gp_Vec DN (const Standard_Real U, const Standard_Integer N) const Standard_OVERRIDE;
  
  //! Raised if FromK1 = ToK2.
  Standard_EXPORT gp_Pnt LocalValue (const Standard_Real U, const Standard_Integer FromK1, const Standard_Integer ToK2) const;
  
  //! Raised if FromK1 = ToK2.
  Standard_EXPORT void LocalD0 (const Standard_Real U, const Standard_Integer FromK1, const Standard_Integer ToK2, gp_Pnt& P) const;
  

  //! Raised if the local continuity of the curve is not C1
  //! between the knot K1 and the knot K2.
  //! Raised if FromK1 = ToK2.
  Standard_EXPORT void LocalD1 (const Standard_Real U, const Standard_Integer FromK1, const Standard_Integer ToK2, gp_Pnt& P, gp_Vec& V1) const;
  

  //! Raised if the local continuity of the curve is not C2
  //! between the knot K1 and the knot K2.
  //! Raised if FromK1 = ToK2.
  Standard_EXPORT void LocalD2 (const Standard_Real U, const Standard_Integer FromK1, const Standard_Integer ToK2, gp_Pnt& P, gp_Vec& V1, gp_Vec& V2) const;
  

  //! Raised if the local continuity of the curve is not C3
  //! between the knot K1 and the knot K2.
  //! Raised if FromK1 = ToK2.
  Standard_EXPORT void LocalD3 (const Standard_Real U, const Standard_Integer FromK1, const Standard_Integer ToK2, gp_Pnt& P, gp_Vec& V1, gp_Vec& V2, gp_Vec& V3) const;
  

  //! Raised if the local continuity of the curve is not CN
  //! between the knot K1 and the knot K2.
  //! Raised if FromK1 = ToK2.
  //! Raised if N < 1.
  Standard_EXPORT gp_Vec LocalDN (const Standard_Real U, const Standard_Integer FromK1, const Standard_Integer ToK2, const Standard_Integer N) const;
  

  //! Returns the last point of the curve.
  //! Warnings :
  //! The last point of the curve is different from the last
  //! pole of the curve if the multiplicity of the last knot
  //! is lower than Degree.
  Standard_EXPORT gp_Pnt EndPoint() const Standard_OVERRIDE;
  
  //! Returns the index in the knot array of the knot
  //! corresponding to the first or last parameter of this BSpline curve.
  //! For a BSpline curve, the first (or last) parameter
  //! (which gives the start (or end) point of the curve) is a
  //! knot value. However, if the multiplicity of the first (or
  //! last) knot is less than Degree + 1, where
  //! Degree is the degree of the curve, it is not the first
  //! (or last) knot of the curve.
  Standard_EXPORT Standard_Integer FirstUKnotIndex() const;
  
  //! Returns the value of the first parameter of this
  //! BSpline curve. This is a knot value.
  //! The first parameter is the one of the start point of the BSpline curve.
  Standard_EXPORT Standard_Real FirstParameter() const Standard_OVERRIDE;
  

  //! Returns the knot of range Index. When there is a knot
  //! with a multiplicity greater than 1 the knot is not repeated.
  //! The method Multiplicity can be used to get the multiplicity
  //! of the Knot.
  //! Raised if Index < 1 or Index > NbKnots
  Standard_EXPORT Standard_Real Knot (const Standard_Integer Index) const;
  
  //! returns the knot values of the B-spline curve;
  //! Warning
  //! A knot with a multiplicity greater than 1 is not
  //! repeated in the knot table. The Multiplicity function
  //! can be used to obtain the multiplicity of each knot.
  //!
  //! Raised K.Lower() is less than number of first knot or
  //! K.Upper() is more than number of last knot.
  Standard_EXPORT void Knots (TColStd_Array1OfReal& K) const;
  
  //! returns the knot values of the B-spline curve;
  //! Warning
  //! A knot with a multiplicity greater than 1 is not
  //! repeated in the knot table. The Multiplicity function
  //! can be used to obtain the multiplicity of each knot.
  Standard_EXPORT const TColStd_Array1OfReal& Knots() const;
  
  //! Returns K, the knots sequence of this BSpline curve.
  //! In this sequence, knots with a multiplicity greater than 1 are repeated.
  //! In the case of a non-periodic curve the length of the
  //! sequence must be equal to the sum of the NbKnots
  //! multiplicities of the knots of the curve (where
  //! NbKnots is the number of knots of this BSpline
  //! curve). This sum is also equal to : NbPoles + Degree + 1
  //! where NbPoles is the number of poles and
  //! Degree the degree of this BSpline curve.
  //! In the case of a periodic curve, if there are k periodic
  //! knots, the period is Knot(k+1) - Knot(1).
  //! The initial sequence is built by writing knots 1 to k+1,
  //! which are repeated according to their corresponding multiplicities.
  //! If Degree is the degree of the curve, the degree of
  //! continuity of the curve at the knot of index 1 (or k+1)
  //! is equal to c = Degree + 1 - Mult(1). c
  //! knots are then inserted at the beginning and end of
  //! the initial sequence:
  //! - the c values of knots preceding the first item
  //! Knot(k+1) in the initial sequence are inserted
  //! at the beginning; the period is subtracted from these c values;
  //! - the c values of knots following the last item
  //! Knot(1) in the initial sequence are inserted at
  //! the end; the period is added to these c values.
  //! The length of the sequence must therefore be equal to:
  //! NbPoles + 2*Degree - Mult(1) + 2.
  //! Example
  //! For a non-periodic BSpline curve of degree 2 where:
  //! - the array of knots is: { k1 k2 k3 k4 },
  //! - with associated multiplicities: { 3 1 2 3 },
  //! the knot sequence is:
  //! K = { k1 k1 k1 k2 k3 k3 k4 k4 k4 }
  //! For a periodic BSpline curve of degree 4 , which is
  //! "C1" continuous at the first knot, and where :
  //! - the periodic knots are: { k1 k2 k3 (k4) }
  //! (3 periodic knots: the points of parameter k1 and k4
  //! are identical, the period is p = k4 - k1),
  //! - with associated multiplicities: { 3 1 2 (3) },
  //! the degree of continuity at knots k1 and k4 is:
  //! Degree + 1 - Mult(i) = 2.
  //! 2 supplementary knots are added at the beginning
  //! and end of the sequence:
  //! - at the beginning: the 2 knots preceding k4 minus
  //! the period; in this example, this is k3 - p both times;
  //! - at the end: the 2 knots following k1 plus the period;
  //! in this example, this is k2 + p and k3 + p.
  //! The knot sequence is therefore:
  //! K = { k3-p k3-p k1 k1 k1 k2 k3 k3
  //! k4 k4 k4 k2+p k3+p }
  //! Exceptions
  //! Raised if K.Lower() is less than number of first knot
  //! in knot sequence with repetitions or K.Upper() is more
  //! than number of last knot in knot sequence with repetitions.
  Standard_EXPORT void KnotSequence (TColStd_Array1OfReal& K) const;
  
  //! returns the knots of the B-spline curve.
  //! Knots with multiplicit greater than 1 are repeated
  Standard_EXPORT const TColStd_Array1OfReal& KnotSequence() const;
  

  //! Returns NonUniform or Uniform or QuasiUniform or PiecewiseBezier.
  //! If all the knots differ by a positive constant from the
  //! preceding knot the BSpline Curve can be :
  //! - Uniform if all the knots are of multiplicity 1,
  //! - QuasiUniform if all the knots are of multiplicity 1 except for
  //! the first and last knot which are of multiplicity Degree + 1,
  //! - PiecewiseBezier if the first and last knots have multiplicity
  //! Degree + 1 and if interior knots have multiplicity Degree
  //! A piecewise Bezier with only two knots is a BezierCurve.
  //! else the curve is non uniform.
  //! The tolerance criterion is Epsilon from class Real.
  Standard_EXPORT GeomAbs_BSplKnotDistribution KnotDistribution() const;
  

  //! For a BSpline curve the last parameter (which gives the
  //! end point of the curve) is a knot value but if the
  //! multiplicity of the last knot index is lower than
  //! Degree + 1 it is not the last knot of the curve. This
  //! method computes the index of the knot corresponding to
  //! the last parameter.
  Standard_EXPORT Standard_Integer LastUKnotIndex() const;
  

  //! Computes the parametric value of the end point of the curve.
  //! It is a knot value.
  Standard_EXPORT Standard_Real LastParameter() const Standard_OVERRIDE;
  

  //! Locates the parametric value U in the sequence of knots.
  //! If "WithKnotRepetition" is True we consider the knot's
  //! representation with repetition of multiple knot value,
  //! otherwise  we consider the knot's representation with
  //! no repetition of multiple knot values.
  //! Knots (I1) <= U <= Knots (I2)
  //! . if I1 = I2  U is a knot value (the tolerance criterion
  //! ParametricTolerance is used).
  //! . if I1 < 1  => U < Knots (1) - Abs(ParametricTolerance)
  //! . if I2 > NbKnots => U > Knots (NbKnots) + Abs(ParametricTolerance)
  Standard_EXPORT void LocateU (const Standard_Real U, const Standard_Real ParametricTolerance, Standard_Integer& I1, Standard_Integer& I2, const Standard_Boolean WithKnotRepetition = Standard_False) const;
  

  //! Returns the multiplicity of the knots of range Index.
  //! Raised if Index < 1 or Index > NbKnots
  Standard_EXPORT Standard_Integer Multiplicity (const Standard_Integer Index) const;
  

  //! Returns the multiplicity of the knots of the curve.
  //!
  //! Raised if the length of M is not equal to NbKnots.
  Standard_EXPORT void Multiplicities (TColStd_Array1OfInteger& M) const;
  
  //! returns the multiplicity of the knots of the curve.
  Standard_EXPORT const TColStd_Array1OfInteger& Multiplicities() const;
  

  //! Returns the number of knots. This method returns the number of
  //! knot without repetition of multiple knots.
  Standard_EXPORT Standard_Integer NbKnots() const;
  
  //! Returns the number of poles
  Standard_EXPORT Standard_Integer NbPoles() const;
  
  //! Returns the pole of range Index.
  //! Raised if Index < 1 or Index > NbPoles.
  Standard_EXPORT const gp_Pnt& Pole(const Standard_Integer Index) const;
  
  //! Returns the poles of the B-spline curve;
  //!
  //! Raised if the length of P is not equal to the number of poles.
  Standard_EXPORT void Poles (TColgp_Array1OfPnt& P) const;
  
  //! Returns the poles of the B-spline curve;
  Standard_EXPORT const TColgp_Array1OfPnt& Poles() const;
  

  //! Returns the start point of the curve.
  //! Warnings :
  //! This point is different from the first pole of the curve if the
  //! multiplicity of the first knot is lower than Degree.
  Standard_EXPORT gp_Pnt StartPoint() const Standard_OVERRIDE;
  
  //! Returns the weight of the pole of range Index .
  //! Raised if Index < 1 or Index > NbPoles.
  Standard_EXPORT Standard_Real Weight (const Standard_Integer Index) const;
  
  //! Returns the weights of the B-spline curve;
  //!
  //! Raised if the length of W is not equal to NbPoles.
  Standard_EXPORT void Weights (TColStd_Array1OfReal& W) const;
  
  //! Returns the weights of the B-spline curve;
  Standard_EXPORT const TColStd_Array1OfReal* Weights() const;
  
  //! Applies the transformation T to this BSpline curve.
  Standard_EXPORT void Transform (const gp_Trsf& T) Standard_OVERRIDE;
  

  //! Returns the value of the maximum degree of the normalized
  //! B-spline basis functions in this package.
  Standard_EXPORT static Standard_Integer MaxDegree();
  
  //! Computes for this BSpline curve the parametric
  //! tolerance UTolerance for a given 3D tolerance Tolerance3D.
  //! If f(t) is the equation of this BSpline curve,
  //! UTolerance ensures that:
  //! | t1 - t0| < Utolerance ===>
  //! |f(t1) - f(t0)| < Tolerance3D
  Standard_EXPORT void Resolution (const Standard_Real Tolerance3D, Standard_Real& UTolerance);
  
  //! Creates a new object which is a copy of this BSpline curve.
  Standard_EXPORT Handle(Geom_Geometry) Copy() const Standard_OVERRIDE;
  
  //! Comapare two Bspline curve on identity;
  Standard_EXPORT Standard_Boolean IsEqual (const Handle(Geom_BSplineCurve)& theOther, const Standard_Real thePreci) const;




  DEFINE_STANDARD_RTTIEXT(Geom_BSplineCurve,Geom_BoundedCurve)

protected:




private:

  
  //! Recompute  the  flatknots,  the knotsdistribution, the continuity.
  Standard_EXPORT void UpdateKnots();

  Standard_Boolean rational;
  Standard_Boolean periodic;
  GeomAbs_BSplKnotDistribution knotSet;
  GeomAbs_Shape smooth;
  Standard_Integer deg;
  Handle(TColgp_HArray1OfPnt) poles;
  Handle(TColStd_HArray1OfReal) weights;
  Handle(TColStd_HArray1OfReal) flatknots;
  Handle(TColStd_HArray1OfReal) knots;
  Handle(TColStd_HArray1OfInteger) mults;
  Standard_Real maxderivinv;
  Standard_Boolean maxderivinvok;


};







#endif // _Geom_BSplineCurve_HeaderFile