BSplCLib Class Reference

BSplCLib   B-spline curve Library. <br>

The BSplCLib package is a basic library for BSplines. It
provides three categories of functions.

More...

#include <BSplCLib.hxx>

Static Public Member Functions
static void	Hunt (const TColStd_Array1OfReal &XX, const Standard_Real X, Standard_Integer &Iloc)
	This routine searches the position of the real <br> value X in the ordered set of real values XX. <br> The elements in the table XX are either monotonically increasing or monotonically decreasing. The input value Iloc is used to initialize the algorithm : if Iloc is outside of the bounds [XX.Lower(), – XX.Upper()] the bisection algorithm is used else the routine searches from a previous known position by increasing steps then converges by bisection. This routine is used to locate a knot value in a set of knots. More...
static Standard_Integer	FirstUKnotIndex (const Standard_Integer Degree, const TColStd_Array1OfInteger &Mults)
	Computes the index of the knots value which gives the start point of the curve. More...
static Standard_Integer	LastUKnotIndex (const Standard_Integer Degree, const TColStd_Array1OfInteger &Mults)
	Computes the index of the knots value which gives the end point of the curve. More...
static Standard_Integer	FlatIndex (const Standard_Integer Degree, const Standard_Integer Index, const TColStd_Array1OfInteger &Mults, const Standard_Boolean Periodic)
	Computes the index of the flats knots sequence corresponding to <Index> in the knots sequence which multiplicities are <Mults>. More...
static void	LocateParameter (const Standard_Integer Degree, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const Standard_Real U, const Standard_Boolean IsPeriodic, const Standard_Integer FromK1, const Standard_Integer ToK2, Standard_Integer &KnotIndex, Standard_Real &NewU)
	Locates the parametric value U in the knots <br> sequence between the knot K1 and the knot K2. <br> The value return in Index verifies. <br> Knots(Index) <= U < Knots(Index + 1) if U <= Knots (K1) then Index = K1 if U >= Knots (K2) then Index = K2 - 1 If Periodic is True U may be modified to fit in the range Knots(K1), Knots(K2). In any case the correct value is returned in NewU. Warnings :Index is used as input data to initialize the searching function. Warning: Knots have to be "withe repetitions" More...
static void	LocateParameter (const Standard_Integer Degree, const TColStd_Array1OfReal &Knots, const Standard_Real U, const Standard_Boolean IsPeriodic, const Standard_Integer FromK1, const Standard_Integer ToK2, Standard_Integer &KnotIndex, Standard_Real &NewU)
	Locates the parametric value U in the knots <br> sequence between the knot K1 and the knot K2. <br> The value return in Index verifies. <br> Knots(Index) <= U < Knots(Index + 1) if U <= Knots (K1) then Index = K1 if U >= Knots (K2) then Index = K2 - 1 If Periodic is True U may be modified to fit in the range Knots(K1), Knots(K2). In any case the correct value is returned in NewU. Warnings :Index is used as input data to initialize the searching function. Warning: Knots have to be "flat" More...
static void	LocateParameter (const Standard_Integer Degree, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const Standard_Real U, const Standard_Boolean Periodic, Standard_Integer &Index, Standard_Real &NewU)
static Standard_Integer	MaxKnotMult (const TColStd_Array1OfInteger &Mults, const Standard_Integer K1, const Standard_Integer K2)
	Finds the greatest multiplicity in a set of knots between K1 and K2. Mults is the multiplicity associated with each knot value. More...
static Standard_Integer	MinKnotMult (const TColStd_Array1OfInteger &Mults, const Standard_Integer K1, const Standard_Integer K2)
	Finds the lowest multiplicity in a set of knots between K1 and K2. Mults is the multiplicity associated with each knot value. More...
static Standard_Integer	NbPoles (const Standard_Integer Degree, const Standard_Boolean Periodic, const TColStd_Array1OfInteger &Mults)
	Returns the number of poles of the curve. Returns 0 if <br> one of the multiplicities is incorrect. <br> More...
static Standard_Integer	KnotSequenceLength (const TColStd_Array1OfInteger &Mults, const Standard_Integer Degree, const Standard_Boolean Periodic)
	Returns the length of the sequence of knots with <br> repetition. <br> Periodic : Sum(Mults(i), i = Mults.Lower(); i <= Mults.Upper()); Non Periodic : Sum(Mults(i); i = Mults.Lower(); i < Mults.Upper()) More...
static void	KnotSequence (const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColStd_Array1OfReal &KnotSeq)
static void	KnotSequence (const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const Standard_Integer Degree, const Standard_Boolean Periodic, TColStd_Array1OfReal &KnotSeq)
	Computes the sequence of knots KnotSeq with <br> repetition of the knots of multiplicity greater <br> than 1. <br> Length of KnotSeq must be KnotSequenceLength(Mults,Degree,Periodic) More...
static Standard_Integer	KnotsLength (const TColStd_Array1OfReal &KnotSeq, const Standard_Boolean Periodic=Standard_False)
	Returns the length of the sequence of knots (and Mults) without repetition. More...
static void	Knots (const TColStd_Array1OfReal &KnotSeq, TColStd_Array1OfReal &Knots, TColStd_Array1OfInteger &Mults, const Standard_Boolean Periodic=Standard_False)
	Computes the sequence of knots Knots without <br> repetition of the knots of multiplicity greater <br> than 1. <br> Length of <Knots> and <Mults> must be KnotsLength(KnotSequence,Periodic) More...
static BSplCLib_KnotDistribution	KnotForm (const TColStd_Array1OfReal &Knots, const Standard_Integer FromK1, const Standard_Integer ToK2)
	Analyses if the knots distribution is "Uniform" or "NonUniform" between the knot FromK1 and the knot ToK2. There is no repetition of knot in the knots'sequence <Knots>. More...
static BSplCLib_MultDistribution	MultForm (const TColStd_Array1OfInteger &Mults, const Standard_Integer FromK1, const Standard_Integer ToK2)
	Analyses the distribution of multiplicities between the knot FromK1 and the Knot ToK2. More...
static void	KnotAnalysis (const Standard_Integer Degree, const Standard_Boolean Periodic, const TColStd_Array1OfReal &CKnots, const TColStd_Array1OfInteger &CMults, GeomAbs_BSplKnotDistribution &KnotForm, Standard_Integer &MaxKnotMult)
	Analyzes the array of knots. Returns the form and the maximum knot multiplicity. More...
static void	Reparametrize (const Standard_Real U1, const Standard_Real U2, TColStd_Array1OfReal &Knots)
	Reparametrizes a B-spline curve to [U1, U2]. The knot values are recomputed such that Knots (Lower) = U1 and Knots (Upper) = U2 but the knot form is not modified. Warnings : In the array Knots the values must be in ascending order. U1 must not be equal to U2 to avoid division by zero. More...
static void	Reverse (TColStd_Array1OfReal &Knots)
	Reverses the array knots to become the knots sequence of the reversed curve. More...
static void	Reverse (TColStd_Array1OfInteger &Mults)
	Reverses the array of multiplicities. More...
static void	Reverse (TColgp_Array1OfPnt &Poles, const Standard_Integer Last)
	Reverses the array of poles. Last is the index of <br> the new first pole. On a non periodic curve last <br> is Poles.Upper(). On a periodic curve last is <br> (number of flat knots - degree - 1) or (sum of multiplicities(but for the last) + degree More...
static void	Reverse (TColgp_Array1OfPnt2d &Poles, const Standard_Integer Last)
	Reverses the array of poles. More...
static void	Reverse (TColStd_Array1OfReal &Weights, const Standard_Integer Last)
	Reverses the array of poles. More...
static Standard_Boolean	IsRational (const TColStd_Array1OfReal &Weights, const Standard_Integer I1, const Standard_Integer I2, const Standard_Real Epsilon=0.0)
	Returns False if all the weights of the array <Weights> between I1 an I2 are identic. Epsilon is used for comparing weights. If Epsilon is 0. the Epsilon of the first weight is used. More...
static Standard_Integer	MaxDegree ()
	returns the degree maxima for a BSplineCurve. More...
static void	Eval (const Standard_Real U, const Standard_Integer Degree, Standard_Real &Knots, const Standard_Integer Dimension, Standard_Real &Poles)
	Perform the Boor algorithm to evaluate a point at <br> parameter <U>, with <Degree> and <Dimension>. <br> Poles is an array of Reals of size <Dimension> * <Degree>+1 Containing the poles. At the end <Poles> contains the current point. More...
static void	BoorScheme (const Standard_Real U, const Standard_Integer Degree, Standard_Real &Knots, const Standard_Integer Dimension, Standard_Real &Poles, const Standard_Integer Depth, const Standard_Integer Length)
	Performs the Boor Algorithm at parameter <U> with <br> the given <Degree> and the array of <Knots> on the <br> poles <Poles> of dimension <Dimension>. The schema <br> is computed until level <Depth> on a basis of <br> <Length+1> poles. <br> More...
static Standard_Boolean	AntiBoorScheme (const Standard_Real U, const Standard_Integer Degree, Standard_Real &Knots, const Standard_Integer Dimension, Standard_Real &Poles, const Standard_Integer Depth, const Standard_Integer Length, const Standard_Real Tolerance)
	Compute the content of Pole before the BoorScheme. <br> This method is used to remove poles. <br> U is the poles to remove, Knots should contains the knots of the curve after knot removal. The first and last poles do not change, the other poles are computed by averaging two possible values. The distance between the two possible poles is computed, if it is higher than <Tolerance> False is returned. More...
static void	Derivative (const Standard_Integer Degree, Standard_Real &Knots, const Standard_Integer Dimension, const Standard_Integer Length, const Standard_Integer Order, Standard_Real &Poles)
	Computes the poles of the BSpline giving the <br> derivatives of order <Order>. <br> The formula for the first order is Pole(i) = Degree * (Pole(i+1) - Pole(i)) / (Knots(i+Degree+1) - Knots(i+1)) This formula is repeated (Degree is decremented at each step). More...
static void	Bohm (const Standard_Real U, const Standard_Integer Degree, const Standard_Integer N, Standard_Real &Knots, const Standard_Integer Dimension, Standard_Real &Poles)
	Performs the Bohm Algorithm at parameter <U>. This <br> algorithm computes the value and all the derivatives <br> up to order N (N <= Degree). <br> <Poles> is the original array of poles. The result in <Poles> is the value and the derivatives. Poles[0] is the value, Poles[Degree] is the last derivative. More...
static TColStd_Array1OfReal &	NoWeights ()
	Used as argument for a non rational curve. <br> More...
static TColStd_Array1OfInteger &	NoMults ()
	Used as argument for a flatknots evaluation. <br> More...
static void	BuildKnots (const Standard_Integer Degree, const Standard_Integer Index, const Standard_Boolean Periodic, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, Standard_Real &LK)
	Stores in LK the usefull knots for the BoorSchem on the span Knots(Index) - Knots(Index+1) More...
static Standard_Integer	PoleIndex (const Standard_Integer Degree, const Standard_Integer Index, const Standard_Boolean Periodic, const TColStd_Array1OfInteger &Mults)
	Return the index of the first Pole to use on the span Mults(Index) - Mults(Index+1). This index must be added to Poles.Lower(). More...
static void	BuildEval (const Standard_Integer Degree, const Standard_Integer Index, const TColStd_Array1OfReal &Poles, const TColStd_Array1OfReal &Weights, Standard_Real &LP)
static void	BuildEval (const Standard_Integer Degree, const Standard_Integer Index, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, Standard_Real &LP)
static void	BuildEval (const Standard_Integer Degree, const Standard_Integer Index, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, Standard_Real &LP)
	Copy in <LP> the poles and weights for the Eval scheme. starting from Poles(Poles.Lower()+Index) More...
static void	BuildBoor (const Standard_Integer Index, const Standard_Integer Length, const Standard_Integer Dimension, const TColStd_Array1OfReal &Poles, Standard_Real &LP)
	Copy in <LP> poles for <Dimension> Boor scheme. Starting from <Index> * <Dimension>, copy <Length+1> poles. More...
static Standard_Integer	BoorIndex (const Standard_Integer Index, const Standard_Integer Length, const Standard_Integer Depth)
	Returns the index in the Boor result array of the poles <Index>. If the Boor algorithm was perform with <Length> and <Depth>. More...
static void	GetPole (const Standard_Integer Index, const Standard_Integer Length, const Standard_Integer Depth, const Standard_Integer Dimension, Standard_Real &LocPoles, Standard_Integer &Position, TColStd_Array1OfReal &Pole)
	Copy the pole at position <Index> in the Boor scheme of dimension <Dimension> to <Position> in the array <Pole>. <Position> is updated. More...
static Standard_Boolean	PrepareInsertKnots (const Standard_Integer Degree, const Standard_Boolean Periodic, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const TColStd_Array1OfReal &AddKnots, const TColStd_Array1OfInteger &AddMults, Standard_Integer &NbPoles, Standard_Integer &NbKnots, const Standard_Real Epsilon, const Standard_Boolean Add=Standard_True)
	Returns in <NbPoles, NbKnots> the new number of poles <br> and knots if the sequence of knots <AddKnots, <br> AddMults> is inserted in the sequence <Knots, Mults>. <br> Epsilon is used to compare knots for equality. If Add is True the multiplicities on equal knots are added. If Add is False the max value of the multiplicities is kept. Return False if : The knew knots are knot increasing. The new knots are not in the range. More...
static void	InsertKnots (const Standard_Integer Degree, const Standard_Boolean Periodic, const Standard_Integer Dimension, const TColStd_Array1OfReal &Poles, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const TColStd_Array1OfReal &AddKnots, const TColStd_Array1OfInteger &AddMults, TColStd_Array1OfReal &NewPoles, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults, const Standard_Real Epsilon, const Standard_Boolean Add=Standard_True)
static void	InsertKnots (const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const TColStd_Array1OfReal &AddKnots, const TColStd_Array1OfInteger &AddMults, TColgp_Array1OfPnt &NewPoles, TColStd_Array1OfReal &NewWeights, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults, const Standard_Real Epsilon, const Standard_Boolean Add=Standard_True)
static void	InsertKnots (const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const TColStd_Array1OfReal &AddKnots, const TColStd_Array1OfInteger &AddMults, TColgp_Array1OfPnt2d &NewPoles, TColStd_Array1OfReal &NewWeights, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults, const Standard_Real Epsilon, const Standard_Boolean Add=Standard_True)
	Insert a sequence of knots <AddKnots> with <br> multiplicities <AddMults>. <AddKnots> must be a non decreasing sequence and verifies : Knots(Knots.Lower()) <= AddKnots(AddKnots.Lower()) Knots(Knots.Upper()) >= AddKnots(AddKnots.Upper()) The NewPoles and NewWeights arrays must have a length : Poles.Length() + Sum(AddMults()) When a knot to insert is identic to an existing knot the multiplicities are added. Epsilon is used to test knots for equality. When AddMult is negative or null the knot is not inserted. No multiplicity will becomes higher than the degree. The new Knots and Multiplicities are copied in <NewKnots> and <NewMults>. All the New arrays should be correctly dimensioned. When all the new knots are existing knots, i.e. only the multiplicities will change it is safe to use the same arrays as input and output. More...
static void	InsertKnot (const Standard_Integer UIndex, const Standard_Real U, const Standard_Integer UMult, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColgp_Array1OfPnt &NewPoles, TColStd_Array1OfReal &NewWeights)
static void	InsertKnot (const Standard_Integer UIndex, const Standard_Real U, const Standard_Integer UMult, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColgp_Array1OfPnt2d &NewPoles, TColStd_Array1OfReal &NewWeights)
	Insert a new knot U of multiplicity UMult in the <br> knot sequence. The location of the new Knot should be given as an input data. UIndex locates the new knot U in the knot sequence and Knots (UIndex) < U < Knots (UIndex + 1). The new control points corresponding to this insertion are returned. Knots and Mults are not updated. More...
static void	RaiseMultiplicity (const Standard_Integer KnotIndex, const Standard_Integer Mult, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColgp_Array1OfPnt &NewPoles, TColStd_Array1OfReal &NewWeights)
static void	RaiseMultiplicity (const Standard_Integer KnotIndex, const Standard_Integer Mult, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColgp_Array1OfPnt2d &NewPoles, TColStd_Array1OfReal &NewWeights)
	Raise the multiplicity of knot to <UMult>. <br> The new control points are returned. Knots and Mults are not updated. More...
static Standard_Boolean	RemoveKnot (const Standard_Integer Index, const Standard_Integer Mult, const Standard_Integer Degree, const Standard_Boolean Periodic, const Standard_Integer Dimension, const TColStd_Array1OfReal &Poles, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColStd_Array1OfReal &NewPoles, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults, const Standard_Real Tolerance)
static Standard_Boolean	RemoveKnot (const Standard_Integer Index, const Standard_Integer Mult, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColgp_Array1OfPnt &NewPoles, TColStd_Array1OfReal &NewWeights, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults, const Standard_Real Tolerance)
static Standard_Boolean	RemoveKnot (const Standard_Integer Index, const Standard_Integer Mult, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColgp_Array1OfPnt2d &NewPoles, TColStd_Array1OfReal &NewWeights, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults, const Standard_Real Tolerance)
	Decrement the multiplicity of <Knots(Index)> <br> to <Mult>. If <Mult> is null the knot is <br> removed. <br> As there are two ways to compute the new poles the midlle will be used as long as the distance is lower than Tolerance. If a distance is bigger than tolerance the methods returns False and the new arrays are not modified. A low tolerance can be used to test if the knot can be removed without modifying the curve. A high tolerance can be used to "smooth" the curve. More...
static Standard_Integer	IncreaseDegreeCountKnots (const Standard_Integer Degree, const Standard_Integer NewDegree, const Standard_Boolean Periodic, const TColStd_Array1OfInteger &Mults)
	Returns the number of knots of a curve with multiplicities <Mults> after elevating the degree from <Degree> to <NewDegree>. See the IncreaseDegree method for more comments. More...
static void	IncreaseDegree (const Standard_Integer Degree, const Standard_Integer NewDegree, const Standard_Boolean Periodic, const Standard_Integer Dimension, const TColStd_Array1OfReal &Poles, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColStd_Array1OfReal &NewPoles, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults)
static void	IncreaseDegree (const Standard_Integer Degree, const Standard_Integer NewDegree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColgp_Array1OfPnt &NewPoles, TColStd_Array1OfReal &NewWeights, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults)
static void	IncreaseDegree (const Standard_Integer Degree, const Standard_Integer NewDegree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, TColgp_Array1OfPnt2d &NewPoles, TColStd_Array1OfReal &NewWeights, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults)
static void	IncreaseDegree (const Standard_Integer NewDegree, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, TColgp_Array1OfPnt &NewPoles, TColStd_Array1OfReal &NewWeights)
static void	IncreaseDegree (const Standard_Integer NewDegree, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, TColgp_Array1OfPnt2d &NewPoles, TColStd_Array1OfReal &NewWeights)
	Increase the degree of a bspline (or bezier) curve <br> of dimension <Dimension> form <Degree> to <br> <NewDegree>. <br> The number of poles in the new curve is : Poles.Length() + (NewDegree - Degree) * Number of spans Where the number of spans is : LastUKnotIndex(Mults) - FirstUKnotIndex(Mults) + 1 for a non-periodic curve And Knots.Length() - 1 for a periodic curve. The multiplicities of all knots are increased by the degree elevation. The new knots are usually the same knots with the exception of a non-periodic curve with the first and last multiplicity not equal to Degree+1 where knots are removed form the start and the bottom untils the sum of the multiplicities is equal to NewDegree+1 at the knots corresponding to the first and last parameters of the curve. Example : Suppose a curve of degree 3 starting with following knots and multiplicities : knot : 0. 1. 2. mult : 1 2 1 The FirstUKnot is 2. because the sum of multiplicities is Degree+1 : 1 + 2 + 1 = 4 = 3 + 1 i.e. the first parameter of the curve is 2. and will still be 2. after degree elevation. Let raises this curve to degree 4. The multiplicities are increased by 2. They become 2 3 2. But we need a sum of multiplicities of 5 at knot 2. So the first knot is removed and the new knots are : knot : 1. 2. mult : 3 2 The multipicity of the first knot may also be reduced if the sum is still to big. In the most common situations (periodic curve or curve with first and last multiplicities equals to Degree+1) the knots are knot changes. The method IncreaseDegreeCountKnots can be used to compute the new number of knots. More...
static void	PrepareUnperiodize (const Standard_Integer Degree, const TColStd_Array1OfInteger &Mults, Standard_Integer &NbKnots, Standard_Integer &NbPoles)
	Set in <NbKnots> and <NbPolesToAdd> the number of Knots and Poles of the NotPeriodic Curve identical at the periodic curve with a degree <Degree> , a knots-distribution with Multiplicities <Mults>. More...
static void	Unperiodize (const Standard_Integer Degree, const Standard_Integer Dimension, const TColStd_Array1OfInteger &Mults, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfReal &Poles, TColStd_Array1OfInteger &NewMults, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfReal &NewPoles)
static void	Unperiodize (const Standard_Integer Degree, const TColStd_Array1OfInteger &Mults, const TColStd_Array1OfReal &Knots, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, TColStd_Array1OfInteger &NewMults, TColStd_Array1OfReal &NewKnots, TColgp_Array1OfPnt &NewPoles, TColStd_Array1OfReal &NewWeights)
static void	Unperiodize (const Standard_Integer Degree, const TColStd_Array1OfInteger &Mults, const TColStd_Array1OfReal &Knots, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, TColStd_Array1OfInteger &NewMults, TColStd_Array1OfReal &NewKnots, TColgp_Array1OfPnt2d &NewPoles, TColStd_Array1OfReal &NewWeights)
static void	PrepareTrimming (const Standard_Integer Degree, const Standard_Boolean Periodic, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const Standard_Real U1, const Standard_Real U2, Standard_Integer &NbKnots, Standard_Integer &NbPoles)
	Set in <NbKnots> and <NbPoles> the number of Knots and Poles of the curve resulting of the trimming of the BSplinecurve definded with <degree>, <knots>, <mults> More...
static void	Trimming (const Standard_Integer Degree, const Standard_Boolean Periodic, const Standard_Integer Dimension, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const TColStd_Array1OfReal &Poles, const Standard_Real U1, const Standard_Real U2, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults, TColStd_Array1OfReal &NewPoles)
static void	Trimming (const Standard_Integer Degree, const Standard_Boolean Periodic, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const Standard_Real U1, const Standard_Real U2, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults, TColgp_Array1OfPnt &NewPoles, TColStd_Array1OfReal &NewWeights)
static void	Trimming (const Standard_Integer Degree, const Standard_Boolean Periodic, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const Standard_Real U1, const Standard_Real U2, TColStd_Array1OfReal &NewKnots, TColStd_Array1OfInteger &NewMults, TColgp_Array1OfPnt2d &NewPoles, TColStd_Array1OfReal &NewWeights)
static void	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)
static void	D0 (const Standard_Real U, const Standard_Integer Index, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, gp_Pnt &P)
static void	D0 (const Standard_Real U, const Standard_Integer UIndex, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, gp_Pnt2d &P)
static void	D0 (const Standard_Real U, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &P)
static void	D0 (const Standard_Real U, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &P)
static void	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)
static void	D1 (const Standard_Real U, const Standard_Integer Index, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, gp_Pnt &P, gp_Vec &V)
static void	D1 (const Standard_Real U, const Standard_Integer UIndex, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, gp_Pnt2d &P, gp_Vec2d &V)
static void	D1 (const Standard_Real U, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &P, gp_Vec &V)
static void	D1 (const Standard_Real U, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &P, gp_Vec2d &V)
static void	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)
static void	D2 (const Standard_Real U, const Standard_Integer Index, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, gp_Pnt &P, gp_Vec &V1, gp_Vec &V2)
static void	D2 (const Standard_Real U, const Standard_Integer UIndex, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, gp_Pnt2d &P, gp_Vec2d &V1, gp_Vec2d &V2)
static void	D2 (const Standard_Real U, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &P, gp_Vec &V1, gp_Vec &V2)
static void	D2 (const Standard_Real U, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &P, gp_Vec2d &V1, gp_Vec2d &V2)
static void	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)
static void	D3 (const Standard_Real U, const Standard_Integer Index, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, gp_Pnt &P, gp_Vec &V1, gp_Vec &V2, gp_Vec &V3)
static void	D3 (const Standard_Real U, const Standard_Integer UIndex, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, gp_Pnt2d &P, gp_Vec2d &V1, gp_Vec2d &V2, gp_Vec2d &V3)
static void	D3 (const Standard_Real U, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &P, gp_Vec &V1, gp_Vec &V2, gp_Vec &V3)
static void	D3 (const Standard_Real U, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &P, gp_Vec2d &V1, gp_Vec2d &V2, gp_Vec2d &V3)
static void	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)
static void	DN (const Standard_Real U, const Standard_Integer N, const Standard_Integer Index, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, gp_Vec &VN)
static void	DN (const Standard_Real U, const Standard_Integer N, const Standard_Integer UIndex, const Standard_Integer Degree, const Standard_Boolean Periodic, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &Knots, const TColStd_Array1OfInteger &Mults, gp_Vec2d &V)
static void	DN (const Standard_Real U, const Standard_Integer N, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &P, gp_Vec &VN)
static void	DN (const Standard_Real U, const Standard_Integer N, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &P, gp_Vec2d &VN)
	The above functions compute values and <br> derivatives in the following situations : <br> More...
static Standard_Integer	EvalBsplineBasis (const Standard_Integer Side, const Standard_Integer DerivativeOrder, const Standard_Integer Order, const TColStd_Array1OfReal &FlatKnots, const Standard_Real Parameter, Standard_Integer &FirstNonZeroBsplineIndex, math_Matrix &BsplineBasis)
	This evaluates the Bspline Basis at a given parameter Parameter up to the requested DerivativeOrder and store the result in the array BsplineBasis in the following fashion BSplineBasis(1,1) = value of first non vanishing Bspline function which has Index FirstNonZeroBsplineIndex BsplineBasis(1,2) = value of second non vanishing Bspline function which has Index FirstNonZeroBsplineIndex + 1 BsplineBasis(1,n) = value of second non vanishing non vanishing Bspline function which has Index FirstNonZeroBsplineIndex + n (n <= Order) BSplineBasis(2,1) = value of derivative of first non vanishing Bspline function which has Index FirstNonZeroBsplineIndex BSplineBasis(N,1) = value of Nth derivative of first non vanishing Bspline function which has Index FirstNonZeroBsplineIndex if N <= DerivativeOrder + 1 More...
static Standard_Integer	BuildBSpMatrix (const TColStd_Array1OfReal &Parameters, const TColStd_Array1OfInteger &OrderArray, const TColStd_Array1OfReal &FlatKnots, const Standard_Integer Degree, math_Matrix &Matrix, Standard_Integer &UpperBandWidth, Standard_Integer &LowerBandWidth)
	This Builds a fully blown Matrix of <br> (ni) <br> Bi (tj) <br> with i and j within 1..Order + NumPoles The integer ni is the ith slot of the array OrderArray, tj is the jth slot of the array Parameters More...
static Standard_Integer	FactorBandedMatrix (math_Matrix &Matrix, const Standard_Integer UpperBandWidth, const Standard_Integer LowerBandWidth, Standard_Integer &PivotIndexProblem)
	this factors the Banded Matrix in the LU form with a Banded storage of components of the L matrix WARNING : do not use if the Matrix is totally positive (It is the case for Bspline matrices build as above with parameters being the Schoenberg points More...
static Standard_Integer	SolveBandedSystem (const math_Matrix &Matrix, const Standard_Integer UpperBandWidth, const Standard_Integer LowerBandWidth, const Standard_Integer ArrayDimension, Standard_Real &Array)
	This solves the system Matrix.X = B with when Matrix is factored in LU form The Array is an seen as an Array[1..N][1..ArrayDimension] with N = the rank of the matrix Matrix. The result is stored in Array when each coordinate is solved that is B is the array whose values are B[i] = Array[i][p] for each p in 1..ArrayDimension More...
static Standard_Integer	SolveBandedSystem (const math_Matrix &Matrix, const Standard_Integer UpperBandWidth, const Standard_Integer LowerBandWidth, TColgp_Array1OfPnt2d &Array)
	This solves the system Matrix.X = B with when Matrix is factored in LU form The Array has the length of the rank of the matrix Matrix. The result is stored in Array when each coordinate is solved that is B is the array whose values are B[i] = Array[i][p] for each p in 1..ArrayDimension More...
static Standard_Integer	SolveBandedSystem (const math_Matrix &Matrix, const Standard_Integer UpperBandWidth, const Standard_Integer LowerBandWidth, TColgp_Array1OfPnt &Array)
	This solves the system Matrix.X = B with when Matrix is factored in LU form The Array has the length of the rank of the matrix Matrix. The result is stored in Array when each coordinate is solved that is B is the array whose values are B[i] = Array[i][p] for each p in 1..ArrayDimension More...
static Standard_Integer	SolveBandedSystem (const math_Matrix &Matrix, const Standard_Integer UpperBandWidth, const Standard_Integer LowerBandWidth, const Standard_Boolean HomogenousFlag, const Standard_Integer ArrayDimension, Standard_Real &Array, Standard_Real &Weights)
static Standard_Integer	SolveBandedSystem (const math_Matrix &Matrix, const Standard_Integer UpperBandWidth, const Standard_Integer LowerBandWidth, const Standard_Boolean HomogenousFlag, TColgp_Array1OfPnt2d &Array, TColStd_Array1OfReal &Weights)
	This solves the system Matrix.X = B with when Matrix is factored in LU form The Array is an seen as an Array[1..N][1..ArrayDimension] with N = the rank of the matrix Matrix. The result is stored in Array when each coordinate is solved that is B is the array whose values are B[i] = Array[i][p] for each p in 1..ArrayDimension. If HomogeneousFlag == 0 the Poles are multiplied by the Weights uppon Entry and once interpolation is carried over the result of the poles are divided by the result of the interpolation of the weights. Otherwise if HomogenousFlag == 1 the Poles and Weigths are treated homogenously that is that those are interpolated as they are and result is returned without division by the interpolated weigths. More...
static Standard_Integer	SolveBandedSystem (const math_Matrix &Matrix, const Standard_Integer UpperBandWidth, const Standard_Integer LowerBandWidth, const Standard_Boolean HomogeneousFlag, TColgp_Array1OfPnt &Array, TColStd_Array1OfReal &Weights)
	This solves the system Matrix.X = B with when Matrix is factored in LU form The Array is an seen as an Array[1..N][1..ArrayDimension] with N = the rank of the matrix Matrix. The result is stored in Array when each coordinate is solved that is B is the array whose values are B[i] = Array[i][p] for each p in 1..ArrayDimension If HomogeneousFlag == 0 the Poles are multiplied by the Weights uppon Entry and once interpolation is carried over the result of the poles are divided by the result of the interpolation of the weights. Otherwise if HomogenousFlag == 1 the Poles and Weigths are treated homogenously that is that those are interpolated as they are and result is returned without division by the interpolated weigths. More...
static void	MergeBSplineKnots (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)
	Merges two knot vector by setting the starting and ending values to StartValue and EndValue More...
static void	FunctionReparameterise (const BSplCLib_EvaluatorFunction &Function, 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)
	This function will compose a given Vectorial BSpline F(t) <br> defined by its BSplineDegree and BSplineFlatKnotsl, <br> its Poles array which are coded as an array of Real <br> of the form [1..NumPoles][1..PolesDimension] with a <br> function a(t) which is assumed to satisfy the <br> following: <br> More...
static void	FunctionReparameterise (const BSplCLib_EvaluatorFunction &Function, 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)
	This function will compose a given Vectorial BSpline F(t) <br> defined by its BSplineDegree and BSplineFlatKnotsl, <br> its Poles array which are coded as an array of Real <br> of the form [1..NumPoles][1..PolesDimension] with a <br> function a(t) which is assumed to satisfy the <br> following: <br> More...
static void	FunctionReparameterise (const BSplCLib_EvaluatorFunction &Function, const Standard_Integer BSplineDegree, const TColStd_Array1OfReal &BSplineFlatKnots, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &FlatKnots, const Standard_Integer NewDegree, TColgp_Array1OfPnt &NewPoles, Standard_Integer &Status)
	this will compose a given Vectorial BSpline F(t) defined by its BSplineDegree and BSplineFlatKnotsl, its Poles array which are coded as an array of Real of the form [1..NumPoles][1..PolesDimension] with a function a(t) which is assumed to satisfy the following : 1. F(a(t)) is a polynomial BSpline that can be expressed exactly as a BSpline of degree NewDegree on the knots FlatKnots More...
static void	FunctionReparameterise (const BSplCLib_EvaluatorFunction &Function, const Standard_Integer BSplineDegree, const TColStd_Array1OfReal &BSplineFlatKnots, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &FlatKnots, const Standard_Integer NewDegree, TColgp_Array1OfPnt2d &NewPoles, Standard_Integer &Status)
	this will compose a given Vectorial BSpline F(t) defined by its BSplineDegree and BSplineFlatKnotsl, its Poles array which are coded as an array of Real of the form [1..NumPoles][1..PolesDimension] with a function a(t) which is assumed to satisfy the following : 1. F(a(t)) is a polynomial BSpline that can be expressed exactly as a BSpline of degree NewDegree on the knots FlatKnots More...
static void	FunctionMultiply (const BSplCLib_EvaluatorFunction &Function, 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)
	this will multiply a given Vectorial BSpline F(t) <br> defined by its BSplineDegree and BSplineFlatKnotsl, <br> its Poles array which are coded as an array of Real <br> of the form [1..NumPoles][1..PolesDimension] by a <br> function a(t) which is assumed to satisfy the <br> following : 1. a(t) * F(t) is a polynomial BSpline <br> that can be expressed exactly as a BSpline of degree <br> NewDegree on the knots FlatKnots 2. the range of a(t) <br> is the same as the range of F(t) <br> Warning: it is the caller's responsability to insure that conditions More...
static void	FunctionMultiply (const BSplCLib_EvaluatorFunction &Function, 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)
	this will multiply a given Vectorial BSpline F(t) <br> defined by its BSplineDegree and BSplineFlatKnotsl, <br> its Poles array which are coded as an array of Real <br> of the form [1..NumPoles][1..PolesDimension] by a <br> function a(t) which is assumed to satisfy the <br> following : 1. a(t) * F(t) is a polynomial BSpline <br> that can be expressed exactly as a BSpline of degree <br> NewDegree on the knots FlatKnots 2. the range of a(t) <br> is the same as the range of F(t) <br> Warning: it is the caller's responsability to insure that conditions More...
static void	FunctionMultiply (const BSplCLib_EvaluatorFunction &Function, const Standard_Integer BSplineDegree, const TColStd_Array1OfReal &BSplineFlatKnots, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &FlatKnots, const Standard_Integer NewDegree, TColgp_Array1OfPnt2d &NewPoles, Standard_Integer &Status)
	this will multiply a given Vectorial BSpline F(t) <br> defined by its BSplineDegree and BSplineFlatKnotsl, <br> its Poles array which are coded as an array of Real <br> of the form [1..NumPoles][1..PolesDimension] by a <br> function a(t) which is assumed to satisfy the <br> following : 1. a(t) * F(t) is a polynomial BSpline <br> that can be expressed exactly as a BSpline of degree <br> NewDegree on the knots FlatKnots 2. the range of a(t) <br> is the same as the range of F(t) <br> Warning: it is the caller's responsability to insure that conditions More...
static void	FunctionMultiply (const BSplCLib_EvaluatorFunction &Function, const Standard_Integer BSplineDegree, const TColStd_Array1OfReal &BSplineFlatKnots, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &FlatKnots, const Standard_Integer NewDegree, TColgp_Array1OfPnt &NewPoles, Standard_Integer &Status)
	this will multiply a given Vectorial BSpline F(t) <br> defined by its BSplineDegree and BSplineFlatKnotsl, <br> its Poles array which are coded as an array of Real <br> of the form [1..NumPoles][1..PolesDimension] by a <br> function a(t) which is assumed to satisfy the <br> following : 1. a(t) * F(t) is a polynomial BSpline <br> that can be expressed exactly as a BSpline of degree <br> NewDegree on the knots FlatKnots 2. the range of a(t) <br> is the same as the range of F(t) <br> Warning: it is the caller's responsability to insure that conditions More...
static void	Eval (const Standard_Real U, const Standard_Boolean PeriodicFlag, const Standard_Integer DerivativeRequest, Standard_Integer &ExtrapMode, const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const Standard_Integer ArrayDimension, Standard_Real &Poles, Standard_Real &Result)
	Perform the De Boor algorithm to evaluate a point at <br> parameter <U>, with <Degree> and <Dimension>. <br> Poles is an array of Reals of size <Dimension> * <Degree>+1 Containing the poles. At the end <Poles> contains the current point. Poles Contain all the poles of the BsplineCurve, Knots also Contains all the knots of the BsplineCurve. ExtrapMode has two slots [0] = Degree used to extrapolate before the first knot [1] = Degre used to extrapolate after the last knot has to be between 1 and Degree More...
static void	Eval (const Standard_Real U, const Standard_Boolean PeriodicFlag, const Standard_Integer DerivativeRequest, Standard_Integer &ExtrapMode, const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const Standard_Integer ArrayDimension, Standard_Real &Poles, Standard_Real &Weights, Standard_Real &PolesResult, Standard_Real &WeightsResult)
	Perform the De Boor algorithm to evaluate a point at <br> parameter <U>, with <Degree> and <Dimension>. <br> Evaluates by multiplying the Poles by the Weights and <br> gives the homogeneous result in PolesResult that is <br> the results of the evaluation of the numerator once it <br> has been multiplied by the weights and in <br> WeightsResult one has the result of the evaluation of <br> the denominator <br> Warning: <PolesResult> and <WeightsResult> must be dimensionned properly. More...
static void	Eval (const Standard_Real U, const Standard_Boolean PeriodicFlag, const Standard_Boolean HomogeneousFlag, Standard_Integer &ExtrapMode, const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &Point, Standard_Real &Weight)
	Perform the evaluation of the Bspline Basis and then multiplies by the weights this just evaluates the current point More...
static void	Eval (const Standard_Real U, const Standard_Boolean PeriodicFlag, const Standard_Boolean HomogeneousFlag, Standard_Integer &ExtrapMode, const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &Point, Standard_Real &Weight)
	Perform the evaluation of the Bspline Basis <br> and then multiplies by the weights <br> this just evaluates the current point <br> More...
static void	TangExtendToConstraint (const TColStd_Array1OfReal &FlatKnots, const Standard_Real C1Coefficient, const Standard_Integer NumPoles, Standard_Real &Poles, const Standard_Integer Dimension, const Standard_Integer Degree, const TColStd_Array1OfReal &ConstraintPoint, const Standard_Integer Continuity, const Standard_Boolean After, Standard_Integer &NbPolesResult, Standard_Integer &NbKnotsRsult, Standard_Real &KnotsResult, Standard_Real &PolesResult)
	Extend a BSpline nD using the tangency map <br> <C1Coefficient> is the coefficient of reparametrisation <br> <Continuity> must be equal to 1, 2 or 3. <br> <Degree> must be greater or equal than <Continuity> + 1. <br> Warning: <KnotsResult> and <PolesResult> must be dimensionned properly. More...
static void	CacheD0 (const Standard_Real U, const Standard_Integer Degree, const Standard_Real CacheParameter, const Standard_Real SpanLenght, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &Point)
	Perform the evaluation of the of the cache the parameter must be normalized between the 0 and 1 for the span. The Cache must be valid when calling this routine. Geom Package will insure that. and then multiplies by the weights this just evaluates the current point the CacheParameter is where the Cache was constructed the SpanLength is to normalize the polynomial in the cache to avoid bad conditioning effects More...
static void	CacheD0 (const Standard_Real U, const Standard_Integer Degree, const Standard_Real CacheParameter, const Standard_Real SpanLenght, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &Point)
	Perform the evaluation of the Bspline Basis and then multiplies by the weights this just evaluates the current point the parameter must be normalized between the 0 and 1 for the span. The Cache must be valid when calling this routine. Geom Package will insure that. and then multiplies by the weights ththe CacheParameter is where the Cache was constructed the SpanLength is to normalize the polynomial in the cache to avoid bad conditioning effectsis just evaluates the current point More...
static void	CoefsD0 (const Standard_Real U, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &Point)
	Calls CacheD0 for Bezier Curves Arrays computed with <br> the method PolesCoefficients. <br> Warning: To be used for Beziercurves ONLY!!! More...
static void	CoefsD0 (const Standard_Real U, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &Point)
	Calls CacheD0 for Bezier Curves Arrays computed with <br> the method PolesCoefficients. <br> Warning: To be used for Beziercurves ONLY!!! More...
static void	CacheD1 (const Standard_Real U, const Standard_Integer Degree, const Standard_Real CacheParameter, const Standard_Real SpanLenght, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &Point, gp_Vec &Vec)
	Perform the evaluation of the of the cache the parameter must be normalized between the 0 and 1 for the span. The Cache must be valid when calling this routine. Geom Package will insure that. and then multiplies by the weights this just evaluates the current point the CacheParameter is where the Cache was constructed the SpanLength is to normalize the polynomial in the cache to avoid bad conditioning effects More...
static void	CacheD1 (const Standard_Real U, const Standard_Integer Degree, const Standard_Real CacheParameter, const Standard_Real SpanLenght, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &Point, gp_Vec2d &Vec)
	Perform the evaluation of the Bspline Basis and then multiplies by the weights this just evaluates the current point the parameter must be normalized between the 0 and 1 for the span. The Cache must be valid when calling this routine. Geom Package will insure that. and then multiplies by the weights ththe CacheParameter is where the Cache was constructed the SpanLength is to normalize the polynomial in the cache to avoid bad conditioning effectsis just evaluates the current point More...
static void	CoefsD1 (const Standard_Real U, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &Point, gp_Vec &Vec)
	Calls CacheD1 for Bezier Curves Arrays computed with <br> the method PolesCoefficients. <br> Warning: To be used for Beziercurves ONLY!!! More...
static void	CoefsD1 (const Standard_Real U, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &Point, gp_Vec2d &Vec)
	Calls CacheD1 for Bezier Curves Arrays computed with <br> the method PolesCoefficients. <br> Warning: To be used for Beziercurves ONLY!!! More...
static void	CacheD2 (const Standard_Real U, const Standard_Integer Degree, const Standard_Real CacheParameter, const Standard_Real SpanLenght, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &Point, gp_Vec &Vec1, gp_Vec &Vec2)
	Perform the evaluation of the of the cache the parameter must be normalized between the 0 and 1 for the span. The Cache must be valid when calling this routine. Geom Package will insure that. and then multiplies by the weights this just evaluates the current point the CacheParameter is where the Cache was constructed the SpanLength is to normalize the polynomial in the cache to avoid bad conditioning effects More...
static void	CacheD2 (const Standard_Real U, const Standard_Integer Degree, const Standard_Real CacheParameter, const Standard_Real SpanLenght, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &Point, gp_Vec2d &Vec1, gp_Vec2d &Vec2)
	Perform the evaluation of the Bspline Basis and then multiplies by the weights this just evaluates the current point the parameter must be normalized between the 0 and 1 for the span. The Cache must be valid when calling this routine. Geom Package will insure that. and then multiplies by the weights ththe CacheParameter is where the Cache was constructed the SpanLength is to normalize the polynomial in the cache to avoid bad conditioning effectsis just evaluates the current point More...
static void	CoefsD2 (const Standard_Real U, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &Point, gp_Vec &Vec1, gp_Vec &Vec2)
	Calls CacheD1 for Bezier Curves Arrays computed with <br> the method PolesCoefficients. <br> Warning: To be used for Beziercurves ONLY!!! More...
static void	CoefsD2 (const Standard_Real U, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &Point, gp_Vec2d &Vec1, gp_Vec2d &Vec2)
	Calls CacheD1 for Bezier Curves Arrays computed with <br> the method PolesCoefficients. <br> Warning: To be used for Beziercurves ONLY!!! More...
static void	CacheD3 (const Standard_Real U, const Standard_Integer Degree, const Standard_Real CacheParameter, const Standard_Real SpanLenght, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &Point, gp_Vec &Vec1, gp_Vec &Vec2, gp_Vec &Vec3)
	Perform the evaluation of the of the cache the parameter must be normalized between the 0 and 1 for the span. The Cache must be valid when calling this routine. Geom Package will insure that. and then multiplies by the weights this just evaluates the current point the CacheParameter is where the Cache was constructed the SpanLength is to normalize the polynomial in the cache to avoid bad conditioning effects More...
static void	CacheD3 (const Standard_Real U, const Standard_Integer Degree, const Standard_Real CacheParameter, const Standard_Real SpanLenght, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &Point, gp_Vec2d &Vec1, gp_Vec2d &Vec2, gp_Vec2d &Vec3)
	Perform the evaluation of the Bspline Basis and then multiplies by the weights this just evaluates the current point the parameter must be normalized between the 0 and 1 for the span. The Cache must be valid when calling this routine. Geom Package will insure that. and then multiplies by the weights ththe CacheParameter is where the Cache was constructed the SpanLength is to normalize the polynomial in the cache to avoid bad conditioning effectsis just evaluates the current point More...
static void	CoefsD3 (const Standard_Real U, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt &Point, gp_Vec &Vec1, gp_Vec &Vec2, gp_Vec &Vec3)
	Calls CacheD1 for Bezier Curves Arrays computed with <br> the method PolesCoefficients. <br> Warning: To be used for Beziercurves ONLY!!! More...
static void	CoefsD3 (const Standard_Real U, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, gp_Pnt2d &Point, gp_Vec2d &Vec1, gp_Vec2d &Vec2, gp_Vec2d &Vec3)
	Calls CacheD1 for Bezier Curves Arrays computed with <br> the method PolesCoefficients. <br> Warning: To be used for Beziercurves ONLY!!! More...
static void	BuildCache (const Standard_Real U, const Standard_Real InverseOfSpanDomain, const Standard_Boolean PeriodicFlag, const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, TColgp_Array1OfPnt &CachePoles, TColStd_Array1OfReal &CacheWeights)
	Perform the evaluation of the Taylor expansion of the Bspline normalized between 0 and 1. If rational computes the homogeneous Taylor expension for the numerator and stores it in CachePoles More...
static void	BuildCache (const Standard_Real U, const Standard_Real InverseOfSpanDomain, const Standard_Boolean PeriodicFlag, const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, TColgp_Array1OfPnt2d &CachePoles, TColStd_Array1OfReal &CacheWeights)
	Perform the evaluation of the Taylor expansion of the Bspline normalized between 0 and 1. If rational computes the homogeneous Taylor expension for the numerator and stores it in CachePoles More...
static void	PolesCoefficients (const TColgp_Array1OfPnt2d &Poles, TColgp_Array1OfPnt2d &CachePoles)
static void	PolesCoefficients (const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, TColgp_Array1OfPnt2d &CachePoles, TColStd_Array1OfReal &CacheWeights)
static void	PolesCoefficients (const TColgp_Array1OfPnt &Poles, TColgp_Array1OfPnt &CachePoles)
static void	PolesCoefficients (const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, TColgp_Array1OfPnt &CachePoles, TColStd_Array1OfReal &CacheWeights)
	Encapsulation of BuildCache to perform the <br> evaluation of the Taylor expansion for beziercurves <br> at parameter 0. <br> Warning: To be used for Beziercurves ONLY!!! More...
static const Standard_Real &	FlatBezierKnots (const Standard_Integer Degree)
	Returns pointer to statically allocated array representing flat knots for bezier curve of the specified degree. Raises OutOfRange if Degree > MaxDegree() More...
static void	BuildSchoenbergPoints (const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, TColStd_Array1OfReal &Parameters)
	builds the Schoenberg points from the flat knot used to interpolate a BSpline since the BSpline matrix is invertible. More...
static void	Interpolate (const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const TColStd_Array1OfReal &Parameters, const TColStd_Array1OfInteger &ContactOrderArray, TColgp_Array1OfPnt &Poles, Standard_Integer &InversionProblem)
	Performs the interpolation of the data given in <br> the Poles array according to the requests in <br> ContactOrderArray that is : if <br> ContactOrderArray(i) has value d it means that <br> Poles(i) containes the dth derivative of the <br> function to be interpolated. The length L of the <br> following arrays must be the same : <br> Parameters, ContactOrderArray, Poles, <br> The length of FlatKnots is Degree + L + 1 <br> Warning: the method used to do that interpolation is gauss elimination WITHOUT pivoting. Thus if the diagonal is not dominant there is no guarantee that the algorithm will work. Nevertheless for Cubic interpolation or interpolation at Scheonberg points the method will work The InversionProblem will report 0 if there was no problem else it will give the index of the faulty pivot More...
static void	Interpolate (const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const TColStd_Array1OfReal &Parameters, const TColStd_Array1OfInteger &ContactOrderArray, TColgp_Array1OfPnt2d &Poles, Standard_Integer &InversionProblem)
	Performs the interpolation of the data given in <br> the Poles array according to the requests in <br> ContactOrderArray that is : if <br> ContactOrderArray(i) has value d it means that <br> Poles(i) containes the dth derivative of the <br> function to be interpolated. The length L of the <br> following arrays must be the same : <br> Parameters, ContactOrderArray, Poles, <br> The length of FlatKnots is Degree + L + 1 <br> Warning: the method used to do that interpolation is gauss elimination WITHOUT pivoting. Thus if the diagonal is not dominant there is no guarantee that the algorithm will work. Nevertheless for Cubic interpolation at knots or interpolation at Scheonberg points the method will work. The InversionProblem w ll report 0 if there was no problem else it will give the index of the faulty pivot More...
static void	Interpolate (const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const TColStd_Array1OfReal &Parameters, const TColStd_Array1OfInteger &ContactOrderArray, TColgp_Array1OfPnt &Poles, TColStd_Array1OfReal &Weights, Standard_Integer &InversionProblem)
	Performs the interpolation of the data given in <br> the Poles array according to the requests in <br> ContactOrderArray that is : if <br> ContactOrderArray(i) has value d it means that <br> Poles(i) containes the dth derivative of the <br> function to be interpolated. The length L of the <br> following arrays must be the same : <br> Parameters, ContactOrderArray, Poles, <br> The length of FlatKnots is Degree + L + 1 <br> Warning: the method used to do that interpolation is gauss elimination WITHOUT pivoting. Thus if the diagonal is not dominant there is no guarantee that the algorithm will work. Nevertheless for Cubic interpolation at knots or interpolation at Scheonberg points the method will work. The InversionProblem will report 0 if there was no problem else it will give the index of the faulty pivot More...
static void	Interpolate (const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const TColStd_Array1OfReal &Parameters, const TColStd_Array1OfInteger &ContactOrderArray, TColgp_Array1OfPnt2d &Poles, TColStd_Array1OfReal &Weights, Standard_Integer &InversionProblem)
	Performs the interpolation of the data given in <br> the Poles array according to the requests in <br> ContactOrderArray that is : if <br> ContactOrderArray(i) has value d it means that <br> Poles(i) containes the dth derivative of the <br> function to be interpolated. The length L of the <br> following arrays must be the same : <br> Parameters, ContactOrderArray, Poles, <br> The length of FlatKnots is Degree + L + 1 <br> Warning: the method used to do that interpolation is gauss elimination WITHOUT pivoting. Thus if the diagonal is not dominant there is no guarantee that the algorithm will work. Nevertheless for Cubic interpolation at knots or interpolation at Scheonberg points the method will work. The InversionProblem w ll report 0 if there was no problem else it will give the i More...
static void	Interpolate (const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const TColStd_Array1OfReal &Parameters, const TColStd_Array1OfInteger &ContactOrderArray, const Standard_Integer ArrayDimension, Standard_Real &Poles, Standard_Integer &InversionProblem)
	Performs the interpolation of the data given in <br> the Poles array according to the requests in <br> ContactOrderArray that is : if <br> ContactOrderArray(i) has value d it means that <br> Poles(i) containes the dth derivative of the <br> function to be interpolated. The length L of the <br> following arrays must be the same : <br> Parameters, ContactOrderArray <br> The length of FlatKnots is Degree + L + 1 <br> The PolesArray is an seen as an <br> Array[1..N][1..ArrayDimension] with N = tge length <br> of the parameters array <br> Warning: the method used to do that interpolation is gauss elimination WITHOUT pivoting. Thus if the diagonal is not dominant there is no guarantee that the algorithm will work. Nevertheless for Cubic interpolation or interpolation at Scheonberg points the method will work The InversionProblem will report 0 if there was no problem else it will give the index of the faulty pivot More...
static void	Interpolate (const Standard_Integer Degree, const TColStd_Array1OfReal &FlatKnots, const TColStd_Array1OfReal &Parameters, const TColStd_Array1OfInteger &ContactOrderArray, const Standard_Integer ArrayDimension, Standard_Real &Poles, Standard_Real &Weights, Standard_Integer &InversionProblem)
static void	MovePoint (const Standard_Real U, const gp_Vec2d &Displ, const Standard_Integer Index1, const Standard_Integer Index2, const Standard_Integer Degree, const Standard_Boolean Rational, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &FlatKnots, Standard_Integer &FirstIndex, Standard_Integer &LastIndex, TColgp_Array1OfPnt2d &NewPoles)
	Find the new poles which allows an old point (with a given u as parameter) to reach a new position Index1 and Index2 indicate the range of poles we can move (1, NbPoles-1) or (2, NbPoles) -> no constraint for one side don't enter (1,NbPoles) -> error: rigid move (2, NbPoles-1) -> the ends are enforced (3, NbPoles-2) -> the ends and the tangency are enforced if Problem in BSplineBasis calculation, no change for the curve and FirstIndex, LastIndex = 0 More...
static void	MovePoint (const Standard_Real U, const gp_Vec &Displ, const Standard_Integer Index1, const Standard_Integer Index2, const Standard_Integer Degree, const Standard_Boolean Rational, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &FlatKnots, Standard_Integer &FirstIndex, Standard_Integer &LastIndex, TColgp_Array1OfPnt &NewPoles)
	Find the new poles which allows an old point (with a given u as parameter) to reach a new position Index1 and Index2 indicate the range of poles we can move (1, NbPoles-1) or (2, NbPoles) -> no constraint for one side don't enter (1,NbPoles) -> error: rigid move (2, NbPoles-1) -> the ends are enforced (3, NbPoles-2) -> the ends and the tangency are enforced if Problem in BSplineBasis calculation, no change for the curve and FirstIndex, LastIndex = 0 More...
static void	MovePointAndTangent (const Standard_Real U, const Standard_Integer ArrayDimension, Standard_Real &Delta, Standard_Real &DeltaDerivative, 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)
	This is the dimension free version of the utility <br> U is the parameter must be within the first FlatKnots and the last FlatKnots Delta is the amount the curve has to be moved DeltaDerivative is the amount the derivative has to be moved. Delta and DeltaDerivative must be array of dimension ArrayDimension Degree is the degree of the BSpline and the FlatKnots are the knots of the BSpline Starting Condition if = -1 means the starting point of the curve can move = 0 means the starting point of the cuve cannot move but tangen starting point of the curve cannot move = 1 means the starting point and tangents cannot move = 2 means the starting point tangent and curvature cannot move = ... Same holds for EndingCondition Poles are the poles of the curve Weights are the weights of the curve if Rational = Standard_True NewPoles are the poles of the deformed curve ErrorStatus will be 0 if no error happened 1 if there are not enough knots/poles the imposed conditions The way to solve this problem is to add knots to the BSpline If StartCondition = 1 and EndCondition = 1 then you need at least 4 + 2 = 6 poles so for example to have a C1 cubic you will need have at least 2 internal knots. More...
static void	MovePointAndTangent (const Standard_Real U, const gp_Vec &Delta, const gp_Vec &DeltaDerivative, const Standard_Real Tolerance, const Standard_Integer Degree, const Standard_Boolean Rational, const Standard_Integer StartingCondition, const Standard_Integer EndingCondition, const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &FlatKnots, TColgp_Array1OfPnt &NewPoles, Standard_Integer &ErrorStatus)
	This is the dimension free version of the utility <br> U is the parameter must be within the first FlatKnots and the last FlatKnots Delta is the amount the curve has to be moved DeltaDerivative is the amount the derivative has to be moved. Delta and DeltaDerivative must be array of dimension ArrayDimension Degree is the degree of the BSpline and the FlatKnots are the knots of the BSpline Starting Condition if = -1 means the starting point of the curve can move = 0 means the starting point of the cuve cannot move but tangen starting point of the curve cannot move = 1 means the starting point and tangents cannot move = 2 means the starting point tangent and curvature cannot move = ... Same holds for EndingCondition Poles are the poles of the curve Weights are the weights of the curve if Rational = Standard_True NewPoles are the poles of the deformed curve ErrorStatus will be 0 if no error happened 1 if there are not enough knots/poles the imposed conditions The way to solve this problem is to add knots to the BSpline If StartCondition = 1 and EndCondition = 1 then you need at least 4 + 2 = 6 poles so for example to have a C1 cubic you will need have at least 2 internal knots. More...
static void	MovePointAndTangent (const Standard_Real U, const gp_Vec2d &Delta, const gp_Vec2d &DeltaDerivative, const Standard_Real Tolerance, const Standard_Integer Degree, const Standard_Boolean Rational, const Standard_Integer StartingCondition, const Standard_Integer EndingCondition, const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &FlatKnots, TColgp_Array1OfPnt2d &NewPoles, Standard_Integer &ErrorStatus)
	This is the dimension free version of the utility <br> U is the parameter must be within the first FlatKnots and the last FlatKnots Delta is the amount the curve has to be moved DeltaDerivative is the amount the derivative has to be moved. Delta and DeltaDerivative must be array of dimension ArrayDimension Degree is the degree of the BSpline and the FlatKnots are the knots of the BSpline Starting Condition if = -1 means the starting point of the curve can move = 0 means the starting point of the cuve cannot move but tangen starting point of the curve cannot move = 1 means the starting point and tangents cannot move = 2 means the starting point tangent and curvature cannot move = ... Same holds for EndingCondition Poles are the poles of the curve Weights are the weights of the curve if Rational = Standard_True NewPoles are the poles of the deformed curve ErrorStatus will be 0 if no error happened 1 if there are not enough knots/poles the imposed conditions The way to solve this problem is to add knots to the BSpline If StartCondition = 1 and EndCondition = 1 then you need at least 4 + 2 = 6 poles so for example to have a C1 cubic you will need have at least 2 internal knots. More...
static void	Resolution (Standard_Real &PolesArray, const Standard_Integer ArrayDimension, const Standard_Integer NumPoles, const TColStd_Array1OfReal &Weights, const TColStd_Array1OfReal &FlatKnots, const Standard_Integer Degree, const Standard_Real Tolerance3D, Standard_Real &UTolerance)
	given a tolerance in 3D space returns a tolerance in U parameter space such that all u1 and u0 in the domain of the curve f(u) | u1 - u0 | < UTolerance and we have |f (u1) - f (u0)| < Tolerance3D More...
static void	Resolution (const TColgp_Array1OfPnt &Poles, const TColStd_Array1OfReal &Weights, const Standard_Integer NumPoles, const TColStd_Array1OfReal &FlatKnots, const Standard_Integer Degree, const Standard_Real Tolerance3D, Standard_Real &UTolerance)
	given a tolerance in 3D space returns a tolerance in U parameter space such that all u1 and u0 in the domain of the curve f(u) | u1 - u0 | < UTolerance and we have |f (u1) - f (u0)| < Tolerance3D More...
static void	Resolution (const TColgp_Array1OfPnt2d &Poles, const TColStd_Array1OfReal &Weights, const Standard_Integer NumPoles, const TColStd_Array1OfReal &FlatKnots, const Standard_Integer Degree, const Standard_Real Tolerance3D, Standard_Real &UTolerance)
	given a tolerance in 3D space returns a tolerance in U parameter space such that all u1 and u0 in the domain of the curve f(u) | u1 - u0 | < UTolerance and we have |f (u1) - f (u0)| < Tolerance3D More...

Detailed Description

BSplCLib   B-spline curve Library. <br>

The BSplCLib package is a basic library for BSplines. It
provides three categories of functions.

• Management methods to process knots and multiplicities.
  
  
• Multi-Dimensions spline methods. BSpline methods where
  poles have an arbitrary number of dimensions. They divides
  in two groups :
  
  
  • Global methods modifying the whole set of poles. The
    poles are described by an array of Reals and a
    Dimension. Example : Inserting knots.
    
    
  • Local methods computing points and derivatives. The
    poles are described by a pointer on a local array of
    Reals and a Dimension. The local array is modified.
    
    
• 2D and 3D spline curves methods.
  
  Methods for 2d and 3d BSplines curves rational or not
  rational.
  
  Those methods have the following structure :
  
  
  • They extract the pole informations in a working array.
    
    
  • They process the working array with the
    multi-dimension methods. (for example a 3d rational
    curve is processed as a 4 dimension curve).
    
    
  • They get back the result in the original dimension.
    
    Note that the bspline surface methods found in the
    package BSplSLib uses the same structure and rely on
    BSplCLib.
    
    In the following list of methods the 2d and 3d curve
    methods will be described with the corresponding
    multi-dimension method.
    
    The 3d or 2d B-spline curve is defined with :
    
    . its control points : TColgp_Array1OfPnt(2d) Poles
    . its weights : TColStd_Array1OfReal Weights
    . its knots : TColStd_Array1OfReal Knots
    . its multiplicities : TColStd_Array1OfInteger Mults
    . its degree : Standard_Integer Degree
    . its periodicity : Standard_Boolean Periodic
    
    Warnings :
    The bounds of Poles and Weights should be the same.
    The bounds of Knots and Mults should be the same.
    
    Weights can be a null reference (BSplCLib::NoWeights())
    the curve is non rational.
    
    Mults can be a null reference (BSplCLib::NoMults())
    the knots are "flat" knots.
    
    KeyWords :
    B-spline curve, Functions, Library
    
    References :
    . A survey of curves and surfaces 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
    

Member Function Documentation

static Standard_Boolean BSplCLib::AntiBoorScheme ( const Standard_Real  U, const Standard_Integer  Degree, Standard_Real &  Knots, const Standard_Integer  Dimension, Standard_Real &  Poles, const Standard_Integer  Depth, const Standard_Integer  Length, const Standard_Real  Tolerance  )

static

 Compute  the content of  Pole before the BoorScheme. <br>
     This method is used to remove poles. <br>

U is the poles to remove, Knots should contains the
knots of the curve after knot removal.

The first and last poles do not change, the other
poles are computed by averaging two possible values.
The distance between the two possible poles is
computed, if it is higher than <Tolerance> False is
returned.

static void BSplCLib::Bohm ( const Standard_Real  U, const Standard_Integer  Degree, const Standard_Integer  N, Standard_Real &  Knots, const Standard_Integer  Dimension, Standard_Real &  Poles  )

static

 Performs the Bohm  Algorithm at  parameter <U>. This <br>
     algorithm computes the value and all the derivatives <br>
     up to order N (N <= Degree). <br>

<Poles> is the original array of poles.

The result in <Poles> is the value and the
derivatives. Poles[0] is the value, Poles[Degree]
is the last derivative.

static Standard_Integer BSplCLib::BoorIndex ( const Standard_Integer  Index, const Standard_Integer  Length, const Standard_Integer  Depth  )

static

Returns the index in the Boor result array of the
poles <Index>. If the Boor algorithm was perform
with <Length> and <Depth>.

static void BSplCLib::BoorScheme ( const Standard_Real  U, const Standard_Integer  Degree, Standard_Real &  Knots, const Standard_Integer  Dimension, Standard_Real &  Poles, const Standard_Integer  Depth, const Standard_Integer  Length  )

static

 Performs the  Boor Algorithm  at  parameter <U> with <br>
     the given <Degree> and the  array of <Knots> on  the <br>
     poles <Poles> of dimension  <Dimension>.  The schema <br>
     is  computed  until  level  <Depth>  on a   basis of <br>
     <Length+1> poles. <br>

     * Knots is an array of reals of length : <br>

<Length> + <Degree>

• Poles is an array of reals of length :
  
  (2 * <Length> + 1) * <Dimension>
  
  The poles values must be set in the array at the
  positions.
  
  0..Dimension,
  
  2 * Dimension ..
  3 * Dimension
  
  4 * Dimension ..
  5 * Dimension
  
  ...
  
  The results are found in the array poles depending
  on the Depth. (See the method GetPole).
  

static void BSplCLib::BuildBoor ( const Standard_Integer  Index, const Standard_Integer  Length, const Standard_Integer  Dimension, const TColStd_Array1OfReal &  Poles, Standard_Real &  LP  )

static

Copy in <LP> poles for <Dimension> Boor scheme.
Starting from <Index> * <Dimension>, copy
<Length+1> poles.

static Standard_Integer BSplCLib::BuildBSpMatrix ( const TColStd_Array1OfReal &  Parameters, const TColStd_Array1OfInteger &  OrderArray, const TColStd_Array1OfReal &  FlatKnots, const Standard_Integer  Degree, math_Matrix &  Matrix, Standard_Integer &  UpperBandWidth, Standard_Integer &  LowerBandWidth  )

static

 This Builds   a fully  blown   Matrix of <br>
       (ni) <br>
     Bi    (tj) <br>

with i and j within 1..Order + NumPoles
The integer ni is the ith slot of the
array OrderArray, tj is the jth slot of
the array Parameters

static void BSplCLib::BuildCache ( const Standard_Real  U, const Standard_Real  InverseOfSpanDomain, const Standard_Boolean  PeriodicFlag, const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, TColgp_Array1OfPnt &  CachePoles, TColStd_Array1OfReal &  CacheWeights  )

static

Perform the evaluation of the Taylor expansion
of the Bspline normalized between 0 and 1.
If rational computes the homogeneous Taylor expension
for the numerator and stores it in CachePoles

static void BSplCLib::BuildCache ( const Standard_Real  U, const Standard_Real  InverseOfSpanDomain, const Standard_Boolean  PeriodicFlag, const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, TColgp_Array1OfPnt2d &  CachePoles, TColStd_Array1OfReal &  CacheWeights  )

static

Perform the evaluation of the Taylor expansion
of the Bspline normalized between 0 and 1.
If rational computes the homogeneous Taylor expension
for the numerator and stores it in CachePoles

static void BSplCLib::BuildEval ( const Standard_Integer  Degree, const Standard_Integer  Index, const TColStd_Array1OfReal &  Poles, const TColStd_Array1OfReal &  Weights, Standard_Real &  LP  )

static

static void BSplCLib::BuildEval ( const Standard_Integer  Degree, const Standard_Integer  Index, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, Standard_Real &  LP  )

static

static void BSplCLib::BuildEval ( const Standard_Integer  Degree, const Standard_Integer  Index, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, Standard_Real &  LP  )

static

Copy in <LP> the poles and weights for the Eval
scheme. starting from Poles(Poles.Lower()+Index)

static void BSplCLib::BuildKnots ( const Standard_Integer  Degree, const Standard_Integer  Index, const Standard_Boolean  Periodic, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, Standard_Real &  LK  )

static

Stores in LK the usefull knots for the BoorSchem
on the span Knots(Index) - Knots(Index+1)

static void BSplCLib::BuildSchoenbergPoints ( const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, TColStd_Array1OfReal &  Parameters  )

static

builds the Schoenberg points from the flat knot
used to interpolate a BSpline since the
BSpline matrix is invertible.

static void BSplCLib::CacheD0 ( const Standard_Real  U, const Standard_Integer  Degree, const Standard_Real  CacheParameter, const Standard_Real  SpanLenght, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  Point  )

static

Perform the evaluation of the of the cache
the parameter must be normalized between
the 0 and 1 for the span.
The Cache must be valid when calling this
routine. Geom Package will insure that.
and then multiplies by the weights
this just evaluates the current point
the CacheParameter is where the Cache was
constructed the SpanLength is to normalize
the polynomial in the cache to avoid bad conditioning
effects

static void BSplCLib::CacheD0 ( const Standard_Real  U, const Standard_Integer  Degree, const Standard_Real  CacheParameter, const Standard_Real  SpanLenght, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  Point  )

static

Perform the evaluation of the Bspline Basis
and then multiplies by the weights
this just evaluates the current point
the parameter must be normalized between
the 0 and 1 for the span.
The Cache must be valid when calling this
routine. Geom Package will insure that.
and then multiplies by the weights
ththe CacheParameter is where the Cache was
constructed the SpanLength is to normalize
the polynomial in the cache to avoid bad conditioning
effectsis just evaluates the current point

static void BSplCLib::CacheD1 ( const Standard_Real  U, const Standard_Integer  Degree, const Standard_Real  CacheParameter, const Standard_Real  SpanLenght, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  Point, gp_Vec &  Vec  )

static

Perform the evaluation of the of the cache
the parameter must be normalized between
the 0 and 1 for the span.
The Cache must be valid when calling this
routine. Geom Package will insure that.
and then multiplies by the weights
this just evaluates the current point
the CacheParameter is where the Cache was
constructed the SpanLength is to normalize
the polynomial in the cache to avoid bad conditioning
effects

static void BSplCLib::CacheD1 ( const Standard_Real  U, const Standard_Integer  Degree, const Standard_Real  CacheParameter, const Standard_Real  SpanLenght, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  Point, gp_Vec2d &  Vec  )

static

Perform the evaluation of the Bspline Basis
and then multiplies by the weights
this just evaluates the current point
the parameter must be normalized between
the 0 and 1 for the span.
The Cache must be valid when calling this
routine. Geom Package will insure that.
and then multiplies by the weights
ththe CacheParameter is where the Cache was
constructed the SpanLength is to normalize
the polynomial in the cache to avoid bad conditioning
effectsis just evaluates the current point

static void BSplCLib::CacheD2 ( const Standard_Real  U, const Standard_Integer  Degree, const Standard_Real  CacheParameter, const Standard_Real  SpanLenght, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  Point, gp_Vec &  Vec1, gp_Vec &  Vec2  )

static

Perform the evaluation of the of the cache
the parameter must be normalized between
the 0 and 1 for the span.
The Cache must be valid when calling this
routine. Geom Package will insure that.
and then multiplies by the weights
this just evaluates the current point
the CacheParameter is where the Cache was
constructed the SpanLength is to normalize
the polynomial in the cache to avoid bad conditioning
effects

static void BSplCLib::CacheD2 ( const Standard_Real  U, const Standard_Integer  Degree, const Standard_Real  CacheParameter, const Standard_Real  SpanLenght, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  Point, gp_Vec2d &  Vec1, gp_Vec2d &  Vec2  )

static

Perform the evaluation of the Bspline Basis
and then multiplies by the weights
this just evaluates the current point
the parameter must be normalized between
the 0 and 1 for the span.
The Cache must be valid when calling this
routine. Geom Package will insure that.
and then multiplies by the weights
ththe CacheParameter is where the Cache was
constructed the SpanLength is to normalize
the polynomial in the cache to avoid bad conditioning
effectsis just evaluates the current point

static void BSplCLib::CacheD3 ( const Standard_Real  U, const Standard_Integer  Degree, const Standard_Real  CacheParameter, const Standard_Real  SpanLenght, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  Point, gp_Vec &  Vec1, gp_Vec &  Vec2, gp_Vec &  Vec3  )

static

Perform the evaluation of the of the cache
the parameter must be normalized between
the 0 and 1 for the span.
The Cache must be valid when calling this
routine. Geom Package will insure that.
and then multiplies by the weights
this just evaluates the current point
the CacheParameter is where the Cache was
constructed the SpanLength is to normalize
the polynomial in the cache to avoid bad conditioning
effects

static void BSplCLib::CacheD3 ( const Standard_Real  U, const Standard_Integer  Degree, const Standard_Real  CacheParameter, const Standard_Real  SpanLenght, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  Point, gp_Vec2d &  Vec1, gp_Vec2d &  Vec2, gp_Vec2d &  Vec3  )

static

Perform the evaluation of the Bspline Basis
and then multiplies by the weights
this just evaluates the current point
the parameter must be normalized between
the 0 and 1 for the span.
The Cache must be valid when calling this
routine. Geom Package will insure that.
and then multiplies by the weights
ththe CacheParameter is where the Cache was
constructed the SpanLength is to normalize
the polynomial in the cache to avoid bad conditioning
effectsis just evaluates the current point

static void BSplCLib::CoefsD0 ( const Standard_Real  U, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  Point  )

static

Calls CacheD0 for Bezier  Curves Arrays computed with <br>
    the method PolesCoefficients. <br>

Warning: To be used for Beziercurves ONLY!!!

static void BSplCLib::CoefsD0 ( const Standard_Real  U, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  Point  )

static

Calls CacheD0 for Bezier  Curves Arrays computed with <br>
    the method PolesCoefficients. <br>

Warning: To be used for Beziercurves ONLY!!!

static void BSplCLib::CoefsD1 ( const Standard_Real  U, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  Point, gp_Vec &  Vec  )

static

Calls CacheD1 for Bezier  Curves Arrays computed with <br>
    the method PolesCoefficients. <br>

Warning: To be used for Beziercurves ONLY!!!

static void BSplCLib::CoefsD1 ( const Standard_Real  U, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  Point, gp_Vec2d &  Vec  )

static

Calls CacheD1 for Bezier  Curves Arrays computed with <br>
    the method PolesCoefficients. <br>

Warning: To be used for Beziercurves ONLY!!!

static void BSplCLib::CoefsD2 ( const Standard_Real  U, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  Point, gp_Vec &  Vec1, gp_Vec &  Vec2  )

static

Calls CacheD1 for Bezier  Curves Arrays computed with <br>
    the method PolesCoefficients. <br>

Warning: To be used for Beziercurves ONLY!!!

static void BSplCLib::CoefsD2 ( const Standard_Real  U, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  Point, gp_Vec2d &  Vec1, gp_Vec2d &  Vec2  )

static

Calls CacheD1 for Bezier  Curves Arrays computed with <br>
    the method PolesCoefficients. <br>

Warning: To be used for Beziercurves ONLY!!!

static void BSplCLib::CoefsD3 ( const Standard_Real  U, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  Point, gp_Vec &  Vec1, gp_Vec &  Vec2, gp_Vec &  Vec3  )

static

Calls CacheD1 for Bezier  Curves Arrays computed with <br>
    the method PolesCoefficients. <br>

Warning: To be used for Beziercurves ONLY!!!

static void BSplCLib::CoefsD3 ( const Standard_Real  U, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  Point, gp_Vec2d &  Vec1, gp_Vec2d &  Vec2, gp_Vec2d &  Vec3  )

static

Calls CacheD1 for Bezier  Curves Arrays computed with <br>
    the method PolesCoefficients. <br>

Warning: To be used for Beziercurves ONLY!!!

static 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  )

static

static void BSplCLib::D0 ( const Standard_Real  U, const Standard_Integer  Index, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, gp_Pnt &  P  )

static

static void BSplCLib::D0 ( const Standard_Real  U, const Standard_Integer  UIndex, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, gp_Pnt2d &  P  )

static

static void BSplCLib::D0 ( const Standard_Real  U, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  P  )

static

static void BSplCLib::D0 ( const Standard_Real  U, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  P  )

static

static 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  )

static

static void BSplCLib::D1 ( const Standard_Real  U, const Standard_Integer  Index, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, gp_Pnt &  P, gp_Vec &  V  )

static

static void BSplCLib::D1 ( const Standard_Real  U, const Standard_Integer  UIndex, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, gp_Pnt2d &  P, gp_Vec2d &  V  )

static

static void BSplCLib::D1 ( const Standard_Real  U, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  P, gp_Vec &  V  )

static

static void BSplCLib::D1 ( const Standard_Real  U, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  P, gp_Vec2d &  V  )

static

static 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  )

static

static void BSplCLib::D2 ( const Standard_Real  U, const Standard_Integer  Index, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, gp_Pnt &  P, gp_Vec &  V1, gp_Vec &  V2  )

static

static void BSplCLib::D2 ( const Standard_Real  U, const Standard_Integer  UIndex, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, gp_Pnt2d &  P, gp_Vec2d &  V1, gp_Vec2d &  V2  )

static

static void BSplCLib::D2 ( const Standard_Real  U, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  P, gp_Vec &  V1, gp_Vec &  V2  )

static

static void BSplCLib::D2 ( const Standard_Real  U, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  P, gp_Vec2d &  V1, gp_Vec2d &  V2  )

static

static 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  )

static

static void BSplCLib::D3 ( const Standard_Real  U, const Standard_Integer  Index, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, gp_Pnt &  P, gp_Vec &  V1, gp_Vec &  V2, gp_Vec &  V3  )

static

static void BSplCLib::D3 ( const Standard_Real  U, const Standard_Integer  UIndex, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, gp_Pnt2d &  P, gp_Vec2d &  V1, gp_Vec2d &  V2, gp_Vec2d &  V3  )

static

static void BSplCLib::D3 ( const Standard_Real  U, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  P, gp_Vec &  V1, gp_Vec &  V2, gp_Vec &  V3  )

static

static void BSplCLib::D3 ( const Standard_Real  U, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  P, gp_Vec2d &  V1, gp_Vec2d &  V2, gp_Vec2d &  V3  )

static

static void BSplCLib::Derivative ( const Standard_Integer  Degree, Standard_Real &  Knots, const Standard_Integer  Dimension, const Standard_Integer  Length, const Standard_Integer  Order, Standard_Real &  Poles  )

static

 Computes   the   poles of  the    BSpline  giving the <br>
     derivatives of order <Order>. <br>

The formula for the first order is

Pole(i) = Degree * (Pole(i+1) - Pole(i)) /
(Knots(i+Degree+1) - Knots(i+1))

This formula is repeated (Degree is decremented at
each step).

static 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  )

static

static void BSplCLib::DN ( const Standard_Real  U, const Standard_Integer  N, const Standard_Integer  Index, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, gp_Vec &  VN  )

static

static void BSplCLib::DN ( const Standard_Real  U, const Standard_Integer  N, const Standard_Integer  UIndex, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, gp_Vec2d &  V  )

static

static void BSplCLib::DN ( const Standard_Real  U, const Standard_Integer  N, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  P, gp_Vec &  VN  )

static

static void BSplCLib::DN ( const Standard_Real  U, const Standard_Integer  N, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  P, gp_Vec2d &  VN  )

static

  The  above  functions  compute   values and <br>
    derivatives in the following situations : <br>

    * 3D, 2D and 1D <br>

• Rational or not Rational.
  
  
• Knots and multiplicities or "flat knots" without
  multiplicities.
  
  
• The <Index> is the the localization of the
  parameter in the knot sequence. If <Index> is out
  of range the correct value will be searched.
  
  
  VERY IMPORTANT!!!
  USE BSplCLib::NoWeights() as Weights argument for non
  rational curves computations.
  

static void BSplCLib::Eval ( const Standard_Real  U, const Standard_Integer  Degree, Standard_Real &  Knots, const Standard_Integer  Dimension, Standard_Real &  Poles  )

static

 Perform the Boor  algorithm  to  evaluate a point at <br>
     parameter <U>, with <Degree> and <Dimension>. <br>

Poles is an array of Reals of size

<Dimension> * <Degree>+1

Containing the poles. At the end <Poles> contains
the current point.

static void BSplCLib::Eval ( const Standard_Real  U, const Standard_Boolean  PeriodicFlag, const Standard_Integer  DerivativeRequest, Standard_Integer &  ExtrapMode, const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const Standard_Integer  ArrayDimension, Standard_Real &  Poles, Standard_Real &  Result  )

static

 Perform the De Boor   algorithm  to  evaluate a point at <br>
     parameter <U>, with <Degree> and <Dimension>. <br>

Poles is an array of Reals of size

<Dimension> * <Degree>+1

Containing the poles. At the end <Poles> contains
the current point. Poles Contain all the poles of
the BsplineCurve, Knots also Contains all the knots
of the BsplineCurve. ExtrapMode has two slots [0] =
Degree used to extrapolate before the first knot [1]
= Degre used to extrapolate after the last knot has
to be between 1 and Degree

static void BSplCLib::Eval ( const Standard_Real  U, const Standard_Boolean  PeriodicFlag, const Standard_Integer  DerivativeRequest, Standard_Integer &  ExtrapMode, const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const Standard_Integer  ArrayDimension, Standard_Real &  Poles, Standard_Real &  Weights, Standard_Real &  PolesResult, Standard_Real &  WeightsResult  )

static

 Perform the  De Boor algorithm  to evaluate a point at <br>
     parameter   <U>,  with   <Degree>    and  <Dimension>. <br>
     Evaluates by multiplying the  Poles by the Weights and <br>
     gives  the homogeneous  result  in PolesResult that is <br>
     the results of the evaluation of the numerator once it <br>
     has     been  multiplied   by  the     weights and  in <br>
     WeightsResult one has  the result of the evaluation of <br>
     the denominator <br>

Warning: <PolesResult> and <WeightsResult> must be dimensionned
properly.

static void BSplCLib::Eval ( const Standard_Real  U, const Standard_Boolean  PeriodicFlag, const Standard_Boolean  HomogeneousFlag, Standard_Integer &  ExtrapMode, const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt &  Point, Standard_Real &  Weight  )

static

Perform the evaluation of the Bspline Basis
and then multiplies by the weights
this just evaluates the current point

static void BSplCLib::Eval ( const Standard_Real  U, const Standard_Boolean  PeriodicFlag, const Standard_Boolean  HomogeneousFlag, Standard_Integer &  ExtrapMode, const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, gp_Pnt2d &  Point, Standard_Real &  Weight  )

static

 Perform the evaluation of the Bspline Basis <br>
     and then multiplies by the weights <br>
     this just evaluates the current point <br>

static Standard_Integer BSplCLib::EvalBsplineBasis ( const Standard_Integer  Side, const Standard_Integer  DerivativeOrder, const Standard_Integer  Order, const TColStd_Array1OfReal &  FlatKnots, const Standard_Real  Parameter, Standard_Integer &  FirstNonZeroBsplineIndex, math_Matrix &  BsplineBasis  )

static

This evaluates the Bspline Basis at a
given parameter Parameter up to the
requested DerivativeOrder and store the
result in the array BsplineBasis in the
following fashion
BSplineBasis(1,1) =
value of first non vanishing
Bspline function which has Index FirstNonZeroBsplineIndex
BsplineBasis(1,2) =
value of second non vanishing
Bspline function which has Index
FirstNonZeroBsplineIndex + 1
BsplineBasis(1,n) =
value of second non vanishing non vanishing
Bspline function which has Index
FirstNonZeroBsplineIndex + n (n <= Order)
BSplineBasis(2,1) =
value of derivative of first non vanishing
Bspline function which has Index FirstNonZeroBsplineIndex
BSplineBasis(N,1) =
value of Nth derivative of first non vanishing
Bspline function which has Index FirstNonZeroBsplineIndex
if N <= DerivativeOrder + 1

static Standard_Integer BSplCLib::FactorBandedMatrix ( math_Matrix &  Matrix, const Standard_Integer  UpperBandWidth, const Standard_Integer  LowerBandWidth, Standard_Integer &  PivotIndexProblem  )

static

this factors the Banded Matrix in
the LU form with a Banded storage of
components of the L matrix
WARNING : do not use if the Matrix is
totally positive (It is the case for
Bspline matrices build as above with
parameters being the Schoenberg points

static Standard_Integer BSplCLib::FirstUKnotIndex ( const Standard_Integer  Degree, const TColStd_Array1OfInteger &  Mults  )

static

Computes the index of the knots value which gives
the start point of the curve.

static const Standard_Real& BSplCLib::FlatBezierKnots ( const Standard_Integer  Degree )

static

Returns pointer to statically allocated array representing
flat knots for bezier curve of the specified degree.
Raises OutOfRange if Degree > MaxDegree()

static Standard_Integer BSplCLib::FlatIndex ( const Standard_Integer  Degree, const Standard_Integer  Index, const TColStd_Array1OfInteger &  Mults, const Standard_Boolean  Periodic  )

static

Computes the index of the flats knots sequence
corresponding to <Index> in the knots sequence
which multiplicities are <Mults>.

static void BSplCLib::FunctionMultiply ( const BSplCLib_EvaluatorFunction &  Function, 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  )

static

this will  multiply a given Vectorial BSpline F(t) <br>
    defined  by its  BSplineDegree and BSplineFlatKnotsl, <br>
    its Poles  array which are coded as  an array of Real <br>
    of  the  form  [1..NumPoles][1..PolesDimension] by  a <br>
    function     a(t) which is   assumed to   satisfy the <br>
    following  : 1. a(t)  * F(t)  is a polynomial BSpline <br>
    that can be expressed  exactly as a BSpline of degree <br>
    NewDegree on the knots FlatKnots 2. the range of a(t) <br>
     is the same as the  range of F(t) <br>

Warning: it is
the caller's responsability to insure that conditions

     1. and  2. above are  satisfied : no check whatsoever <br>
     is made in this method <br>

Status will return 0 if OK else it will return the pivot index
of the matrix that was inverted to compute the multiplied
BSpline : the method used is interpolation at Schoenenberg
points of a(t)*F(t)

static void BSplCLib::FunctionMultiply ( const BSplCLib_EvaluatorFunction &  Function, 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  )

static

this will  multiply a given Vectorial BSpline F(t) <br>
    defined  by its  BSplineDegree and BSplineFlatKnotsl, <br>
    its Poles  array which are coded as  an array of Real <br>
    of  the  form  [1..NumPoles][1..PolesDimension] by  a <br>
    function     a(t) which is   assumed to   satisfy the <br>
    following  : 1. a(t)  * F(t)  is a polynomial BSpline <br>
    that can be expressed  exactly as a BSpline of degree <br>
    NewDegree on the knots FlatKnots 2. the range of a(t) <br>
     is the same as the  range of F(t) <br>

Warning: it is
the caller's responsability to insure that conditions

     1. and  2. above are  satisfied : no check whatsoever <br>
     is made in this method <br>

Status will return 0 if OK else it will return the pivot index
of the matrix that was inverted to compute the multiplied
BSpline : the method used is interpolation at Schoenenberg
points of a(t)*F(t)

static void BSplCLib::FunctionMultiply ( const BSplCLib_EvaluatorFunction &  Function, const Standard_Integer  BSplineDegree, const TColStd_Array1OfReal &  BSplineFlatKnots, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  FlatKnots, const Standard_Integer  NewDegree, TColgp_Array1OfPnt2d &  NewPoles, Standard_Integer &  Status  )

static

this will  multiply a given Vectorial BSpline F(t) <br>
    defined  by its  BSplineDegree and BSplineFlatKnotsl, <br>
    its Poles  array which are coded as  an array of Real <br>
    of  the  form  [1..NumPoles][1..PolesDimension] by  a <br>
    function     a(t) which is   assumed to   satisfy the <br>
    following  : 1. a(t)  * F(t)  is a polynomial BSpline <br>
    that can be expressed  exactly as a BSpline of degree <br>
    NewDegree on the knots FlatKnots 2. the range of a(t) <br>
     is the same as the  range of F(t) <br>

Warning: it is
the caller's responsability to insure that conditions

     1. and  2. above are  satisfied : no check whatsoever <br>
     is made in this method <br>

Status will return 0 if OK else it will return the pivot index
of the matrix that was inverted to compute the multiplied
BSpline : the method used is interpolation at Schoenenberg
points of a(t)*F(t)

static void BSplCLib::FunctionMultiply ( const BSplCLib_EvaluatorFunction &  Function, const Standard_Integer  BSplineDegree, const TColStd_Array1OfReal &  BSplineFlatKnots, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  FlatKnots, const Standard_Integer  NewDegree, TColgp_Array1OfPnt &  NewPoles, Standard_Integer &  Status  )

static

this will  multiply a given Vectorial BSpline F(t) <br>
    defined  by its  BSplineDegree and BSplineFlatKnotsl, <br>
    its Poles  array which are coded as  an array of Real <br>
    of  the  form  [1..NumPoles][1..PolesDimension] by  a <br>
    function     a(t) which is   assumed to   satisfy the <br>
    following  : 1. a(t)  * F(t)  is a polynomial BSpline <br>
    that can be expressed  exactly as a BSpline of degree <br>
    NewDegree on the knots FlatKnots 2. the range of a(t) <br>
     is the same as the  range of F(t) <br>

Warning: it is
the caller's responsability to insure that conditions

     1. and  2. above are  satisfied : no check whatsoever <br>
     is made in this method <br>

Status will return 0 if OK else it will return the pivot index
of the matrix that was inverted to compute the multiplied
BSpline : the method used is interpolation at Schoenenberg
points of a(t)*F(t)

static void BSplCLib::FunctionReparameterise ( const BSplCLib_EvaluatorFunction &  Function, 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  )

static

 This function will compose  a given Vectorial BSpline F(t) <br>
     defined  by its  BSplineDegree and BSplineFlatKnotsl, <br>
     its Poles  array which are coded as  an array of Real <br>
     of  the  form  [1..NumPoles][1..PolesDimension] with  a <br>
     function     a(t) which is   assumed to   satisfy the <br>
     following: <br>

  1. F(a(t))  is a polynomial BSpline <br>
     that can be expressed  exactly as a BSpline of degree <br>
     NewDegree on the knots FlatKnots <br>

1. a(t) defines a differentiable
   isomorphism between the range of FlatKnots to the range
   of BSplineFlatKnots which is the
   same as the range of F(t)
   
   Warning: it is
   the caller's responsability to insure that conditions
   
   1. and 2. above are satisfied : no check whatsoever
      is made in this method
      
      Status will return 0 if OK else it will return the pivot index
      of the matrix that was inverted to compute the multiplied
      BSpline : the method used is interpolation at Schoenenberg
      points of F(a(t))
      

static void BSplCLib::FunctionReparameterise ( const BSplCLib_EvaluatorFunction &  Function, 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  )

static

 This function will compose  a given Vectorial BSpline F(t) <br>
     defined  by its  BSplineDegree and BSplineFlatKnotsl, <br>
     its Poles  array which are coded as  an array of Real <br>
     of  the  form  [1..NumPoles][1..PolesDimension] with  a <br>
     function     a(t) which is   assumed to   satisfy the <br>
     following: <br>

  1. F(a(t))  is a polynomial BSpline <br>
     that can be expressed  exactly as a BSpline of degree <br>
     NewDegree on the knots FlatKnots <br>

1. a(t) defines a differentiable
   isomorphism between the range of FlatKnots to the range
   of BSplineFlatKnots which is the
   same as the range of F(t)
   
   Warning: it is
   the caller's responsability to insure that conditions
   
   1. and 2. above are satisfied : no check whatsoever
      is made in this method
      
      Status will return 0 if OK else it will return the pivot index
      of the matrix that was inverted to compute the multiplied
      BSpline : the method used is interpolation at Schoenenberg
      points of F(a(t))
      

static void BSplCLib::FunctionReparameterise ( const BSplCLib_EvaluatorFunction &  Function, const Standard_Integer  BSplineDegree, const TColStd_Array1OfReal &  BSplineFlatKnots, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  FlatKnots, const Standard_Integer  NewDegree, TColgp_Array1OfPnt &  NewPoles, Standard_Integer &  Status  )

static

this will compose a given Vectorial BSpline F(t)
defined by its BSplineDegree and BSplineFlatKnotsl,
its Poles array which are coded as an array of Real
of the form [1..NumPoles][1..PolesDimension] with a
function a(t) which is assumed to satisfy the
following : 1. F(a(t)) is a polynomial BSpline
that can be expressed exactly as a BSpline of degree
NewDegree on the knots FlatKnots

                  2. a(t) defines a differentiable <br>
     isomorphism between the range of FlatKnots to the range <br>
     of BSplineFlatKnots which is the <br>
     same as the  range of F(t) <br>

Warning: it is
the caller's responsability to insure that conditions

1. and 2. above are satisfied : no check whatsoever
   is made in this method
   Status will return 0 if OK else it will return the pivot index
   of the matrix that was inverted to compute the multiplied
   BSpline : the method used is interpolation at Schoenenberg
   points of F(a(t))
   

static void BSplCLib::FunctionReparameterise ( const BSplCLib_EvaluatorFunction &  Function, const Standard_Integer  BSplineDegree, const TColStd_Array1OfReal &  BSplineFlatKnots, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  FlatKnots, const Standard_Integer  NewDegree, TColgp_Array1OfPnt2d &  NewPoles, Standard_Integer &  Status  )

static

this will compose a given Vectorial BSpline F(t)
defined by its BSplineDegree and BSplineFlatKnotsl,
its Poles array which are coded as an array of Real
of the form [1..NumPoles][1..PolesDimension] with a
function a(t) which is assumed to satisfy the
following : 1. F(a(t)) is a polynomial BSpline
that can be expressed exactly as a BSpline of degree
NewDegree on the knots FlatKnots

                  2. a(t) defines a differentiable <br>
     isomorphism between the range of FlatKnots to the range <br>
     of BSplineFlatKnots which is the <br>
     same as the  range of F(t) <br>

Warning: it is
the caller's responsability to insure that conditions

1. and 2. above are satisfied : no check whatsoever
   is made in this method
   Status will return 0 if OK else it will return the pivot index
   of the matrix that was inverted to compute the multiplied
   BSpline : the method used is interpolation at Schoenenberg
   points of F(a(t))
   

static void BSplCLib::GetPole ( const Standard_Integer  Index, const Standard_Integer  Length, const Standard_Integer  Depth, const Standard_Integer  Dimension, Standard_Real &  LocPoles, Standard_Integer &  Position, TColStd_Array1OfReal &  Pole  )

static

Copy the pole at position <Index> in the Boor
scheme of dimension <Dimension> to <Position> in
the array <Pole>. <Position> is updated.

static void BSplCLib::Hunt ( const TColStd_Array1OfReal &  XX, const Standard_Real  X, Standard_Integer &  Iloc  )

static

  This routine searches the position of the real <br>
    value X in  the ordered set of  real  values XX. <br>

The elements in the table XX are either
monotonically increasing or monotonically
decreasing.

The input value Iloc is used to initialize the
algorithm : if Iloc is outside of the bounds
[XX.Lower(), – XX.Upper()] the bisection algorithm
is used else the routine searches from a previous
known position by increasing steps then converges
by bisection.

This routine is used to locate a knot value in a
set of knots.

static void BSplCLib::IncreaseDegree ( const Standard_Integer  Degree, const Standard_Integer  NewDegree, const Standard_Boolean  Periodic, const Standard_Integer  Dimension, const TColStd_Array1OfReal &  Poles, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColStd_Array1OfReal &  NewPoles, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults  )

static

static void BSplCLib::IncreaseDegree ( const Standard_Integer  Degree, const Standard_Integer  NewDegree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColgp_Array1OfPnt &  NewPoles, TColStd_Array1OfReal &  NewWeights, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults  )

static

static void BSplCLib::IncreaseDegree ( const Standard_Integer  Degree, const Standard_Integer  NewDegree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColgp_Array1OfPnt2d &  NewPoles, TColStd_Array1OfReal &  NewWeights, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults  )

static

static void BSplCLib::IncreaseDegree ( const Standard_Integer  NewDegree, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, TColgp_Array1OfPnt &  NewPoles, TColStd_Array1OfReal &  NewWeights  )

static

static void BSplCLib::IncreaseDegree ( const Standard_Integer  NewDegree, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, TColgp_Array1OfPnt2d &  NewPoles, TColStd_Array1OfReal &  NewWeights  )

static

 Increase the degree of a bspline (or bezier) curve <br>
     of   dimension  <Dimension>  form <Degree>      to <br>
     <NewDegree>. <br>

The number of poles in the new curve is :

Poles.Length() + (NewDegree - Degree) * Number of spans

Where the number of spans is :

LastUKnotIndex(Mults) - FirstUKnotIndex(Mults) + 1

for a non-periodic curve

And Knots.Length() - 1 for a periodic curve.

The multiplicities of all knots are increased by
the degree elevation.

The new knots are usually the same knots with the
exception of a non-periodic curve with the first
and last multiplicity not equal to Degree+1 where
knots are removed form the start and the bottom
untils the sum of the multiplicities is equal to
NewDegree+1 at the knots corresponding to the
first and last parameters of the curve.

Example : Suppose a curve of degree 3 starting
with following knots and multiplicities :

knot : 0. 1. 2.
mult : 1 2 1

The FirstUKnot is 2. because the sum of
multiplicities is Degree+1 : 1 + 2 + 1 = 4 = 3 + 1

i.e. the first parameter of the curve is 2. and
will still be 2. after degree elevation. Let
raises this curve to degree 4. The multiplicities
are increased by 2.

They become 2 3 2. But we need a sum of
multiplicities of 5 at knot 2. So the first knot
is removed and the new knots are :

knot : 1. 2.
mult : 3 2

The multipicity of the first knot may also be
reduced if the sum is still to big.

In the most common situations (periodic curve or
curve with first and last multiplicities equals to
Degree+1) the knots are knot changes.

The method IncreaseDegreeCountKnots can be used to
compute the new number of knots.

static Standard_Integer BSplCLib::IncreaseDegreeCountKnots ( const Standard_Integer  Degree, const Standard_Integer  NewDegree, const Standard_Boolean  Periodic, const TColStd_Array1OfInteger &  Mults  )

static

Returns the number of knots of a curve with
multiplicities <Mults> after elevating the degree from
<Degree> to <NewDegree>. See the IncreaseDegree method
for more comments.

static void BSplCLib::InsertKnot ( const Standard_Integer  UIndex, const Standard_Real  U, const Standard_Integer  UMult, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColgp_Array1OfPnt &  NewPoles, TColStd_Array1OfReal &  NewWeights  )

static

static void BSplCLib::InsertKnot ( const Standard_Integer  UIndex, const Standard_Real  U, const Standard_Integer  UMult, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColgp_Array1OfPnt2d &  NewPoles, TColStd_Array1OfReal &  NewWeights  )

static

 Insert a new knot U of multiplicity UMult in the <br>

knot sequence.

The location of the new Knot should be given as an input
data. UIndex locates the new knot U in the knot sequence
and Knots (UIndex) < U < Knots (UIndex + 1).

The new control points corresponding to this insertion are
returned. Knots and Mults are not updated.

static void BSplCLib::InsertKnots ( const Standard_Integer  Degree, const Standard_Boolean  Periodic, const Standard_Integer  Dimension, const TColStd_Array1OfReal &  Poles, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const TColStd_Array1OfReal &  AddKnots, const TColStd_Array1OfInteger &  AddMults, TColStd_Array1OfReal &  NewPoles, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults, const Standard_Real  Epsilon, const Standard_Boolean  Add = Standard_True  )

static

static void BSplCLib::InsertKnots ( const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const TColStd_Array1OfReal &  AddKnots, const TColStd_Array1OfInteger &  AddMults, TColgp_Array1OfPnt &  NewPoles, TColStd_Array1OfReal &  NewWeights, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults, const Standard_Real  Epsilon, const Standard_Boolean  Add = Standard_True  )

static

static void BSplCLib::InsertKnots ( const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const TColStd_Array1OfReal &  AddKnots, const TColStd_Array1OfInteger &  AddMults, TColgp_Array1OfPnt2d &  NewPoles, TColStd_Array1OfReal &  NewWeights, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults, const Standard_Real  Epsilon, const Standard_Boolean  Add = Standard_True  )

static

 Insert   a  sequence  of  knots <AddKnots> with <br>

multiplicities <AddMults>. <AddKnots> must be a non
decreasing sequence and verifies :

Knots(Knots.Lower()) <= AddKnots(AddKnots.Lower())
Knots(Knots.Upper()) >= AddKnots(AddKnots.Upper())

The NewPoles and NewWeights arrays must have a length :
Poles.Length() + Sum(AddMults())

When a knot to insert is identic to an existing knot the
multiplicities are added.

Epsilon is used to test knots for equality.

When AddMult is negative or null the knot is not inserted.
No multiplicity will becomes higher than the degree.

The new Knots and Multiplicities are copied in <NewKnots>
and <NewMults>.

All the New arrays should be correctly dimensioned.

When all the new knots are existing knots, i.e. only the
multiplicities will change it is safe to use the same
arrays as input and output.

static void BSplCLib::Interpolate ( const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const TColStd_Array1OfReal &  Parameters, const TColStd_Array1OfInteger &  ContactOrderArray, TColgp_Array1OfPnt &  Poles, Standard_Integer &  InversionProblem  )

static

Performs the interpolation of  the data given in <br>
    the Poles  array  according  to the  requests in <br>
    ContactOrderArray    that is      :           if <br>
    ContactOrderArray(i) has value  d it means  that <br>
    Poles(i)   containes the dth  derivative of  the <br>
    function to be interpolated. The length L of the <br>
    following arrays must be the same : <br>
    Parameters, ContactOrderArray, Poles, <br>
    The length of FlatKnots is Degree + L + 1 <br>

Warning:
the method used to do that interpolation is
gauss elimination WITHOUT pivoting. Thus if the
diagonal is not dominant there is no guarantee
that the algorithm will work. Nevertheless for
Cubic interpolation or interpolation at Scheonberg
points the method will work
The InversionProblem will report 0 if there was no
problem else it will give the index of the faulty
pivot

static void BSplCLib::Interpolate ( const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const TColStd_Array1OfReal &  Parameters, const TColStd_Array1OfInteger &  ContactOrderArray, TColgp_Array1OfPnt2d &  Poles, Standard_Integer &  InversionProblem  )

static

Performs the interpolation of  the data given in <br>
    the Poles  array  according  to the  requests in <br>
    ContactOrderArray    that is      :           if <br>
    ContactOrderArray(i) has value  d it means  that <br>
    Poles(i)   containes the dth  derivative of  the <br>
    function to be interpolated. The length L of the <br>
    following arrays must be the same : <br>
    Parameters, ContactOrderArray, Poles, <br>
    The length of FlatKnots is Degree + L + 1 <br>

Warning:
the method used to do that interpolation is
gauss elimination WITHOUT pivoting. Thus if the
diagonal is not dominant there is no guarantee
that the algorithm will work. Nevertheless for
Cubic interpolation at knots or interpolation at Scheonberg
points the method will work.
The InversionProblem w
ll report 0 if there was no
problem else it will give the index of the faulty
pivot

static void BSplCLib::Interpolate ( const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const TColStd_Array1OfReal &  Parameters, const TColStd_Array1OfInteger &  ContactOrderArray, TColgp_Array1OfPnt &  Poles, TColStd_Array1OfReal &  Weights, Standard_Integer &  InversionProblem  )

static

 Performs the interpolation of  the data given in <br>
     the Poles  array  according  to the  requests in <br>
     ContactOrderArray    that is      :           if <br>
     ContactOrderArray(i) has value  d it means  that <br>
     Poles(i)   containes the dth  derivative of  the <br>
     function to be interpolated. The length L of the <br>
     following arrays must be the same : <br>
     Parameters, ContactOrderArray, Poles, <br>
     The length of FlatKnots is Degree + L + 1 <br>

Warning:
the method used to do that interpolation is
gauss elimination WITHOUT pivoting. Thus if the
diagonal is not dominant there is no guarantee
that the algorithm will work. Nevertheless for
Cubic interpolation at knots or interpolation at Scheonberg
points the method will work.
The InversionProblem will report 0 if there was no
problem else it will give the index of the faulty
pivot

static void BSplCLib::Interpolate ( const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const TColStd_Array1OfReal &  Parameters, const TColStd_Array1OfInteger &  ContactOrderArray, TColgp_Array1OfPnt2d &  Poles, TColStd_Array1OfReal &  Weights, Standard_Integer &  InversionProblem  )

static

Performs the interpolation of  the data given in <br>
    the Poles  array  according  to the  requests in <br>
    ContactOrderArray    that is      :           if <br>
    ContactOrderArray(i) has value  d it means  that <br>
    Poles(i)   containes the dth  derivative of  the <br>
    function to be interpolated. The length L of the <br>
    following arrays must be the same : <br>
    Parameters, ContactOrderArray, Poles, <br>
    The length of FlatKnots is Degree + L + 1 <br>

Warning:
the method used to do that interpolation is
gauss elimination WITHOUT pivoting. Thus if the
diagonal is not dominant there is no guarantee
that the algorithm will work. Nevertheless for
Cubic interpolation at knots or interpolation at Scheonberg
points the method will work.
The InversionProblem w
ll report 0 if there was no
problem else it will give the i

static void BSplCLib::Interpolate ( const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const TColStd_Array1OfReal &  Parameters, const TColStd_Array1OfInteger &  ContactOrderArray, const Standard_Integer  ArrayDimension, Standard_Real &  Poles, Standard_Integer &  InversionProblem  )

static

 Performs the interpolation of  the data given in <br>
     the Poles  array  according  to the  requests in <br>
     ContactOrderArray    that is      :           if <br>
     ContactOrderArray(i) has value  d it means  that <br>
     Poles(i)   containes the dth  derivative of  the <br>
     function to be interpolated. The length L of the <br>
     following arrays must be the same : <br>
     Parameters, ContactOrderArray <br>
     The length of FlatKnots is Degree + L + 1 <br>
     The  PolesArray   is    an   seen   as    an <br>
     Array[1..N][1..ArrayDimension] with N = tge length <br>
     of the parameters array <br>

Warning:
the method used to do that interpolation is
gauss elimination WITHOUT pivoting. Thus if the
diagonal is not dominant there is no guarantee
that the algorithm will work. Nevertheless for
Cubic interpolation or interpolation at Scheonberg
points the method will work
The InversionProblem will report 0 if there was no
problem else it will give the index of the faulty
pivot

static void BSplCLib::Interpolate ( const Standard_Integer  Degree, const TColStd_Array1OfReal &  FlatKnots, const TColStd_Array1OfReal &  Parameters, const TColStd_Array1OfInteger &  ContactOrderArray, const Standard_Integer  ArrayDimension, Standard_Real &  Poles, Standard_Real &  Weights, Standard_Integer &  InversionProblem  )

static

static Standard_Boolean BSplCLib::IsRational ( const TColStd_Array1OfReal &  Weights, const Standard_Integer  I1, const Standard_Integer  I2, const Standard_Real  Epsilon = 0.0  )

static

Returns False if all the weights of the array <Weights>
between I1 an I2 are identic. Epsilon is used for
comparing weights. If Epsilon is 0. the Epsilon of the
first weight is used.

static void BSplCLib::KnotAnalysis ( const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColStd_Array1OfReal &  CKnots, const TColStd_Array1OfInteger &  CMults, GeomAbs_BSplKnotDistribution &  KnotForm, Standard_Integer &  MaxKnotMult  )

static

Analyzes the array of knots.
Returns the form and the maximum knot multiplicity.

static BSplCLib_KnotDistribution BSplCLib::KnotForm ( const TColStd_Array1OfReal &  Knots, const Standard_Integer  FromK1, const Standard_Integer  ToK2  )

static

Analyses if the knots distribution is "Uniform"
or "NonUniform" between the knot FromK1 and the
knot ToK2. There is no repetition of knot in the
knots'sequence <Knots>.

static void BSplCLib::Knots ( const TColStd_Array1OfReal &  KnotSeq, TColStd_Array1OfReal &  Knots, TColStd_Array1OfInteger &  Mults, const Standard_Boolean  Periodic = Standard_False  )

static

  Computes  the  sequence   of knots Knots  without <br>
    repetition  of the  knots  of multiplicity  greater <br>
    than 1. <br>

Length of <Knots> and <Mults> must be
KnotsLength(KnotSequence,Periodic)

static void BSplCLib::KnotSequence ( const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColStd_Array1OfReal &  KnotSeq  )

static

static void BSplCLib::KnotSequence ( const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const Standard_Integer  Degree, const Standard_Boolean  Periodic, TColStd_Array1OfReal &  KnotSeq  )

static

 Computes  the  sequence   of knots KnotSeq  with <br>
    repetition  of the  knots  of multiplicity  greater <br>
    than 1. <br>

Length of KnotSeq must be KnotSequenceLength(Mults,Degree,Periodic)

static Standard_Integer BSplCLib::KnotSequenceLength ( const TColStd_Array1OfInteger &  Mults, const Standard_Integer  Degree, const Standard_Boolean  Periodic  )

static

 Returns the length  of the sequence  of knots with <br>
     repetition. <br>

Periodic :

Sum(Mults(i), i = Mults.Lower(); i <= Mults.Upper());

Non Periodic :

Sum(Mults(i); i = Mults.Lower(); i < Mults.Upper())

• 2 * Degree
  

static Standard_Integer BSplCLib::KnotsLength ( const TColStd_Array1OfReal &  KnotSeq, const Standard_Boolean  Periodic = Standard_False  )

static

Returns the length of the sequence of knots (and
Mults) without repetition.

static Standard_Integer BSplCLib::LastUKnotIndex ( const Standard_Integer  Degree, const TColStd_Array1OfInteger &  Mults  )

static

Computes the index of the knots value which gives
the end point of the curve.

static void BSplCLib::LocateParameter ( const Standard_Integer  Degree, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const Standard_Real  U, const Standard_Boolean  IsPeriodic, const Standard_Integer  FromK1, const Standard_Integer  ToK2, Standard_Integer &  KnotIndex, Standard_Real &  NewU  )

static

  Locates  the parametric value    U  in the knots <br>
    sequence  between  the  knot K1   and the knot  K2. <br>
    The value return in Index verifies. <br>

Knots(Index) <= U < Knots(Index + 1)
if U <= Knots (K1) then Index = K1
if U >= Knots (K2) then Index = K2 - 1

If Periodic is True U may be modified to fit in
the range Knots(K1), Knots(K2). In any case the
correct value is returned in NewU.

Warnings :Index is used as input data to initialize the
searching function.
Warning: Knots have to be "withe repetitions"

static void BSplCLib::LocateParameter ( const Standard_Integer  Degree, const TColStd_Array1OfReal &  Knots, const Standard_Real  U, const Standard_Boolean  IsPeriodic, const Standard_Integer  FromK1, const Standard_Integer  ToK2, Standard_Integer &  KnotIndex, Standard_Real &  NewU  )

static

  Locates  the parametric value    U  in the knots <br>
    sequence  between  the  knot K1   and the knot  K2. <br>
    The value return in Index verifies. <br>

Knots(Index) <= U < Knots(Index + 1)
if U <= Knots (K1) then Index = K1
if U >= Knots (K2) then Index = K2 - 1

If Periodic is True U may be modified to fit in
the range Knots(K1), Knots(K2). In any case the
correct value is returned in NewU.

Warnings :Index is used as input data to initialize the
searching function.
Warning: Knots have to be "flat"

static void BSplCLib::LocateParameter ( const Standard_Integer  Degree, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const Standard_Real  U, const Standard_Boolean  Periodic, Standard_Integer &  Index, Standard_Real &  NewU  )

static

static Standard_Integer BSplCLib::MaxDegree ( )

static

returns the degree maxima for a BSplineCurve.

static Standard_Integer BSplCLib::MaxKnotMult ( const TColStd_Array1OfInteger &  Mults, const Standard_Integer  K1, const Standard_Integer  K2  )

static

Finds the greatest multiplicity in a set of knots
between K1 and K2. Mults is the multiplicity
associated with each knot value.

static void BSplCLib::MergeBSplineKnots ( 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  )

static

Merges two knot vector by setting the starting and
ending values to StartValue and EndValue

static Standard_Integer BSplCLib::MinKnotMult ( const TColStd_Array1OfInteger &  Mults, const Standard_Integer  K1, const Standard_Integer  K2  )

static

Finds the lowest multiplicity in a set of knots
between K1 and K2. Mults is the multiplicity
associated with each knot value.

static void BSplCLib::MovePoint ( const Standard_Real  U, const gp_Vec2d &  Displ, const Standard_Integer  Index1, const Standard_Integer  Index2, const Standard_Integer  Degree, const Standard_Boolean  Rational, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  FlatKnots, Standard_Integer &  FirstIndex, Standard_Integer &  LastIndex, TColgp_Array1OfPnt2d &  NewPoles  )

static

Find the new poles which allows an old point (with a
given u as parameter) to reach a new position
Index1 and Index2 indicate the range of poles we can move
(1, NbPoles-1) or (2, NbPoles) -> no constraint for one side
don't enter (1,NbPoles) -> error: rigid move
(2, NbPoles-1) -> the ends are enforced
(3, NbPoles-2) -> the ends and the tangency are enforced
if Problem in BSplineBasis calculation, no change for the curve
and FirstIndex, LastIndex = 0

static void BSplCLib::MovePoint ( const Standard_Real  U, const gp_Vec &  Displ, const Standard_Integer  Index1, const Standard_Integer  Index2, const Standard_Integer  Degree, const Standard_Boolean  Rational, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  FlatKnots, Standard_Integer &  FirstIndex, Standard_Integer &  LastIndex, TColgp_Array1OfPnt &  NewPoles  )

static

Find the new poles which allows an old point (with a
given u as parameter) to reach a new position
Index1 and Index2 indicate the range of poles we can move
(1, NbPoles-1) or (2, NbPoles) -> no constraint for one side
don't enter (1,NbPoles) -> error: rigid move
(2, NbPoles-1) -> the ends are enforced
(3, NbPoles-2) -> the ends and the tangency are enforced
if Problem in BSplineBasis calculation, no change for the curve
and FirstIndex, LastIndex = 0

static void BSplCLib::MovePointAndTangent ( const Standard_Real  U, const Standard_Integer  ArrayDimension, Standard_Real &  Delta, Standard_Real &  DeltaDerivative, 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  )

static

 This is the dimension free version of the utility <br>

U is the parameter must be within the first FlatKnots and the
last FlatKnots Delta is the amount the curve has to be moved
DeltaDerivative is the amount the derivative has to be moved.
Delta and DeltaDerivative must be array of dimension
ArrayDimension Degree is the degree of the BSpline and the
FlatKnots are the knots of the BSpline Starting Condition if =
-1 means the starting point of the curve can move
= 0 means the
starting point of the cuve cannot move but tangen starting
point of the curve cannot move
= 1 means the starting point and tangents cannot move
= 2 means the starting point tangent and curvature cannot move
= ...
Same holds for EndingCondition
Poles are the poles of the curve
Weights are the weights of the curve if Rational = Standard_True
NewPoles are the poles of the deformed curve
ErrorStatus will be 0 if no error happened
1 if there are not enough knots/poles
the imposed conditions
The way to solve this problem is to add knots to the BSpline
If StartCondition = 1 and EndCondition = 1 then you need at least
4 + 2 = 6 poles so for example to have a C1 cubic you will need
have at least 2 internal knots.

static void BSplCLib::MovePointAndTangent ( const Standard_Real  U, const gp_Vec &  Delta, const gp_Vec &  DeltaDerivative, const Standard_Real  Tolerance, const Standard_Integer  Degree, const Standard_Boolean  Rational, const Standard_Integer  StartingCondition, const Standard_Integer  EndingCondition, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  FlatKnots, TColgp_Array1OfPnt &  NewPoles, Standard_Integer &  ErrorStatus  )

static

 This is the dimension free version of the utility <br>

U is the parameter must be within the first FlatKnots and the
last FlatKnots Delta is the amount the curve has to be moved
DeltaDerivative is the amount the derivative has to be moved.
Delta and DeltaDerivative must be array of dimension
ArrayDimension Degree is the degree of the BSpline and the
FlatKnots are the knots of the BSpline Starting Condition if =
-1 means the starting point of the curve can move
= 0 means the
starting point of the cuve cannot move but tangen starting
point of the curve cannot move
= 1 means the starting point and tangents cannot move
= 2 means the starting point tangent and curvature cannot move
= ...
Same holds for EndingCondition
Poles are the poles of the curve
Weights are the weights of the curve if Rational = Standard_True
NewPoles are the poles of the deformed curve
ErrorStatus will be 0 if no error happened
1 if there are not enough knots/poles
the imposed conditions
The way to solve this problem is to add knots to the BSpline
If StartCondition = 1 and EndCondition = 1 then you need at least
4 + 2 = 6 poles so for example to have a C1 cubic you will need
have at least 2 internal knots.

static void BSplCLib::MovePointAndTangent ( const Standard_Real  U, const gp_Vec2d &  Delta, const gp_Vec2d &  DeltaDerivative, const Standard_Real  Tolerance, const Standard_Integer  Degree, const Standard_Boolean  Rational, const Standard_Integer  StartingCondition, const Standard_Integer  EndingCondition, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  FlatKnots, TColgp_Array1OfPnt2d &  NewPoles, Standard_Integer &  ErrorStatus  )

static

 This is the dimension free version of the utility <br>

U is the parameter must be within the first FlatKnots and the
last FlatKnots Delta is the amount the curve has to be moved
DeltaDerivative is the amount the derivative has to be moved.
Delta and DeltaDerivative must be array of dimension
ArrayDimension Degree is the degree of the BSpline and the
FlatKnots are the knots of the BSpline Starting Condition if =
-1 means the starting point of the curve can move
= 0 means the
starting point of the cuve cannot move but tangen starting
point of the curve cannot move
= 1 means the starting point and tangents cannot move
= 2 means the starting point tangent and curvature cannot move
= ...
Same holds for EndingCondition
Poles are the poles of the curve
Weights are the weights of the curve if Rational = Standard_True
NewPoles are the poles of the deformed curve
ErrorStatus will be 0 if no error happened
1 if there are not enough knots/poles
the imposed conditions
The way to solve this problem is to add knots to the BSpline
If StartCondition = 1 and EndCondition = 1 then you need at least
4 + 2 = 6 poles so for example to have a C1 cubic you will need
have at least 2 internal knots.

static BSplCLib_MultDistribution BSplCLib::MultForm ( const TColStd_Array1OfInteger &  Mults, const Standard_Integer  FromK1, const Standard_Integer  ToK2  )

static

Analyses the distribution of multiplicities between
the knot FromK1 and the Knot ToK2.

static Standard_Integer BSplCLib::NbPoles ( const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColStd_Array1OfInteger &  Mults  )

static

 Returns the number of poles of the curve. Returns 0 if <br>
     one of the multiplicities is incorrect. <br>

     * Non positive. <br>

• Greater than Degree, or Degree+1 at the first and
  last knot of a non periodic curve.
  
  
• The last periodicity on a periodic curve is not
  equal to the first.
  

static TColStd_Array1OfInteger& BSplCLib::NoMults ( )

static

 Used as argument for a flatknots evaluation. <br>

static TColStd_Array1OfReal& BSplCLib::NoWeights ( )

static

 Used as argument for a non rational curve. <br>

static Standard_Integer BSplCLib::PoleIndex ( const Standard_Integer  Degree, const Standard_Integer  Index, const Standard_Boolean  Periodic, const TColStd_Array1OfInteger &  Mults  )

static

Return the index of the first Pole to use on the
span Mults(Index) - Mults(Index+1). This index
must be added to Poles.Lower().

static void BSplCLib::PolesCoefficients ( const TColgp_Array1OfPnt2d &  Poles, TColgp_Array1OfPnt2d &  CachePoles  )

static

static void BSplCLib::PolesCoefficients ( const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, TColgp_Array1OfPnt2d &  CachePoles, TColStd_Array1OfReal &  CacheWeights  )

static

static void BSplCLib::PolesCoefficients ( const TColgp_Array1OfPnt &  Poles, TColgp_Array1OfPnt &  CachePoles  )

static

static void BSplCLib::PolesCoefficients ( const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, TColgp_Array1OfPnt &  CachePoles, TColStd_Array1OfReal &  CacheWeights  )

static

Encapsulation   of  BuildCache    to   perform   the <br>
    evaluation  of the Taylor expansion for beziercurves <br>
    at parameter 0. <br>

Warning: To be used for Beziercurves ONLY!!!

static Standard_Boolean BSplCLib::PrepareInsertKnots ( const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const TColStd_Array1OfReal &  AddKnots, const TColStd_Array1OfInteger &  AddMults, Standard_Integer &  NbPoles, Standard_Integer &  NbKnots, const Standard_Real  Epsilon, const Standard_Boolean  Add = Standard_True  )

static

 Returns in <NbPoles, NbKnots> the  new number of poles <br>
     and  knots    if  the  sequence   of  knots <AddKnots, <br>
     AddMults> is inserted in the sequence <Knots, Mults>. <br>

Epsilon is used to compare knots for equality.

If Add is True the multiplicities on equal knots are
added.

If Add is False the max value of the multiplicities is
kept.

Return False if :
The knew knots are knot increasing.
The new knots are not in the range.

static void BSplCLib::PrepareTrimming ( const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const Standard_Real  U1, const Standard_Real  U2, Standard_Integer &  NbKnots, Standard_Integer &  NbPoles  )

static

Set in <NbKnots> and <NbPoles> the number of Knots and
Poles of the curve resulting of the trimming of the
BSplinecurve definded with <degree>, <knots>, <mults>

static void BSplCLib::PrepareUnperiodize ( const Standard_Integer  Degree, const TColStd_Array1OfInteger &  Mults, Standard_Integer &  NbKnots, Standard_Integer &  NbPoles  )

static

Set in <NbKnots> and <NbPolesToAdd> the number of Knots and
Poles of the NotPeriodic Curve identical at the
periodic curve with a degree <Degree> , a
knots-distribution with Multiplicities <Mults>.

static void BSplCLib::RaiseMultiplicity ( const Standard_Integer  KnotIndex, const Standard_Integer  Mult, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColgp_Array1OfPnt &  NewPoles, TColStd_Array1OfReal &  NewWeights  )

static

static void BSplCLib::RaiseMultiplicity ( const Standard_Integer  KnotIndex, const Standard_Integer  Mult, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColgp_Array1OfPnt2d &  NewPoles, TColStd_Array1OfReal &  NewWeights  )

static

 Raise the multiplicity of knot to <UMult>. <br>

The new control points are returned. Knots and Mults are
not updated.

static Standard_Boolean BSplCLib::RemoveKnot ( const Standard_Integer  Index, const Standard_Integer  Mult, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const Standard_Integer  Dimension, const TColStd_Array1OfReal &  Poles, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColStd_Array1OfReal &  NewPoles, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults, const Standard_Real  Tolerance  )

static

static Standard_Boolean BSplCLib::RemoveKnot ( const Standard_Integer  Index, const Standard_Integer  Mult, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColgp_Array1OfPnt &  NewPoles, TColStd_Array1OfReal &  NewWeights, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults, const Standard_Real  Tolerance  )

static

static Standard_Boolean BSplCLib::RemoveKnot ( const Standard_Integer  Index, const Standard_Integer  Mult, const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, TColgp_Array1OfPnt2d &  NewPoles, TColStd_Array1OfReal &  NewWeights, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults, const Standard_Real  Tolerance  )

static

 Decrement the  multiplicity  of <Knots(Index)> <br>
     to <Mult>. If <Mult>   is  null the   knot  is <br>
     removed. <br>

As there are two ways to compute the new poles
the midlle will be used as long as the
distance is lower than Tolerance.

If a distance is bigger than tolerance the
methods returns False and the new arrays are
not modified.

A low tolerance can be used to test if the
knot can be removed without modifying the
curve.

A high tolerance can be used to "smooth" the
curve.

static void BSplCLib::Reparametrize ( const Standard_Real  U1, const Standard_Real  U2, TColStd_Array1OfReal &  Knots  )

static

Reparametrizes a B-spline curve to [U1, U2].
The knot values are recomputed such that Knots (Lower) = U1
and Knots (Upper) = U2 but the knot form is not modified.
Warnings :
In the array Knots the values must be in ascending order.
U1 must not be equal to U2 to avoid division by zero.

static void BSplCLib::Resolution ( Standard_Real &  PolesArray, const Standard_Integer  ArrayDimension, const Standard_Integer  NumPoles, const TColStd_Array1OfReal &  Weights, const TColStd_Array1OfReal &  FlatKnots, const Standard_Integer  Degree, const Standard_Real  Tolerance3D, Standard_Real &  UTolerance  )

static

given a tolerance in 3D space returns a
tolerance in U parameter space such that
all u1 and u0 in the domain of the curve f(u)
| u1 - u0 | < UTolerance and
we have |f (u1) - f (u0)| < Tolerance3D

static void BSplCLib::Resolution ( const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const Standard_Integer  NumPoles, const TColStd_Array1OfReal &  FlatKnots, const Standard_Integer  Degree, const Standard_Real  Tolerance3D, Standard_Real &  UTolerance  )

static

given a tolerance in 3D space returns a
tolerance in U parameter space such that
all u1 and u0 in the domain of the curve f(u)
| u1 - u0 | < UTolerance and
we have |f (u1) - f (u0)| < Tolerance3D

static void BSplCLib::Resolution ( const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const Standard_Integer  NumPoles, const TColStd_Array1OfReal &  FlatKnots, const Standard_Integer  Degree, const Standard_Real  Tolerance3D, Standard_Real &  UTolerance  )

static

given a tolerance in 3D space returns a
tolerance in U parameter space such that
all u1 and u0 in the domain of the curve f(u)
| u1 - u0 | < UTolerance and
we have |f (u1) - f (u0)| < Tolerance3D

static void BSplCLib::Reverse ( TColStd_Array1OfReal &  Knots )

static

Reverses the array knots to become the knots
sequence of the reversed curve.

static void BSplCLib::Reverse ( TColStd_Array1OfInteger &  Mults )

static

Reverses the array of multiplicities.

static void BSplCLib::Reverse ( TColgp_Array1OfPnt &  Poles, const Standard_Integer  Last  )

static

 Reverses the array of poles. Last is the  index of <br>
     the new first pole. On  a  non periodic curve last <br>
     is Poles.Upper(). On a periodic curve last is <br>

(number of flat knots - degree - 1)

or

(sum of multiplicities(but for the last) + degree

• 1)
  

static void BSplCLib::Reverse ( TColgp_Array1OfPnt2d &  Poles, const Standard_Integer  Last  )

static

Reverses the array of poles.

static void BSplCLib::Reverse ( TColStd_Array1OfReal &  Weights, const Standard_Integer  Last  )

static

Reverses the array of poles.

static Standard_Integer BSplCLib::SolveBandedSystem ( const math_Matrix &  Matrix, const Standard_Integer  UpperBandWidth, const Standard_Integer  LowerBandWidth, const Standard_Integer  ArrayDimension, Standard_Real &  Array  )

static

This solves the system Matrix.X = B
with when Matrix is factored in LU form
The Array is an seen as an
Array[1..N][1..ArrayDimension] with N =
the rank of the matrix Matrix. The
result is stored in Array when each
coordinate is solved that is B is the
array whose values are
B[i] = Array[i][p] for each p in 1..ArrayDimension

static Standard_Integer BSplCLib::SolveBandedSystem ( const math_Matrix &  Matrix, const Standard_Integer  UpperBandWidth, const Standard_Integer  LowerBandWidth, TColgp_Array1OfPnt2d &  Array  )

static

This solves the system Matrix.X = B
with when Matrix is factored in LU form
The Array has the length of
the rank of the matrix Matrix. The
result is stored in Array when each
coordinate is solved that is B is the
array whose values are
B[i] = Array[i][p] for each p in 1..ArrayDimension

static Standard_Integer BSplCLib::SolveBandedSystem ( const math_Matrix &  Matrix, const Standard_Integer  UpperBandWidth, const Standard_Integer  LowerBandWidth, TColgp_Array1OfPnt &  Array  )

static

This solves the system Matrix.X = B
with when Matrix is factored in LU form
The Array has the length of
the rank of the matrix Matrix. The
result is stored in Array when each
coordinate is solved that is B is the
array whose values are
B[i] = Array[i][p] for each p in 1..ArrayDimension

static Standard_Integer BSplCLib::SolveBandedSystem ( const math_Matrix &  Matrix, const Standard_Integer  UpperBandWidth, const Standard_Integer  LowerBandWidth, const Standard_Boolean  HomogenousFlag, const Standard_Integer  ArrayDimension, Standard_Real &  Array, Standard_Real &  Weights  )

static

static Standard_Integer BSplCLib::SolveBandedSystem ( const math_Matrix &  Matrix, const Standard_Integer  UpperBandWidth, const Standard_Integer  LowerBandWidth, const Standard_Boolean  HomogenousFlag, TColgp_Array1OfPnt2d &  Array, TColStd_Array1OfReal &  Weights  )

static

This solves the system Matrix.X = B
with when Matrix is factored in LU form
The Array is an seen as an
Array[1..N][1..ArrayDimension] with N =
the rank of the matrix Matrix. The
result is stored in Array when each
coordinate is solved that is B is the
array whose values are B[i] =
Array[i][p] for each p in
1..ArrayDimension. If HomogeneousFlag ==
0 the Poles are multiplied by the
Weights uppon Entry and once
interpolation is carried over the
result of the poles are divided by the
result of the interpolation of the
weights. Otherwise if HomogenousFlag == 1
the Poles and Weigths are treated homogenously
that is that those are interpolated as they
are and result is returned without division
by the interpolated weigths.

static Standard_Integer BSplCLib::SolveBandedSystem ( const math_Matrix &  Matrix, const Standard_Integer  UpperBandWidth, const Standard_Integer  LowerBandWidth, const Standard_Boolean  HomogeneousFlag, TColgp_Array1OfPnt &  Array, TColStd_Array1OfReal &  Weights  )

static

This solves the system Matrix.X = B
with when Matrix is factored in LU form
The Array is an seen as an
Array[1..N][1..ArrayDimension] with N =
the rank of the matrix Matrix. The
result is stored in Array when each
coordinate is solved that is B is the
array whose values are
B[i] = Array[i][p] for each p in 1..ArrayDimension
If HomogeneousFlag ==
0 the Poles are multiplied by the
Weights uppon Entry and once
interpolation is carried over the
result of the poles are divided by the
result of the interpolation of the
weights. Otherwise if HomogenousFlag == 1
the Poles and Weigths are treated homogenously
that is that those are interpolated as they
are and result is returned without division
by the interpolated weigths.

static void BSplCLib::TangExtendToConstraint ( const TColStd_Array1OfReal &  FlatKnots, const Standard_Real  C1Coefficient, const Standard_Integer  NumPoles, Standard_Real &  Poles, const Standard_Integer  Dimension, const Standard_Integer  Degree, const TColStd_Array1OfReal &  ConstraintPoint, const Standard_Integer  Continuity, const Standard_Boolean  After, Standard_Integer &  NbPolesResult, Standard_Integer &  NbKnotsRsult, Standard_Real &  KnotsResult, Standard_Real &  PolesResult  )

static

 Extend a BSpline nD using the tangency map <br>
     <C1Coefficient> is the coefficient of reparametrisation <br>
     <Continuity> must be equal to 1, 2 or 3. <br>
     <Degree> must be greater or equal than <Continuity> + 1. <br>

Warning: <KnotsResult> and <PolesResult> must be dimensionned
properly.

static void BSplCLib::Trimming ( const Standard_Integer  Degree, const Standard_Boolean  Periodic, const Standard_Integer  Dimension, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const TColStd_Array1OfReal &  Poles, const Standard_Real  U1, const Standard_Real  U2, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults, TColStd_Array1OfReal &  NewPoles  )

static

static void BSplCLib::Trimming ( const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, const Standard_Real  U1, const Standard_Real  U2, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults, TColgp_Array1OfPnt &  NewPoles, TColStd_Array1OfReal &  NewWeights  )

static

static void BSplCLib::Trimming ( const Standard_Integer  Degree, const Standard_Boolean  Periodic, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfInteger &  Mults, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, const Standard_Real  U1, const Standard_Real  U2, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfInteger &  NewMults, TColgp_Array1OfPnt2d &  NewPoles, TColStd_Array1OfReal &  NewWeights  )

static

static void BSplCLib::Unperiodize ( const Standard_Integer  Degree, const Standard_Integer  Dimension, const TColStd_Array1OfInteger &  Mults, const TColStd_Array1OfReal &  Knots, const TColStd_Array1OfReal &  Poles, TColStd_Array1OfInteger &  NewMults, TColStd_Array1OfReal &  NewKnots, TColStd_Array1OfReal &  NewPoles  )

static

static void BSplCLib::Unperiodize ( const Standard_Integer  Degree, const TColStd_Array1OfInteger &  Mults, const TColStd_Array1OfReal &  Knots, const TColgp_Array1OfPnt &  Poles, const TColStd_Array1OfReal &  Weights, TColStd_Array1OfInteger &  NewMults, TColStd_Array1OfReal &  NewKnots, TColgp_Array1OfPnt &  NewPoles, TColStd_Array1OfReal &  NewWeights  )

static

static void BSplCLib::Unperiodize ( const Standard_Integer  Degree, const TColStd_Array1OfInteger &  Mults, const TColStd_Array1OfReal &  Knots, const TColgp_Array1OfPnt2d &  Poles, const TColStd_Array1OfReal &  Weights, TColStd_Array1OfInteger &  NewMults, TColStd_Array1OfReal &  NewKnots, TColgp_Array1OfPnt2d &  NewPoles, TColStd_Array1OfReal &  NewWeights  )

static

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The documentation for this class was generated from the following file:

• BSplCLib.hxx