Definition of the B_spline curve. A B-spline curve can be Uniform or non-uniform Rational or non-rational Periodic or non-periodic. More...

#include <Geom_BSplineCurve.hxx>

Inheritance diagram for Geom_BSplineCurve:

[legend]

Public Member Functions
	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 non-rational B_spline curve on the basis <Knots, Multiplicities> of degree <Degree>. More...

	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)
	Creates a rational B_spline curve on the basis <Knots, Multiplicities> of degree <Degree>. Raises ConstructionError subject to the following conditions 0 < Degree <= MaxDegree. More...

void	IncreaseDegree (const Standard_Integer Degree)
	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(). More...

void	IncreaseMultiplicity (const Standard_Integer Index, const Standard_Integer M)
	Increases the multiplicity of the knot <Index> to <M>. More...

void	IncreaseMultiplicity (const Standard_Integer I1, const Standard_Integer I2, const Standard_Integer M)
	Increases the multiplicities of the knots in [I1,I2] to <M>. More...

void	IncrementMultiplicity (const Standard_Integer I1, const Standard_Integer I2, const Standard_Integer M)
	Increment the multiplicities of the knots in [I1,I2] by <M>. More...

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 knot value in the sequence of knots. If <U> is an existing knot the multiplicity is increased by <M>. More...

void	InsertKnots (const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const Standard_Real ParametricTolerance=0.0, const Standard_Boolean Add=Standard_False)
	Inserts a set of knots values in the sequence of knots. More...

Standard_Boolean	RemoveKnot (const Standard_Integer Index, const Standard_Integer M, const Standard_Real Tolerance)
	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. More...

void	Reverse ()
	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. More...

Standard_Real	ReversedParameter (const Standard_Real U) const
	Returns the parameter on the reversed curve for the point of parameter U on <me>. More...

void	Segment (const Standard_Real U1, const Standard_Real U2)
	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. More...

void	SetKnot (const Standard_Integer Index, const Standard_Real K)
	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: More...

void	SetKnots (const TColStd_Array1OfReal &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. More...

void	SetKnot (const Standard_Integer Index, const Standard_Real K, const Standard_Integer M)
	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. More...

void	PeriodicNormalization (Standard_Real &U) const
	returns the parameter normalized within the period if the curve is periodic : otherwise does not do anything More...

void	SetPeriodic ()
	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. More...

void	SetOrigin (const Standard_Integer Index)
	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. More...

void	SetOrigin (const Standard_Real U, const Standard_Real Tol)
	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. More...

void	SetNotPeriodic ()
	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. More...

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. Exceptions Standard_OutOfRange if Index is outside the bounds of the poles table. Standard_ConstructionError if Weight is negative or null. More...

void	SetPole (const Standard_Integer Index, const gp_Pnt &P, const Standard_Real Weight)
	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. More...

void	SetWeight (const Standard_Integer Index, 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. More...

void	MovePoint (const Standard_Real U, const gp_Pnt &P, const Standard_Integer Index1, const Standard_Integer Index2, Standard_Integer &FirstModifiedPole, Standard_Integer &LastModifiedPole)
	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: More...

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)
	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. More...

Standard_Boolean	IsCN (const Standard_Integer N) const
	Returns the continuity of the curve, the curve is at least C0. Raised if N < 0. More...

Standard_Boolean	IsG1 (const Standard_Real theTf, const Standard_Real theTl, const Standard_Real theAngTol) const
	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. More...

Standard_Boolean	IsClosed () 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. More...

Standard_Boolean	IsPeriodic () const
	Returns True if the curve is periodic. More...

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

GeomAbs_Shape	Continuity () 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. More...

Standard_Integer	Degree () const
	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. More...

void	D0 (const Standard_Real U, gp_Pnt &P) const
	Returns in P the point of parameter U. More...

void	D1 (const Standard_Real U, gp_Pnt &P, gp_Vec &V1) const
	Raised if the continuity of the curve is not C1. More...

void	D2 (const Standard_Real U, gp_Pnt &P, gp_Vec &V1, gp_Vec &V2) const
	Raised if the continuity of the curve is not C2. More...

void	D3 (const Standard_Real U, gp_Pnt &P, gp_Vec &V1, gp_Vec &V2, gp_Vec &V3) const
	Raised if the continuity of the curve is not C3. More...

gp_Vec	DN (const Standard_Real U, const Standard_Integer N) const
	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. More...

gp_Pnt	LocalValue (const Standard_Real U, const Standard_Integer FromK1, const Standard_Integer ToK2) const
	Raised if FromK1 = ToK2. More...

void	LocalD0 (const Standard_Real U, const Standard_Integer FromK1, const Standard_Integer ToK2, gp_Pnt &P) const
	Raised if FromK1 = ToK2. More...

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 C1 between the knot K1 and the knot K2. Raised if FromK1 = ToK2. More...

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 C2 between the knot K1 and the knot K2. Raised if FromK1 = ToK2. More...

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 C3 between the knot K1 and the knot K2. Raised if FromK1 = ToK2. More...

gp_Vec	LocalDN (const Standard_Real U, const Standard_Integer FromK1, const Standard_Integer ToK2, const Standard_Integer N) 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. More...

gp_Pnt	EndPoint () 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. More...

Standard_Integer	FirstUKnotIndex () const
	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. More...

Standard_Real	FirstParameter () 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. More...

Standard_Real	Knot (const Standard_Integer Index) const
	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. More...

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. More...

const TColStd_Array1OfReal &	Knots () 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. More...

void	KnotSequence (TColStd_Array1OfReal &K) 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: More...

const TColStd_Array1OfReal &	KnotSequence () const
	returns the knots of the B-spline curve. Knots with multiplicit greater than 1 are repeated More...

GeomAbs_BSplKnotDistribution	KnotDistribution () 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 : More...

Standard_Integer	LastUKnotIndex () 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. More...

Standard_Real	LastParameter () const
	Computes the parametric value of the end point of the curve. It is a knot value. More...

void	LocateU (const Standard_Real U, const Standard_Real ParametricTolerance, Standard_Integer &I1, Standard_Integer &I2, const Standard_Boolean WithKnotRepetition=Standard_False) const
	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) More...

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

void	Multiplicities (TColStd_Array1OfInteger &M) const
	Returns the multiplicity of the knots of the curve. More...

const TColStd_Array1OfInteger &	Multiplicities () const
	returns the multiplicity of the knots of the curve. More...

Standard_Integer	NbKnots () const
	Returns the number of knots. This method returns the number of knot without repetition of multiple knots. More...

Standard_Integer	NbPoles () const
	Returns the number of poles. More...

gp_Pnt	Pole (const Standard_Integer Index) const
	Returns the pole of range Index. Raised if Index < 1 or Index > NbPoles. More...

void	Poles (TColgp_Array1OfPnt &P) const
	Returns the poles of the B-spline curve;. More...

const TColgp_Array1OfPnt &	Poles () const
	Returns the poles of the B-spline curve;. More...

gp_Pnt	StartPoint () 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. More...

Standard_Real	Weight (const Standard_Integer Index) const
	Returns the weight of the pole of range Index . Raised if Index < 1 or Index > NbPoles. More...

void	Weights (TColStd_Array1OfReal &W) const
	Returns the weights of the B-spline curve;. More...

const TColStd_Array1OfReal &	Weights () const
	Returns the weights of the B-spline curve;. More...

void	Transform (const gp_Trsf &T)
	Applies the transformation T to this BSpline curve. More...

void	Resolution (const Standard_Real Tolerance3D, Standard_Real &UTolerance)
	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. More...

Handle< Geom_Geometry >	Copy () const
	Creates a new object which is a copy of this BSpline curve. More...

Standard_Boolean	IsEqual (const Handle< Geom_BSplineCurve > &theOther, const Standard_Real thePreci) const
	Comapare two Bspline curve on identity;. More...

Public Member Functions inherited from Geom_Curve
virtual Standard_Real	TransformedParameter (const Standard_Real U, const gp_Trsf &T) const
	Returns the parameter on the transformed curve for the transform of the point of parameter U on <me>. More...

virtual Standard_Real	ParametricTransformation (const gp_Trsf &T) const
	Returns a coefficient to compute the parameter on the transformed curve for the transform of the point on <me>. More...

Handle< Geom_Curve >	Reversed () const
	Returns a copy of <me> reversed. More...

virtual Standard_Real	Period () const
	Returns the period of this curve. Exceptions Standard_NoSuchObject if this curve is not periodic. More...

gp_Pnt	Value (const Standard_Real U) const
	Computes the point of parameter U on <me>. If the curve is periodic then the returned point is P(U) with U = Ustart + (U - Uend) where Ustart and Uend are the parametric bounds of the curve. it is implemented with D0. More...

Public Member Functions inherited from Geom_Geometry
void	Mirror (const gp_Pnt &P)
	Performs the symmetrical transformation of a Geometry with respect to the point P which is the center of the symmetry. More...

void	Mirror (const gp_Ax1 &A1)
	Performs the symmetrical transformation of a Geometry with respect to an axis placement which is the axis of the symmetry. More...

void	Mirror (const gp_Ax2 &A2)
	Performs the symmetrical transformation of a Geometry with respect to a plane. The axis placement A2 locates the plane of the symmetry : (Location, XDirection, YDirection). More...

void	Rotate (const gp_Ax1 &A1, const Standard_Real Ang)
	Rotates a Geometry. A1 is the axis of the rotation. Ang is the angular value of the rotation in radians. More...

void	Scale (const gp_Pnt &P, const Standard_Real S)
	Scales a Geometry. S is the scaling value. More...

void	Translate (const gp_Vec &V)
	Translates a Geometry. V is the vector of the tanslation. More...

void	Translate (const gp_Pnt &P1, const gp_Pnt &P2)
	Translates a Geometry from the point P1 to the point P2. More...

Handle< Geom_Geometry >	Mirrored (const gp_Pnt &P) const

Handle< Geom_Geometry >	Mirrored (const gp_Ax1 &A1) const

Handle< Geom_Geometry >	Mirrored (const gp_Ax2 &A2) const

Handle< Geom_Geometry >	Rotated (const gp_Ax1 &A1, const Standard_Real Ang) const

Handle< Geom_Geometry >	Scaled (const gp_Pnt &P, const Standard_Real S) const

Handle< Geom_Geometry >	Transformed (const gp_Trsf &T) const

Handle< Geom_Geometry >	Translated (const gp_Vec &V) const

Handle< Geom_Geometry >	Translated (const gp_Pnt &P1, const gp_Pnt &P2) const

Public Member Functions inherited from MMgt_TShared
virtual void	Delete () const
	Memory deallocator for transient classes. More...

Public Member Functions inherited from Standard_Transient
	Standard_Transient ()
	Empty constructor. More...

	Standard_Transient (const Standard_Transient &)
	Copy constructor – does nothing. More...

Standard_Transient &	operator= (const Standard_Transient &)
	Assignment operator, needed to avoid copying reference counter. More...

virtual	~Standard_Transient ()
	Destructor must be virtual. More...

virtual const Handle_Standard_Type &	DynamicType () const
	Returns a type information object about this object. More...

Standard_Boolean	IsInstance (const Handle_Standard_Type &theType) const
	Returns a true value if this is an instance of Type. More...

Standard_Boolean	IsInstance (const Standard_CString theTypeName) const
	Returns a true value if this is an instance of TypeName. More...

Standard_Boolean	IsKind (const Handle_Standard_Type &theType) const
	Returns true if this is an instance of Type or an instance of any class that inherits from Type. Note that multiple inheritance is not supported by OCCT RTTI mechanism. More...

Standard_Boolean	IsKind (const Standard_CString theTypeName) const
	Returns true if this is an instance of TypeName or an instance of any class that inherits from TypeName. Note that multiple inheritance is not supported by OCCT RTTI mechanism. More...

virtual Handle_Standard_Transient	This () const
	Returns a Handle which references this object. Must never be called to objects created in stack. More...

Standard_Integer	GetRefCount () const
	Get the reference counter of this object. More...

Static Public Member Functions
static Standard_Integer	MaxDegree ()
	Returns the value of the maximum degree of the normalized B-spline basis functions in this package. More...

Detailed Description

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

Constructor & Destructor Documentation

Geom_BSplineCurve::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 non-rational B_spline curve on the basis <Knots, Multiplicities> of degree <Degree>.

Geom_BSplineCurve::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`
	)

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

Member Function Documentation

GeomAbs_Shape Geom_BSplineCurve::Continuity ( ) const

virtual

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.

Implements Geom_Curve.

Handle< Geom_Geometry > Geom_BSplineCurve::Copy ( ) const

virtual

Creates a new object which is a copy of this BSpline curve.

Implements Geom_Geometry.

void Geom_BSplineCurve::D0	(	const Standard_Real	U,
		gp_Pnt &	P
	)		const

virtual

Returns in P the point of parameter U.

Implements Geom_Curve.

void Geom_BSplineCurve::D1	(	const Standard_Real	U,
		gp_Pnt &	P,
		gp_Vec &	V1
	)		const

virtual

Raised if the continuity of the curve is not C1.

Implements Geom_Curve.

void Geom_BSplineCurve::D2	(	const Standard_Real	U,
		gp_Pnt &	P,
		gp_Vec &	V1,
		gp_Vec &	V2
	)		const

virtual

Raised if the continuity of the curve is not C2.

Implements Geom_Curve.

void Geom_BSplineCurve::D3	(	const Standard_Real	U,
		gp_Pnt &	P,
		gp_Vec &	V1,
		gp_Vec &	V2,
		gp_Vec &	V3
	)		const

virtual

Raised if the continuity of the curve is not C3.

Implements Geom_Curve.

Standard_Integer Geom_BSplineCurve::Degree ( ) const

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.

gp_Vec Geom_BSplineCurve::DN	(	const Standard_Real	U,
		const Standard_Integer	N
	)		const

virtual

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.

Implements Geom_Curve.

gp_Pnt Geom_BSplineCurve::EndPoint ( ) const

virtual

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.

Implements Geom_BoundedCurve.

Standard_Real Geom_BSplineCurve::FirstParameter ( ) const

virtual

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.

Implements Geom_Curve.

Standard_Integer Geom_BSplineCurve::FirstUKnotIndex ( ) const

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.

void Geom_BSplineCurve::IncreaseDegree ( const Standard_Integer Degree )

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().

void Geom_BSplineCurve::IncreaseMultiplicity	(	const Standard_Integer	Index,
		const Standard_Integer	M
	)

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]

void Geom_BSplineCurve::IncreaseMultiplicity	(	const Standard_Integer	I1,
		const Standard_Integer	I2,
		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]

void Geom_BSplineCurve::IncrementMultiplicity	(	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]

void Geom_BSplineCurve::InsertKnot	(	const Standard_Real	U,
		const Standard_Integer	M = `1`,
		const Standard_Real	ParametricTolerance = `0.0`,
		const Standard_Boolean	Add = `Standard_True`
	)

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.

void Geom_BSplineCurve::InsertKnots	(	const TColStd_Array1OfReal &	Knots,
		const TColStd_Array1OfInteger &	Mults,
		const Standard_Real	ParametricTolerance = `0.0`,
		const Standard_Boolean	Add = `Standard_False`
	)

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_Boolean Geom_BSplineCurve::IsClosed ( ) const

virtual

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.

Implements Geom_Curve.

Standard_Boolean Geom_BSplineCurve::IsCN ( const Standard_Integer N ) const

virtual

Returns the continuity of the curve, the curve is at least C0. Raised if N < 0.

Implements Geom_Curve.

Standard_Boolean Geom_BSplineCurve::IsEqual	(	const Handle< Geom_BSplineCurve > &	theOther,
		const Standard_Real	thePreci
	)		const

Comapare two Bspline curve on identity;.

Standard_Boolean Geom_BSplineCurve::IsG1	(	const Standard_Real	theTf,
		const Standard_Real	theTl,
		const Standard_Real	theAngTol
	)		const

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_Boolean Geom_BSplineCurve::IsPeriodic ( ) const

virtual

Returns True if the curve is periodic.

Implements Geom_Curve.

Standard_Boolean Geom_BSplineCurve::IsRational ( ) const

Returns True if the weights are not identical. The tolerance criterion is Epsilon of the class Real.

Standard_Real Geom_BSplineCurve::Knot ( const Standard_Integer Index ) const

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.

GeomAbs_BSplKnotDistribution Geom_BSplineCurve::KnotDistribution ( ) 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.

void Geom_BSplineCurve::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.

Raised if the length of K is not equal to the number of knots.

const TColStd_Array1OfReal& Geom_BSplineCurve::Knots ( ) 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.

void Geom_BSplineCurve::KnotSequence ( TColStd_Array1OfReal & K ) 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 Standard_DimensionError if the array K is not of the appropriate length.Returns the knots sequence.

const TColStd_Array1OfReal& Geom_BSplineCurve::KnotSequence ( ) const

returns the knots of the B-spline curve. Knots with multiplicit greater than 1 are repeated

Standard_Real Geom_BSplineCurve::LastParameter ( ) const

virtual

Computes the parametric value of the end point of the curve. It is a knot value.

Implements Geom_Curve.

Standard_Integer Geom_BSplineCurve::LastUKnotIndex ( ) 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.

void Geom_BSplineCurve::LocalD0	(	const Standard_Real	U,
		const Standard_Integer	FromK1,
		const Standard_Integer	ToK2,
		gp_Pnt &	P
	)		const

Raised if FromK1 = ToK2.