KernelCAD Documentation

Represents a bspline curve. To create a bspline curve use a call similar to iDIObjGenerator.Create3("KO_Geom_BSplineCurve") where iDIObjGenerator has IDIObjGenerator type.

Definition of the B_spline curve. A B-spline curve can be Uniform or non-uniform Rational or non-rational Periodic or non-periodic a b-spline curve is defined by : its degree; the degree for a Geom_BSplineCurve is limited to a value (25) which is defined and controlled by the system. This value is returned by the function MaxDegree; its periodic or non-periodic nature; a table of poles (also called control points), with their associated weights if the BSpline curve is rational. The poles of the curve are "control points" used to deform the curve. If the curve is non-periodic, the first pole is the start point of the curve, and the last pole is the end point of the curve. The segment which joins the first pole to the second pole is the tangent to the curve at its start point, and the segment which joins the last pole to the second-from-last pole is the tangent to the curve at its end point. If the curve is periodic, these geometric properties are not verified. It is more difficult to give a geometric signification to the weights but are useful for providing exact representations of the arcs of a circle or ellipse. Moreover, if the weights of all the poles are equal, the curve has a polynomial equation; it is therefore a non-rational curve. a table of knots with their multiplicities. For a Geom_BSplineCurve, the table of knots is an increasing sequence of reals without repetition; the multiplicities define the repetition of the knots. A BSpline curve is a piecewise polynomial or rational curve.

The knots are the parameters of junction points between two pieces. The multiplicity Mult(i) of the knot Knot(i) of the BSpline curve is related to the degree of continuity of the curve at the knot Knot(i), which is equal to Degree - Mult(i) where Degree is the degree of the BSpline curve. If the knots are regularly spaced (i.e. the difference between two consecutive knots is a constant), three specific and frequently used cases of knot distribution can be identified: "uniform" if all multiplicities are equal to 1, "quasi-uniform" if all multiplicities are equal to 1, except the first and the last knot which have a multiplicity of Degree + 1, where Degree is the degree of the BSpline curve, "Piecewise Bezier" if all multiplicities are equal to Degree except the first and last knot which have a multiplicity of Degree + 1, where Degree is the degree of the BSpline curve. A curve of this type is a concatenation of arcs of Bezier curves. If the BSpline curve is not periodic: the bounds of the Poles and Weights tables are 1 and NbPoles, where NbPoles is the number of poles of the BSpline curve, the bounds of the Knots and Multiplicities tables are 1 and NbKnots, where NbKnots is the number of knots of the BSpline curve. If the BSpline curve is periodic, and if there are k periodic knots and p periodic poles, the period is: period = Knot(k + 1) - Knot(1) and the poles and knots tables can be considered as infinite tables, verifying: Knot(i+k) = Knot(i) + period Pole(i+p) = Pole(i)

Note: data structures of a periodic BSpline curve are more complex than those of a non-periodic one. Warning In this class, weight value is considered to be zero if the weight is less than or equal to gp::Resolution(). References : . A survey of curve and surface methods in CADG Wolfgang BOHM CAGD 1 (1984) . On de Boor-like algorithms and blossoming Wolfgang BOEHM cagd 5 (1988) . Blossoming and knot insertion algorithms for B-spline curves Ronald N. GOLDMAN . Modelisation des surfaces en CAO, Henri GIAUME Peugeot SA . Curves and Surfaces for Computer Aided Geometric Design, a practical guide Gerald Farin

Query IKO_gp_Object from this interface to obtain or modify location and orientation of the curve

Query IKO_gp_Transformation to transform position and orientation

IKO_Standard_Object to create a copy or obtain type name

IncreaseMultiplicity2

PeriodicNormalization

HRESULT Init(IKO_TColgp_Array1OfPnt* Poles, IKO_TColStd_Array1OfReal* Knots, IKO_TColStd_Array1OfInteger* Multiplicities, int Degree, VARIANT_BOOL Periodic);

Constructs a non-rational B_spline curve on the basis of degree .

HRESULT Init2(IKO_TColgp_Array1OfPnt* Poles, IKO_TColStd_Array1OfReal* Weights, IKO_TColStd_Array1OfReal* Knots, IKO_TColStd_Array1OfInteger* Multiplicities, int Degree, VARIANT_BOOL Periodic, VARIANT_BOOL CheckRational)

Remarks:

Creates a rational B_spline curve on the basis of degree . Raises ConstructionError subject to the following conditions 0 < Degree <= MaxDegree. Weights.Length() == Poles.Length() Knots.Length() == Mults.Length() >= 2 Knots(i) < Knots(i+1) (Knots are increasing) 1 <= Mults(i) <= Degree On a non periodic curve the first and last multiplicities may be Degree+1 (this is even recommanded if you want the curve to start and finish on the first and last pole). On a periodic curve the first and the last multicities must be the same. on non-periodic curves Poles.Length() == Sum(Mults(i)) - Degree - 1 >= 2 on periodic curves Poles.Length() == Sum(Mults(i)) except the first or last

HRESULT IncreaseDegree(int Degree)

Increases the degree of this BSpline curve to Degree. As a result, the poles, weights and multiplicities tables are modified; the knots table is not changed. Nothing is done if Degree is less than or equal to the current degree. Exceptions Standard_ConstructionError if Degree is greater than Geom_BSplineCurve::MaxDegree().

HRESULT IncreaseMultiplicity(int Index, int M)

Increases the multiplicity of the knot to . If is lower or equal to the current multiplicity nothing is done. If is higher than the degree the degree is used. //! If is not in [FirstUKnotIndex, LastUKnotIndex]

HRESULT IncreaseMultiplicity2(int I1, int I2, int M)

Remarks:

Increases the multiplicities of the knots in [I1,I2] to . For each knot if is lower or equal to the current multiplicity nothing is done. If is higher than the degree the degree is used. //! If are not in [FirstUKnotIndex, LastUKnotIndex]

HRESULT InsertKnot(double U, int M, double ParametricTolerance, VARIANT_BOOL Add)

Inserts a knot value in the sequence of knots. If is an existing knot the multiplicity is increased by . If U is not on the parameter range nothing is done. If the multiplicity is negative or null nothing is done. The new multiplicity is limited to the degree. The tolerance criterion for knots equality is the max of Epsilon(U) . If U is not on the parameter range nothing is done. If the multiplicity is negative or null nothing is done. The new multiplicity is limited to the degree. The tolerance criterion for knots equality is the max of Epsilon(U) and ParametricTolerance.

HRESULT InsertKnots(IKO_TColStd_Array1OfReal* Knots, IKO_TColStd_Array1OfInteger* Mults, double ParametricTolerance, VARIANT_BOOL Add)

Inserts a set of knots values in the sequence of knots. For each U = Knots(i), M = Mults(i) If is an existing knot the multiplicity is increased by if is True, increased to if is False. If U is not on the parameter range nothing is done. If the multiplicity is negative or null nothing is done. The new multiplicity is limited to the degree. The tolerance criterion for knots equality is the max of Epsilon(U) and ParametricTolerance.

HRESULT RemoveKnot (int Index, int M, double Tolerance)

Reduces the multiplicity of the knot of index Index to M. If M is equal to 0, the knot is removed. With a modification of this type, the array of poles is also modified. Two different algorithms are systematically used to compute the new poles of the curve. If, for each pole, the distance between the pole calculated using the first algorithm and the same pole calculated using the second algorithm, is less than Tolerance, this ensures that the curve is not modified by more than Tolerance. Under these conditions, true is returned; otherwise, false is returned. A low tolerance is used to prevent modification of the curve. A high tolerance is used to "smooth" the curve. Exceptions Standard_OutOfRange if Index is outside the bounds of the knots table. //! pole insertion and pole removing this operation is limited to the Uniform or QuasiUniform BSplineCurve. The knot values are modified . If the BSpline is NonUniform or Piecewise Bezier an exception Construction error is raised.

HRESULT Reverse()

Changes the direction of parametrization of . The Knot sequence is modified, the FirstParameter and the LastParameter are not modified. The StartPoint of the initial curve becomes the EndPoint of the reversed curve and the EndPoint of the initial curve becomes the StartPoint of the reversed curve.

HRESULT SetKnot(int Index, double K)

Modifies this BSModifies this BSpline curve by assigning the value K to the knot of index Index in the knots table. This is a relatively local modification because K must be such that: Knots(Index - 1) < K < Knots(Index + 1) The second syntax allows you also to increase the multiplicity of the knot to M (but it is not possible to decrease the multiplicity of the knot with this function). Standard_ConstructionError if: K is not such that: Knots(Index - 1) < K < Knots(Index + 1) M is greater than the degree of this BSpline curve or lower than the previous multiplicity of knot of index Index in the knots table. Standard_OutOfRange if Index is outside the bounds of the knots table.

HRESULT SetKnots(IKO_TColStd_Array1OfReal* K)

Modifies this BSpline curve by assigning the array K to its knots table. The multiplicity of the knots is not modified. Exceptions Standard_ConstructionError if the values in the array K are not in ascending order. Standard_OutOfRange if the bounds of the array K are not respectively 1 and the number of knots of this BSpline curve.

HRESULT SetKnot2(int Index, double K, int M)

Changes the knot of range Index with its multiplicity. You can increase the multiplicity of a knot but it is not allowed to decrease the multiplicity of an existing knot. Raised if K >= Knots(Index+1) or K <= Knots(Index-1). Raised if M is greater than Degree or lower than the previous multiplicity of knot of range Index. //! Raised if Index < 1 || Index > NbKnots

HRESULT PeriodicNormalization(double* U)

returns the parameter normalized within the period if the curve is periodic : otherwise does not do anything

HRESULT SetPeriodic()

Changes this BSpline curve into a periodic curve. To become periodic, the curve must first be closed. Next, the knot sequence must be periodic. For this, FirstUKnotIndex and LastUKnotIndex are used to compute I1 and I2, the indexes in the knots array of the knots corresponding to the first and last parameters of this BSpline curve. The period is therefore: Knots(I2) - Knots(I1). Consequently, the knots and poles tables are modified. Exceptions Standard_ConstructionError if this BSpline curve is not closed.

HRESULT SetOrigin(int Index)

Assigns the knot of index Index in the knots table as the origin of this periodic BSpline curve. As a consequence, the knots and poles tables are modified. Exceptions Standard_NoSuchObject if this curve is not periodic. Standard_DomainError if Index is outside the bounds of the knots table.

HRESULT SetOrigin2(double U, double Tol)

Set the origin of a periodic curve at Knot U. If U is not a knot of the BSpline a new knot is inseted. KnotVector and poles are modified.

HRESULT SetNotPeriodic()

Changes this BSpline curve into a non-periodic curve. If this curve is already non-periodic, it is not modified. Note: the poles and knots tables are modified. Warning If this curve is periodic, as the multiplicity of the first and last knots is not modified, and is not equal to Degree + 1, where Degree is the degree of this BSpline curve, the start and end points of the curve are not its first and last poles.

HRESULT SetPole(int Index, DIPoint* P)

Modifies this BSpline curve by assigning P to the pole of index Index in the poles table. Exceptions Standard_OutOfRange if Index is outside the bounds of the poles table. Standard_ConstructionError if Weight is negative or null.

HRESULT SetPole2(int Index, DIPoint* P, double Weight)

Modifies this BSpline curve by assigning P to the pole of index Index in the poles table. This syntax also allows you to modify the weight of the modified pole, which becomes Weight. In this case, if this BSpline curve is non-rational, it can become rational and vice versa. Exceptions Standard_OutOfRange if Index is outside the bounds of the poles table. Standard_ConstructionError if Weight is negative or null.

HRESULT SetWeight(int Index, double Weight)

Changes the weight for the pole of range Index. If the curve was non rational it can become rational. If the curve was rational it can become non rational. Raised if Index < 1 || Index > NbPoles

HRESULT MovePoint(double U, DIPoint* P, int Index1, int Index2, int* HRESULT MovePoint(double U, DIPoint* P, int Index1, int Index2, int* FirstModifiedPole, int* LastModifiedPole)

Moves the point of parameter U of this BSpline curve to P. Index1 and Index2 are the indexes in the table of poles of this BSpline curve of the first and last poles designated to be moved. FirstModifiedPole and LastModifiedPole are the indexes of the first and last poles which are effectively modified. In the event of incompatibility between Index1, Index2 and the value U: no change is made to this BSpline curve, and the FirstModifiedPole and LastModifiedPole are returned null. Exceptions Standard_OutOfRange if: Index1 is greater than or equal to Index2, or Index1 or Index2 is less than 1 or greater than the number of poles of this BSpline curve.

HRESULT MovePointAndTangent(double U, DIPoint* P, DIVect* Tangent, double Tolerance, int StartingCondition, int EndingCondition, int* ErrorStatus)

Move a point with parameter U to P. and makes it tangent at U be Tangent. StartingCondition = -1 means first can move EndingCondition = -1 means last point can move StartingCondition = 0 means the first point cannot move EndingCondition = 0 means the last point cannot move StartingCondition = 1 means the first point and tangent cannot move EndingCondition = 1 means the last point and tangent cannot move and so forth ErrorStatus != 0 means that there are not enought degree of freedom with the constrain to deform the curve accordingly

HRESULT IsCN(int N, VARIANT_BOOL* ret)

Returns the continuity of the curve, the curve is at least C0.

HRESULT IsClosed( VARIANT_BOOL* closed)

Returns true if the distance between the first point and the last point of the curve is lower or equal to Resolution from package gp. Warnings : The first and the last point can be different from the first pole and the last pole of the curve.

HRESULT IsPeriodic( VARIANT_BOOL* periodic)

HRESULT IsRational( VARIANT_BOOL* rational)

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

HRESULT Continuity(int* continuity)

Returns the global continuity of the curve : C0 : only geometric continuity, C1 : continuity of the first derivative all along the Curve, C2 : continuity of the second derivative all along the Curve, C3 : continuity of the third derivative all along the Curve, CN : the order of continuity is infinite. For a B-spline curve of degree d if a knot Ui has a multiplicity p the B-spline curve is only Cd-p continuous at Ui. So the global continuity of the curve can't be greater than Cd-p where p is the maximum multiplicity of the interior Knots. In the interior of a knot span the curve is infinitely continuously differentiable.

HRESULT Degree(int* degree)

Returns the degree of this BSpline curve. The degree of a Geom_BSplineCurve curve cannot be greater than Geom_BSplineCurve::MaxDegree().

HRESULT D0(double U, DIPoint* P)

Returns in P the point of parameter U.

HRESULT D1(double U, DIPoint* P, DIVect* V1)

Returns in P the point of parameter U and its tangential vector

HRESULT D2(double U, DIPoint* P, DIVect* V1, DIVect* V2)

HRESULT D3(double U, DIPoint* P, DIVect* V1, DIVect* V2, DIVect* V3)

HRESULT DN(double U, int N, DIVect* V)

For the point of parameter U of this BSpline curve, computes the vector For the point of parameter U of this BSpline curve, computes the vector corresponding to the Nth derivative. Warning On a point where the continuity of the curve is not the one requested, this function impacts the part defined by the parameter with a value greater than U, i.e. the part of the curve to the "right" of the singularity. Exceptions Standard_RangeError if N is less than 1. The following functions compute the point of parameter U and the derivatives at this point on the B-spline curve arc defined between the knot FromK1 and the knot ToK2. U can be out of bounds [Knot (FromK1), Knot (ToK2)] but for the computation we only use the definition of the curve between these two knots. This method is useful to compute local derivative, if the order of continuity of the whole curve is not greater enough. Inside the parametric domain Knot (FromK1), Knot (ToK2) the evaluations are the same as if we consider the whole definition of the curve. Of course the evaluations are different outside this parametric domain.

HRESULT EndPoint(DIPoint* P)

Returns the last point of the curve. Warnings : The last point of the curve is different from the last pole of the curve if the multiplicity of the last knot is lower than Degree.

HRESULT FirstUKnotIndex(int* ind)

Returns the index in the knot array of the knot corresponding to the first or last parameter of this BSpline curve. For a BSpline curve, the first (or last) parameter (which gives the start (or end) point of the curve) is a knot value. However, if the multiplicity of the first (or last) knot is less than Degree + 1, where Degree is the degree of the curve, it is not the first (or last) knot of the curve.

HRESULT FirstParameter(double* t)

Returns the value of the first parameter of this BSpline curve. This is a knot value. The first parameter is the one of the start point of the BSpline curve.

HRESULT Knot(int Index, double* t)

Returns the knot of range Index. When there is a knot with a multiplicity greater than 1 the knot is not repeated. The method Multiplicity can be used to get the multiplicity of the Knot.

HRESULT Knots(IKO_TColStd_Array1OfReal* K)

returns the knot values of the B-spline curve; Warning A knot with a multiplicity greater than 1 is not repeated in the knot table. The Multiplicity function can be used to obtain the multiplicity of each knot. Raised if the length of K is not equal to the number of knots

HRESULT KnotSequence(IKO_TColStd_Array1OfReal* K)

Returns K, the knots sequence of this BSpline curve. In this sequence, knots with a multiplicity greater than 1 are repeated. In the case of a non-periodic curve the length of the sequence must be equal to the sum of the NbKnots multiplicities of the knots of the curve (where NbKnots is the number of knots of this BSpline curve). This sum is also equal to : NbPoles + Degree + 1 where NbPoles is the number of poles and Degree the degree of this BSpline curve. In the case of a periodic curve, if there are k periodic knots, the period is Knot(k+1) - Knot(1).

The initial sequence is built by writing knots 1 to k+1, which are repeated according to their corresponding multiplicities. If Degree is the degree of the curve, the degree of continuity of the curve at the knot of index 1 (or k+1) is equal to c = Degree + 1 - Mult(1). c knots are then inserted at the beginning and end of the initial sequence: the c values of knots preceding the first item Knot(k+1) in the initial sequence are inserted at the beginning; the period is subtracted from these c values; the c values of knots following the last item Knot(1) in the initial sequence are inserted at the end; the period is added to these c values. The length of the sequence must therefore be equal to: NbPoles + 2*Degree - Mult(1) + 2.

Example For a non-periodic BSpline curve of degree 2 where: the array of knots is: { k1 k2 k3 k4 }, with associated multiplicities: { 3 1 2 3 }, the knot sequence is: K = { k1 k1 k1 k2 k3 k3 k4 k4 k4 } For a periodic BSpline curve of degree 4 , which is "C1" continuous at the first knot, and where : the periodic knots are: { k1 k2 k3 (k4) } (3 periodic knots: the points of parameter k1 and k4 are identical, the period is p = k4 - k1), with associated multiplicities: { 3 1 2 (3) }, the degree of continuity at knots k1 and k4 is: Degree + 1 - Mult(i) = 2. 2 supplementary knots are added at the beginning and end of the sequence: at the beginning: the 2 knots preceding k4 minus the period; in this example, this is k3 - p both times; at the end: the 2 knots following k1 plus the period; in this example, this is k2 + p and k3 + p. The knot sequence is therefore: K = { k3-p k3-p k1 k1 k1 k2 k3 k3 k4 k4 k4 k2+p k3+p } Exceptions Standard_DimensionError if the array K is not of the appropriate length.Returns the knots sequence.

HRESULT KnotDistribution(int* GeomAbs_BSplKnotDistribution_retVal)

Returns type of knot spacing. The returned values ar eenumerated in GeomAbs_BSplKnotDistribution.

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

HRESULT LastUKnotIndex(int* retVal)

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

HRESULT LastParameter(double* t)

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

HRESULT LocateU(double U, double ParametricTolerance, int* I1, int* I2, VARIANT_BOOL WithKnotRepetition)

Locates the parametric value U in the sequence of knots. If "WithKnotRepetition" is True we consider the knot's representation with repetition of multiple knot value, otherwise we consider the knot's representation with no repetition of multiple knot values. Knots (I1) <= U <= Knots (I2) . if I1 = I2 U is a knot value (the tolerance criterion ParametricTolerance is used). . if I1 < 1 => U < Knots (1) - Abs(ParametricTolerance) . if I2 > NbKnots => U > Knots (NbKnots) + Abs(ParametricTolerance)

HRESULT Multiplicity(int Index, int* multiplicity)

Returns the multiplicity of the knots of range Index

HRESULT Multiplicities(IKO_TColStd_Array1OfInteger* iArr)

Returns the multiplicity of the knots of the curve. Raised if the length of M is not equal to NbKnots.

HRESULT NbKnots(int* count)

Returns the number of knots. This method returns the number of knot without repetition of multiple knots.

HRESULT NbPoles(int* count)

Returns the number of poles

HRESHRESULT Pole(int Index, DIPoint* P)

Returns the pole of range Index

HRESULT Poles(IKO_TColgp_Array1OfPnt** iArr)

Returns the poles of the B-spline curve; Raised if the length of P is not equal to the number of poles.

HRESULT StartPoint(DIPoint* P)

Returns the start point of the curve. Warnings : This point is different from the first pole of the curve if the multiplicity of the first knot is lower than Degree.

HRESULT Weight(int Index, double* weight)

Returns the weight of the pole of range Index

HRESULT Weights(IKO_TColStd_Array1OfReal* W)