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PLib::NurbsCurve Class Template Reference

A NURBS curve class. More...

#include <nurbs.h>

Inheritance diagram for PLib::NurbsCurve:

Public Methods
	NurbsCurve ()
	NurbsCurve (const NurbsCurve< T, N > &nurb)
	NurbsCurve (const Vector< HPoint_nD< T, N > > &P1, const Vector< T > &U1, int deg=3)
	NurbsCurve (const Vector< Point_nD< T, N > > &P1, const Vector< T > &W, const Vector< T > &U1, int deg=3)
virtual	~NurbsCurve ()
int	degree () const
const Vector< HPoint_nD< T, N > > &	ctrlPnts () const
const HPoint_nD< T, N >	ctrlPnts (int i) const
const Vector< T > &	knot () const
T	knot (int i) const
void	resize (int n, int Deg)
virtual void	reset (const Vector< HPoint_nD< T, N > > &P1, const Vector< T > &U1, int deg)
virtual NurbsCurve &	operator= (const NurbsCurve< T, N > &)
virtual HPoint_nD< T, N >	operator() (T u) const
HPoint_nD< T, N >	hpointAt (T u) const
HPoint_nD< T, N >	hpointAt (T u, int span) const
void	deriveAtH (T u, int, Vector< HPoint_nD< T, N > > &) const
void	deriveAt (T u, int, Vector< Point_nD< T, N > > &) const
void	deriveAtH (T u, int, int, Vector< HPoint_nD< T, N > > &) const
void	deriveAt (T u, int, int, Vector< Point_nD< T, N > > &) const
Point_nD< T, N >	derive3D (T u, int d) const
HPoint_nD< T, N >	derive (T u, int d) const
Point_nD< T, N >	normal (T u, const Point_nD< T, N > &v) const
HPoint_nD< T, N >	firstD (T u) const
HPoint_nD< T, N >	firstD (T u, int span) const
Point_nD< T, N >	firstDn (T u) const
Point_nD< T, N >	firstDn (T u, int span) const
T	basisFun (T u, int i, int p=-1) const
void	basisFuns (T u, int span, Vector< T > &N) const
void	dersBasisFuns (int n, T u, int span, Matrix< T > &N) const
T	minKnot () const
T	maxKnot () const
int	findSpan (T u) const
void	findMultSpan (T u, int &r, int &s) const
int	findMult (int r) const
int	findKnot (T u) const
T	getRemovalBnd (int r, int s) const
void	removeKnot (int r, int s, int num)
void	removeKnotsBound (const Vector< T > &ub, Vector< T > &ek, T E)
int	knotInsertion (T u, int r, NurbsCurve< T, N > &nc)
void	refineKnotVector (const Vector< T > &X)
void	refineKnotVectorClosed (const Vector< T > &X)
void	mergeKnotVector (const Vector< T > &Um)
void	clamp ()
void	unclamp ()
int	leastSquares (const Vector< Point_nD< T, N > > &Q, int degC, int n)
int	leastSquares (const Vector< Point_nD< T, N > > &Q, int degC, int n, const Vector< T > &ub)
int	leastSquaresH (const Vector< HPoint_nD< T, N > > &Q, int degC, int n, const Vector< T > &ub)
int	leastSquares (const Vector< Point_nD< T, N > > &Q, int degC, int n, const Vector< T > &ub, const Vector< T > &knot)
int	leastSquaresH (const Vector< HPoint_nD< T, N > > &Q, int degC, int n, const Vector< T > &ub, const Vector< T > &knot)
int	leastSquaresClosed (const Vector< Point_nD< T, N > > &Q, int degC, int n)
int	leastSquaresClosed (const Vector< Point_nD< T, N > > &Q, int degC, int n, const Vector< T > &ub)
int	leastSquaresClosedH (const Vector< HPoint_nD< T, N > > &Q, int degC, int n, const Vector< T > &ub)
int	leastSquaresClosed (const Vector< Point_nD< T, N > > &Q, int degC, int n, const Vector< T > &ub, const Vector< T > &knot)
int	leastSquaresClosedH (const Vector< HPoint_nD< T, N > > &Q, int degC, int n, const Vector< T > &ub, const Vector< T > &knot)
void	globalApproxErrBnd (Vector< Point_nD< T, N > > &Q, int deg, T E)
void	globalApproxErrBnd (Vector< Point_nD< T, N > > &Q, Vector< T > &ub, int deg, T E)
void	globalApproxErrBnd2 (Vector< Point_nD< T, N > > &Q, int degC, T E)
void	globalApproxErrBnd3 (Vector< Point_nD< T, N > > &Q, int degC, T E)
void	globalApproxErrBnd3 (Vector< Point_nD< T, N > > &Q, const Vector< T > &ub, int degC, T E)
void	globalInterp (const Vector< Point_nD< T, N > > &Q, int d)
void	globalInterp (const Vector< Point_nD< T, N > > &Q, const Vector< T > &ub, int d)
void	globalInterpH (const Vector< HPoint_nD< T, N > > &Q, int d)
void	globalInterpH (const Vector< HPoint_nD< T, N > > &Q, const Vector< T > &U, int d)
void	globalInterpH (const Vector< HPoint_nD< T, N > > &Q, const Vector< T > &ub, const Vector< T > &U, int d)
void	globalInterpClosed (const Vector< Point_nD< T, N > > &Qw, int d)
void	globalInterpClosed (const Vector< Point_nD< T, N > > &Qw, const Vector< T > &ub, int d)
void	globalInterpClosedH (const Vector< HPoint_nD< T, N > > &Qw, int d)
void	globalInterpClosedH (const Vector< HPoint_nD< T, N > > &Qw, const Vector< T > &U, int d)
void	globalInterpClosedH (const Vector< HPoint_nD< T, N > > &Qw, const Vector< T > &ub, const Vector< T > &U, int d)
void	globalInterpClosed (const Vector< Point_nD< T, N > > &Qw, const Vector< T > &ub, const Vector< T > &Uc, int d)
void	globalInterpD (const Vector< Point_nD< T, N > > &Q, const Vector< Point_nD< T, N > > &D, int d, int unitD, T a=1.0)
void	projectTo (const Point_nD< T, N > &p, T guess, T &u, Point_nD< T, N > &r, T e1=0.001, T e2=0.001, int maxTry=100) const
T	length (T eps=0.001, int n=100) const
T	lengthIn (T us, T ue, T eps=0.001, int n=100) const
T	lengthF (T) const
T	lengthF (T, int) const
void	makeCircle (const Point_nD< T, N > &O, const Point_nD< T, N > &X, const Point_nD< T, N > &Y, T r, double as, double ae)
void	makeCircle (const Point_nD< T, N > &O, T r, double as, double ae)
void	makeCircle (const Point_nD< T, N > &O, T r)
void	makeLine (const Point_nD< T, N > &P0, const Point_nD< T, N > &P1, int d)
virtual void	degreeElevate (int t)
void	decompose (NurbsCurveArray< T, N > &c) const
void	decomposeClosed (NurbsCurveArray< T, N > &c) const
int	splitAt (T u, NurbsCurve< T, N > &cl, NurbsCurve< T, N > &cu) const
int	mergeOf (const NurbsCurve< T, N > &cl, const NurbsCurve< T, N > &cu)
void	transform (const MatrixRT< T > &A)
void	modCP (int i, const HPoint_nD< T, N > &a)
void	modCPby (int i, const HPoint_nD< T, N > &a)
virtual void	modKnot (const Vector< T > &knotU)
int	movePoint (T u, const Point_nD< T, N > &delta)
int	movePoint (T u, const BasicArray< Point_nD< T, N > > &delta)
int	movePoint (const BasicArray< T > &ur, const BasicArray< Point_nD< T, N > > &D)
int	movePoint (const BasicArray< T > &ur, const BasicArray< Point_nD< T, N > > &D, const BasicArray_INT &Dr, const BasicArray_INT &Dk)
int	movePoint (const BasicArray< T > &ur, const BasicArray< Point_nD< T, N > > &D, const BasicArray_INT &Dr, const BasicArray_INT &Dk, const BasicArray_INT &fixCP)
void	setTangent (T u, const Point_nD< T, N > &T0)
void	setTangentAtEnd (const Point_nD< T, N > &T0, const Point_nD< T, N > &T1)
int	read (const char *)
int	write (const char *) const
virtual int	read (ifstream &fin)
int	write (ofstream &fout) const
int	writePS (const char *, int cp=0, T magFact=T(-1), T dash=T(5), bool bOpen=true) const
int	writePSp (const char *, const Vector< Point_nD< T, N > > &, const Vector< Point_nD< T, N > > &, int cp=0, T magFact=0.0, T dash=5.0, bool bOpen=true) const
int	writeVRML (ostream &fout, T radius, int K, const Color &color, int Nu, int Nv, T u_s, T u_e) const
int	writeVRML (const char *filename, T radius, int K, const Color &color, int Nu, int Nv, T u_s, T u_e) const
int	writeVRML (const char *filename, T radius=1, int K=5, const Color &color=whiteColor, int Nu=20, int Nv=20) const
int	writeVRML (ostream &fout, T radius=1, int K=5, const Color &color=whiteColor, int Nu=20, int Nv=20) const
int	writeVRML97 (const char *filename, T radius, int K, const Color &color, int Nu, int Nv, T u_s, T u_e) const
int	writeVRML97 (ostream &fout, T radius, int K, const Color &color, int Nu, int Nv, T u_s, T u_e) const
int	writeVRML97 (const char *filename, T radius=1, int K=5, const Color &color=whiteColor, int Nu=20, int Nv=20) const
int	writeVRML97 (ostream &fout, T radius=1, int K=5, const Color &color=whiteColor, int Nu=20, int Nv=20) const
int	writeDisplayLINE (const char *filename, int iNu, const Color &color=blueColor, T fA=1) const
int	writeDisplayLINE (const char *filename, const Color &color, int iNu, T u_s, T u_e) const
void	drawImg (Image_UBYTE &Img, unsigned char color=255, T step=0.01)
void	drawImg (Image_Color &Img, const Color &color, T step=0.01)
void	drawAaImg (Image_Color &Img, const Color &color, int precision=3, int alpha=1)
void	drawAaImg (Image_Color &Img, const Color &color, const NurbsCurve< T, 3 > &profile, int precision=3, int alpha=1)
NurbsSurface< T, 3 >	drawAaImg (Image_Color &Img, const Color &color, const NurbsCurve< T, 3 > &profile, const NurbsCurve< T, 3 > &scaling, int precision=3, int alpha=1)
BasicList< Point_nD< T, N > >	tesselate (T tolerance, BasicList< T > *uk) const
Protected Attributes
Vector< HPoint_nD< T, N > >	P
Vector< T >	U
int	deg_
Friends
HPoint_nD< T, N >	C (T u, const NurbsCurve< T, N > &nurb)
Point_nD< T, N >	Cp (T u, const NurbsCurve< T, N > &nurb)
void	generateCompatibleCurves (NurbsCurveArray< T, N > &ca)
Related Functions
(Note that these are not member functions.)
T	chordLengthParam (const Vector< Point_nD< T, N > > &Q, Vector< T > &ub)

Detailed Description

template<class T, int N>
class PLib::NurbsCurve< T, N >

A NURBS curve class.

This class is used to represent and manipulate NURBS curve. The curves are composed of points in 4D. They can have any degree and have any number of control points.

Author:: Philippe Lavoie

Date:: 4 Oct. 1996

Definition at line 58 of file nurbs.h.

Constructor & Destructor Documentation

template<class T, int N>

PLib::NurbsCurve< T, N >::NurbsCurve ( )

default constructor.

Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 46 of file nurbs.cpp.

template<class T, int N>

PLib::NurbsCurve< T, N >::NurbsCurve ( const NurbsCurve< T, N > & nurb )

A copy constructor.

Parameters:

nurb the NURBS curve to copy

Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 59 of file nurbs.cpp.

template<class T, int N>

PLib::NurbsCurve< T, N >::NurbsCurve ( const Vector< HPoint_nD< T, N > > & P1,

const Vector< T > & U1,

int Degree = 3

)

Constructor with control points in 4D.

Parameters:

P1 the control points

U1 the knot vector

Degree the degree of the curve

Warning:
The size of P1,U1 and Degree must agree: P.n()+degree+1=U.n()
Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 109 of file nurbs.cpp.

template<class T, int N>

PLib::NurbsCurve< T, N >::NurbsCurve ( const Vector< Point_nD< T, N > > & P1,

const Vector< T > & W,

const Vector< T > & U1,

int Degree = 3

)

Constructor with control points in 3D.

Parameters:

P1 --> the control point vector

W --> the weight for each control points

U1 --> the knot vector

Degree --> the degree of the curve

Warning:
The size of P1,U1 and Degree must agree: P.n()+degree+1=U.n()
Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 138 of file nurbs.cpp.

Member Function Documentation

template<class T, int D>

T PLib::NurbsCurve< T, D >::basisFun ( T u,

int i,

int p = -1

) const

Computes the basis function of the curve.

Computes the i basis function of degree p of the curve at parameter u.
The basis function is noted as .
The B-spline basis function of $p$-degree is defined as

where the define the knot vector as a nondecreasing sequence of real numbers, i.e., for . And is related to the number of control points $n$ and the degree of the curve $p$ with the relation . The knot vector has the form

Parameters:

u the parametric variable

i specifies which basis function to compute

p the degree to which the basis function is computed

Returns:
the value of N_{ip}(u)
Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 840 of file nurbs.cpp.
Referenced by removeKnotsBound, and PLib::NurbsCurveSP::updateMaxU.

template<class T, int D>

void PLib::NurbsCurve< T, D >::basisFuns ( T u,

int i,

Vector< T > & N

) const

computes the non-zero basis functions of the curve.

Computes the non-zero basis functions and puts the result into N. N has a size of deg+1. To relate N to the basis functions, Basis[span -deg +i] = N[i] for i=0...deg.
The B-spline basis function of -degree is defined as

where the define the knot vector as a nondecreasing sequence of real numbers, {i.e.}, for . And is related to the number of control points $n$ and the degree of the curve with the relation . The knot vector has the form

The B-spline basis function are non-zero for at most of the . The relationship between the non-zero basis functions in N and is as follows, for . Where span is the non-zero span of the basis functions. This non-zero span for u can be found by calling {\tt findSpan(u)}.
Parameters:

u the parametric value

i the non-zero span of the basis functions

N the non-zero basis functions

Warning:
u and i must be valid values
Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 1046 of file nurbs.cpp.
Referenced by hpointAt, leastSquares, leastSquaresH, and operator().

template<class T, int N>

HPoint_nD< T, N > PLib::NurbsCurve< T, N >::derive ( T u,

int d

) const

Computes the derivative of degree of the curve at parameter u.

For more information on the algorithm used, see A3.2 on p 93 of the NurbsBook.
Parameters:

u the parametric value to evaluate at

d the degree of the derivative

Returns:
The derivative d in 4D at the parameter u
Warning:
u and d must be in a valid range.
Author:
Philippe Lavoie
Date:
24 January, 1997

Definition at line 589 of file nurbs.cpp.
References deriveAtH.

template<class T, int N>

Point_nD< T, N > PLib::NurbsCurve< T, N >::derive3D ( T u,

int d

) const

Computes the derivative of degree d of the curve at parameter u.

For more information on the algorithm used, see A3.2 on p 93 of the NurbsBook.
Parameters:

u the parametric value to evaluate at

d the degree of the derivative

Returns:
The derivative d in norma space at the parameter u
Warning:
u and d must be in a valid range.
Author:
Philippe Lavoie
Date:
24 January, 1997

Definition at line 566 of file nurbs.cpp.
References deriveAt.

template<class T, int N>

void PLib::NurbsCurve< T, N >::deriveAt ( T u,

int d,

int span,

Vector< Point_nD< T, N > > & ders

) const

Computes the derivative of the curve at the parameter u.

Parameters:

u the parameter at which the derivative is computed

d the degree of derivation

span the span of u.

ders the vector containing the derivatives of the point at u.

Warning:
u and $d$ must be in a valid range.
Author:
Philippe Lavoie
Date:
9 October 1998

Definition at line 755 of file nurbs.cpp.
References deriveAtH.

template<class T, int N>

void PLib::NurbsCurve< T, N >::deriveAt ( T u,

int d,

Vector< Point_nD< T, N > > & ders

) const [virtual]

Computes the derivative at the parameter u.

Parameters:

u the parameter at which the derivative is computed

d the degree of derivation

ders the vector containing the derivatives of the point at u.

\wanring u and d must be in a valid range.
Author:
Philippe Lavoie
Date:
24 January, 1997

Implements PLib::ParaCurve.
Definition at line 715 of file nurbs.cpp.
References deriveAtH.
Referenced by derive3D.

template<class T, int N>

void PLib::NurbsCurve< T, N >::deriveAtH ( T u,

int d,

int span,

Vector< HPoint_nD< T, N > > & ders

) const

Computes the derivative of degree d of the curve at parameter u.

For more information on the algorithm used, see A3.2 on p 93 of the NurbsBook.
Parameters:

u the parametric value to evaluate at

d the degree of the derivative

span the span of u

ders a vector containing the derivatives of the curve at u.

Warning:
u and d must be in a valid range.
Author:
Philippe Lavoie
Date:
9 October, 1998

Definition at line 644 of file nurbs.cpp.
References dersBasisFuns.

template<class T, int N>

void PLib::NurbsCurve< T, N >::deriveAtH ( T u,

int d,

Vector< HPoint_nD< T, N > > & ders

) const [virtual]

Computes the derivative of degree d of the curve at parameter u in the homonegeous domain.

For more information on the algorithm used, see A3.2 on p 93 of the NurbsBook.
Parameters:

u the parametric value to evaluate at

d the degree of the derivative

ders a vector containing the derivatives of the curve at u.

Warning:
u and d must be in a valid range.
Author:
Philippe Lavoie
Date:
24 January, 1997

Implements PLib::ParaCurve.
Definition at line 612 of file nurbs.cpp.
References dersBasisFuns, and findSpan.
Referenced by derive, and deriveAt.

template<class T, int N>

void PLib::NurbsCurve< T, N >::dersBasisFuns ( int n,

T u,

int span,

Matrix< T > & ders

) const

Compute the derivatives functions at u of the NURBS curve.

For information on the algorithm, see A2.3 on p72 of the NURBS book.
The result is stored in the ders matrix, where ders is of size (n+1,deg+1) and the derivative N'_i(u) = ders(1,i=span-deg+j) where j=0...deg+1.
Parameters:

n the degree of the derivation

u the parametric value

span the span for the basis functions

ders A matrix containing the derivatives of the curve.

Warning:
n, u and span must be valid values.
Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 912 of file nurbs.cpp.
Referenced by deriveAtH.

template<class T, int N>

NurbsSurface< T, 3 > PLib::NurbsCurve< T, N >::drawAaImg ( Image_Color & Img,

const Color & color,

const NurbsCurve< T, 3 > & profile,

const NurbsCurve< T, 3 > & scaling,

int precision = 3,

int alpha = 1

)

Draws an anti-aliased NURBS curve on an image.

This will draw the NURBS by using a brush profile. The drawing is performed by averaging the intensity of the profile at the pixels.
This function generates a sweep surface by using the profile given in its argument. The sweep is always performed by following the y-axis of the profile. A scaling function is also used when sweeping. This is used to vary the shape of the profile while it's being swept (see the sweep member function of NurbsSurface<T,N> for more details).
Parameters:

Img draws the nurbs curve to this Image

color the line is drawn in this color

profile the profile of the NURBS curve to draw

scaling the scaling to give the profile while drawing the curve

precision this number influences the number of points used for averaging purposes.

alpha a flag indicating if the profile is used as an alpha chanel. If so, the line doesn't overwrite, it blends the line with the image already present in Img.

Warning:
This routine is very slow; use normal drawing for speed or lower the precision factor.
Author:
Philippe Lavoie
Date:
25 July 1997

Definition at line 385 of file nurbs.cpp.
References PLib::Basic2DArray::cols, PLib::ParaCurve::pointAt, PLib::Basic2DArray::resize, and PLib::Basic2DArray::rows.

template<class T, int N>

void PLib::NurbsCurve< T, N >::drawAaImg ( Image_Color & Img,

const Color & color,

const NurbsCurve< T, 3 > & profile,

int precision = 3,

int alpha = 1

)

draws an anti-aliased NURBS curve on an image.

This will draw the NURBS by using a user-defined brush profile. The drawing is performed by averaging the intensity of the profile at the pixels.
Parameters:

Img draws the nurbs curve to this Image

color the line is drawn in this color

profile the profile of the NURBS curve to draw

precision this number influences the number of points used for averaging purposes.

alpha a flag indicating if the profile is used as an alpha chanel. If so, the line doesn't overwrite, it blends the line with the image already present in Img.

Warning:
This routine is very slow; use normal drawing for speed.
Author:
Philippe Lavoie
Date:
22 August 1997

Definition at line 342 of file nurbs.cpp.
References drawAaImg.

template<class T, int N>

void PLib::NurbsCurve< T, N >::drawAaImg ( Image_Color & Img,

const Color & color,

int precision = 3,

int alpha = 1

)

Draws an anti-aliased NURBS curve on an image.

This will draw the NURBS by using a circular brush profile. The drawing is performed by averaging the intensity of the profile at the pixels.
Parameters:

Img draws the nurbs curve to this Image

color the line is drawn in this color

precision this number influences the number of points used for averaging purposes.

alpha a flag indicating if the profile is used as an alpha chanel. If so, the line doesn't overwrite, it blends the line with the image already present in Img.

Warning:
This routine is very slow; use normal drawing for speed.
Author:
Philippe Lavoie
Date:
25 July 1997

Definition at line 46 of file d_nurbs.cpp.
References C.
Referenced by drawAaImg.

template<class T, int N>

void PLib::NurbsCurve< T, N >::drawImg ( Image_Color & Img,

const Color & color,

T step = 0.01

)

Draws a NURBS curve on an image.

This will draw very primitively the NURBS curve on an image. The drawing assumes the line is only in the xy plane (the z is not used for now).
The algorithm finds the points on the curve at a step parametric intervall between them and join them by a line. No fancy stuff.
Parameters:

Img draws the nurbs curve to this Image

color the line is drawn in this color

step the parametric distance between two computed points.

Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 270 of file nurbs.cpp.
References PLib::Basic2DArray::cols, PLib::MatrixImage::drawLine, PLib::ParaCurve::pointAt, and PLib::Basic2DArray::rows.

template<class T, int N>

void PLib::NurbsCurve< T, N >::drawImg ( Image_UBYTE & Img,

unsigned char color = 255,

T step = 0.01

)

draws a NURBS curve on an image.

This will draw very primitively the NURBS curve on an image. The drawing assumes the line is only in the xy plane (the z is not used for now).
The algorithm finds the points on the curve at a step parametric intervall between them and join them by a line. No fancy stuff.
Parameters:

Img <-- draws the nurbs curve to this Image

color --> the line is drawn in this color

step --> the parametric distance between two computed points.

Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 229 of file nurbs.cpp.
References PLib::Basic2DArray::cols, PLib::MatrixImage::drawLine, PLib::ParaCurve::pointAt, and PLib::Basic2DArray::rows.

template<class T, int N>

int PLib::NurbsCurve< T, N >::findKnot ( T u ) const

Finds the knot k for which u is in the range [u_k,u_{k+1}).

Parameters:

u parametric value

Returns:
the index k
Warning:
u must be in a valid range.
Author:
Philippe Lavoie
Date:
24 January, 1997

Definition at line 1118 of file nurbs.cpp.
Referenced by findMultSpan.

template<class T, int N>

int PLib::NurbsCurve< T, N >::findMult ( int r ) const

Finds the multiplicity of a knot.

Parameters:

r the knot to observe

Returns:
the multiplicity of the knot
Warning:
r must be a valid knot index
Author:
Philippe Lavoie
Date:
24 January, 1997

Definition at line 1139 of file nurbs.cpp.
Referenced by findMultSpan.

template<class T, int N>

void PLib::NurbsCurve< T, N >::findMultSpan ( T u,

int & r,

int & s

) const

Finds the multiplicity of a knot at a parametric value.

Finds the index of the knot at parametric value u and returns its multiplicity.
Parameters:

u the parametric value

r the knot of interest

s the multiplicity of this knot

Warning:
u must be in a valid range.
Author:
Philippe Lavoie
Date:
24 January, 1997

Definition at line 1165 of file nurbs.cpp.
References findKnot, and findMult.

template<class T, int N>

int PLib::NurbsCurve< T, N >::findSpan ( T u ) const

Determines the knot span index.

Determines the knot span for which their exists non-zero basis functions. The span is the index k for which the parameter u is valid in the [u_k,u_{k+1}] range.
Parameters:

u the parametric value

Returns:
the span index at u.
Warning:
u must be in a valid range
Author:
Philippe Lavoie
Date:
24 January 1997 \modified 20 January, 1999 (Alejandro Frangi)

Definition at line 1084 of file nurbs.cpp.
Referenced by deriveAtH, leastSquares, leastSquaresH, operator(), and removeKnotsBound.

template<class T, int N>

T PLib::NurbsCurve< T, N >::getRemovalBnd ( int r,

int s

) const

Get the knot removal error bound for an internal knot.

Get the knot removal error bound for an internal knot r (non-rational). For more information on the algorithm, see A9.8 from the Nurbs book on page 428.
Parameters:

curve a NURBS curve

r the index of the internal knot to check

s the multiplicity of that knot

Returns:
The maximum distance between the new curve and the old one
Author:
Philippe Lavoie
Date:
24 January, 1997

Definition at line 1648 of file nurbs.cpp.
Referenced by removeKnotsBound.

template<class T, int N>

void PLib::NurbsCurve< T, N >::globalApproxErrBnd ( Vector< Point_nD< T, N > > & Q,

int degC,

T E

)

Approximation of a curve bounded to a certain error.

It is a type II approximation: it starts with a lot of control points then tries to eliminate as much as it can as long as the curve stays within a certain error bound.
The method uses least squares fitting along with knot removal techniques. It is the algorithm A9.10 on p 431 of the NURBS book.
Parameters:

Q the points to approximate

degree the degree of the approximation curve

E the maximum error allowed

Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 2056 of file nurbs.cpp.
References chordLengthParam.

template<class T, int N>

HPoint_nD< T, N > PLib::NurbsCurve< T, N >::hpointAt ( T u,

int span

) const [virtual]

Evaluates the curve in homogenous space at parameter u.

It evaluates the NURBS curve in 4D at the parametric point $u$. Using the following equation

where $P_i$ are the control points and $N_{i,p}$ are the $p$th degree B-spline basis functions.
For more details on the algorithm, see A4.1 on page 124 of the Nurbs Book.
Parameters:

u the parametric value at which the curve is evaluated

span the span of u

Returns:
the 4D point at C(u)
Warning:
the parametric value must be in a valid range
Author:
Philippe Lavoie
Date:
24 January, 1997

Implements PLib::ParaCurve.
Definition at line 537 of file nurbs.cpp.
References basisFuns.

template<class T, int D>

int PLib::NurbsCurve< T, D >::leastSquares ( const Vector< Point_nD< T, D > > & Q,

int degC,

int n,

const Vector< T > & ub,

const Vector< T > & knot

)

A least squares curve approximation.

This routine solves the following problem: find the NURBS curve satisfying \begin{itemize} \item and \item the remaining are approximated in the least squares sense, {i.e.} \[ \sum_{k=1}^{m-1} | Q_k-C(\bar{u}_k)|^2 \] in a minimum with respect to the variable ; the are the precomputed parameter values. \end{itemize}
The resulting curve will generally not pass through and is not the closest point on to .
For more details, see section 9.4.1 on page 491 of the NURBS book.
Parameters:

Q the vector of 3D points

degC the degree of the curve

n the number of control points in the new curve

ub the knot coefficients

knot the knot vector to use for the curve

Returns:
1 if succesfull, 0 it the number of points to approximate the curve with is too big compared to the number of points.
Warning:
the variable curve must contain a valid knot vector.
Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 1406 of file nurbs.cpp.
References basisFuns, PLib::Basic2DArray::cols, findSpan, PLib::Matrix::get, knot, and PLib::Basic2DArray::rows.

template<class T, int N>

int PLib::NurbsCurve< T, N >::leastSquares ( const Vector< Point_nD< T, N > > & Q,

int degC,

int n,

const Vector< T > & ub

)

A least squares curve approximation.

This routine solves the following problem: find the NURBS curve satisfying \begin{itemize} \item and \item the remaining are approximated in the least squares sense, {i.e.} \[ \sum_{k=1}^{m-1} | Q_k-C(\bar{u}_k)|^2 \] in a minimum with respect to the variable ; the are the precomputed parameter values. \end{itemize}
The resulting curve will generally not pass through and is not the closest point on to .
For more details, see section 9.4.1 on page 491 of the NURBS book.
Parameters:

Q the vector of 3D points

degC the degree of the curve

n the number of control points in the new curve

ub the knot coefficients

Warning:
the variable curve must contain a valid knot vector.
Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 1262 of file nurbs.cpp.
References leastSquares.

template<class T, int N>

int PLib::NurbsCurve< T, N >::leastSquares ( const Vector< Point_nD< T, N > > & Q,

int degC,

int n

)

A least squares curve approximation.

This routine solves the following problem: find the NURBS curve satisfying

and
\item the remaining are approximated in the least squares sense, {i.e.} \[ \sum_{k=1}^{m-1} | Q_k-C(\bar{u}_k)|^2 \] in a minimum with respect to the variable ; the are the parameter values computed with the chord length method.
The resulting curve will generally not pass through and is not the closest point on to .
For more details, see section 9.4.1 on page 491 of the NURBS book.
Parameters:

Q the vector of 3D points

degC the degree of the curve

n the number of control points in the new curve.

Warning:
deg must be smaller than Q.n().
Author:
Philippe Lavoie
Date:
24 January, 1997

Definition at line 1224 of file nurbs.cpp.
References chordLengthParam.
Referenced by leastSquares.

template<class T, int D>

int PLib::NurbsCurve< T, D >::leastSquaresH ( const Vector< HPoint_nD< T, D > > & Q,

int degC,

int n,

const Vector< T > & ub,

const Vector< T > & knot

)

A least squares curve approximation.

This routine solves the following problem: find the NURBS curve satisfying \begin{itemize} \item and \item the remaining are approximated in the least squares sense, {i.e.} \[ \sum_{k=1}^{m-1} | Q_k-C(\bar{u}_k)|^2 \] in a minimum with respect to the variable ; the are the precomputed parameter values. \end{itemize}
The resulting curve will generally not pass through and is not the closest point on to .
For more details, see section 9.4.1 on page 491 of the NURBS book.
Parameters:

Q the vector of 4D points

degC the degree of the curve

n the number of control points in the new curve

ub the knot coefficients

knot the knot vector to use for the curve

Returns:
1 if succesfull, 0 it the number of points to approximate the curve with is too big compared to the number of points.
Warning:
the variable curve must contain a valid knot vector.
Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 1535 of file nurbs.cpp.
References basisFuns, PLib::Basic2DArray::cols, findSpan, PLib::Matrix::get, knot, and PLib::Basic2DArray::rows.

template<class T, int N>

int PLib::NurbsCurve< T, N >::leastSquaresH ( const Vector< HPoint_nD< T, N > > & Q,

int degC,

int n,

const Vector< T > & ub

)

A least squares curve approximation.

This routine solves the following problem: find the NURBS curve satisfying \begin{itemize} \item and \item the remaining are approximated in the least squares sense, {i.e.} \[ \sum_{k=1}^{m-1} | Q_k-C(\bar{u}_k)|^2 \] in a minimum with respect to the variable ; the are the precomputed parameter values. \end{itemize}
The resulting curve will generally not pass through and is not the closest point on to .
For more details, see section 9.4.1 on page 491 of the NURBS book.
Parameters:

Q the vector of 4D points

degC the degree of the curve

n the number of control points in the new curve

ub the knot coefficients

Warning:
the variable curve must contain a valid knot vector.
Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 1333 of file nurbs.cpp.

template<class T, int N>

Point_nD< T, N > PLib::NurbsCurve< T, N >::normal ( T u,

const Point_nD< T, N > & v

) const

Computes the normal of the curve at u from a vector.

Computes the normal of the curve at u from a vector. If the curve lies only in the xy-plane, then calling the function with the vector v = (0,0,1) (the z$axis) will yield a proper normal for this curve.
Parameters:

u the parameter at which the normal is computed

v the vector to compute the normal with

Returns:
the normal vector in 3D.
Warning:
u must be in a valid range.
Author:
Philippe Lavoie
Date:
2 September, 1997

Definition at line 799 of file nurbs.cpp.

template<class T, int N>

HPoint_nD< T, N > PLib::NurbsCurve< T, N >::operator() ( T u ) const [virtual]

Evaluates the curve in 4D at parameter u.

It evaluates the NURBS curve in 4D at the parametric point $u$. Using the following equation

where $P_i$ are the control points and $N_{i,p}$ are the $p$th degree B-spline basis functions.
For more details on the algorithm, see A4.1 on page 124 of the Nurbs Book.
Parameters:

u the parametric value at which the curve is evaluated

Returns:
the 4D point at C(u)
Warning:
the parametric value must be in a valid range
Author:
Philippe Lavoie
Date:
24 January, 1997

Implements PLib::ParaCurve.
Definition at line 500 of file nurbs.cpp.
References basisFuns, and findSpan.
Referenced by PLib::NurbsCurve< float, N >::hpointAt.

template<class T, int N>

NurbsCurve< T, N > & PLib::NurbsCurve< T, N >::operator= ( const NurbsCurve< T, N > & curve ) [virtual]

The assignment operator for a NURBS curve.

Parameters:

curve the NURBS curve to copy

Returns:
A reference to itself
Warning:
The curve being copied must be valid, otherwise strange results might occur.
Author:
Philippe Lavoie
Date:
24 January 1997

Reimplemented in PLib::NurbsCurveSP.
Definition at line 184 of file nurbs.cpp.

template<class T, int N>

void PLib::NurbsCurve< T, N >::removeKnot ( int r,

int s,

int num

)

Removes an internal knot from a curve. This is A5.8 on p185 from the NURB book modified to not check for tolerance before removing the knot.

Parameters:

r the knot to remove

s the multiplicity of the knot

num the number of times to try to remove the knot

Warning:
r must be an internal knot.
Author:
Philippe Lavoie
Date:
24 January 1997

Reimplemented in PLib::NurbsCurveSP.
Definition at line 1700 of file nurbs.cpp.
References resize.
Referenced by removeKnotsBound.

template<class T, int N>

void PLib::NurbsCurve< T, N >::removeKnotsBound ( const Vector< T > & ub,

Vector< T > & ek,

T E

)

Remove knots from a curve without exceeding an error bound.

For more information about the algorithm, see A9.9 on p429 of the NURB book.
Parameters:

ub the knot coefficients

ek the error after removing

Author:
Philippe Lavoie
Date:
24 January 1997

Reimplemented in PLib::NurbsCurveSP.
Definition at line 1787 of file nurbs.cpp.
References basisFun, findSpan, getRemovalBnd, PLib::Vector< int >::minIndex, PLib::BasicArray< int >::n, removeKnot, PLib::BasicArray< int >::reset, and PLib::BasicArray< int >::resize.

template<class T, int N>

void PLib::NurbsCurve< T, N >::reset ( const Vector< HPoint_nD< T, N > > & P1,

const Vector< T > & U1,

int Degree

) [virtual]

Resets a NURBS curve to new values.

Parameters:

P1 the new values for the control points

U1 the new values for the knot vector

Degree the new degree of the curve

Warning:
The size of P1,U1 and Degree must agree: P.n()+degree+1=U.n()
Author:
Philippe Lavoie
Date:
24 January 1997

Reimplemented in PLib::NurbsCurveSP.
Definition at line 76 of file nurbs.cpp.

template<class T, int N>

void PLib::NurbsCurve< T, N >::resize ( int n,

int Deg

)

Resizes a NURBS curve.

Resizes a NURBS curve. The old values are lost and new ones have to be created.
Parameters:

n the new number of control points for the curve

Deg the new degree for the curve

Author:
Philippe Lavoie
Date:
24 January 1997

Definition at line 1187 of file nurbs.cpp.
Referenced by removeKnot.

template<class T, int N>

void PLib::NurbsCurve< T, N >::transform ( const MatrixRT< T > & A )

Performs geometrical modifications.

Each control points will be modified by a rotation-translation matrix.
Parameters:

A the rotation-translation matrix

Author:
Philippe Lavoie
Date:
22 August 1997

Definition at line 471 of file nurbs.cpp.

Friends And Related Function Documentation

template<class T, int N>

T chordLengthParam ( const Vector< Point_nD< T, N > > & Q,

Vector< T > & ub

) [related]

chord length parameterization.

Performs chord length parameterization from a vector of points. The chord length parameterization works as follows: \begin{itemize} \item The total chord length is defined as

\item and \item

\end{itemize} Where is the size of the vector . \endlatexonly
Parameters:

Q a vector of 3D points

ub the result of chord length parameterization

Returns:
the total chord length of the points.
Author:
Philippe Lavoie
Date:
24 January, 1997

Definition at line 1971 of file nurbs.cpp.
Referenced by globalApproxErrBnd, and leastSquares.

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

Generated on Tue Jun 24 13:27:00 2003 for NURBS++ by

PLib::NurbsCurve Class Template Reference

Public Methods

Protected Attributes

Friends

Related Functions

Detailed Description

template<class T, int N> class PLib::NurbsCurve< T, N >

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation

template<class T, int N>
class PLib::NurbsCurve< T, N >