A class to represent a NURBS surface. More...

Public Types
enum	Method { equallySpaced = 0 , chordLength }

Public Member Functions
	NurbsSurface ()
	default constructor

	NurbsSurface (const NurbsSurface &nS)
	copy constructor

	NurbsSurface (int DegU, int DegV, const fmatvec::VecV &Uk, const fmatvec::VecV &Vk, fmatvec::GeneralMatrix< fmatvec::Vec4 > &Q)
	constructor

virtual	~NurbsSurface ()=default
	Empty desctructor.

const fmatvec::VecV &	knotU () const
	A reference to the U knot vector.

const fmatvec::VecV &	knotV () const
	A reference to the V knot vector.

double	knotU (int i) const
	Returns the i-th knot from U.

double	knotV (int i) const
	Returns the i-th knot from V.

const fmatvec::GeneralMatrix< fmatvec::Vec4 > &	ctrlPnts () const
	A reference to the control points.

const fmatvec::Vec4	ctrlPnts (int i, int j) const
	A reference to the control point at (i,j)

int	degreeU () const
	A reference to the degree in U of the surface.

int	degreeV () const
	A reference to the degree in V of the surface.

void	setKnotU (const fmatvec::VecV &U_)

void	setKnotV (const fmatvec::VecV &V_)

void	setCtrlPnts (const fmatvec::GeneralMatrix< fmatvec::Vec4 > &P_)

void	setDegreeU (int degU_)

void	setDegreeV (int degV_)

virtual NurbsSurface &	operator= (const NurbsSurface &)
	NurbsSurface assignment. More...

void	resize (int Pu, int Pv, int DegU, int DegV)
	Resize the surface. More...

fmatvec::HPoint< 3 >	operator() (double u, double v) const
	Returns the point on the surface at u,v. More...

fmatvec::Point< 3 >	pointAt (double u, double v) const
	rojects the point in the normal space More...

void	deriveAt (double u, double v, int d, fmatvec::GeneralMatrix< fmatvec::Vec3 > &skl) const
	Find the non-zero basis functions in the U and V direction. More...

void	deriveAtH (double u, double v, int d, fmatvec::GeneralMatrix< fmatvec::Vec4 > &skl) const
	computes the point and the derivatives of degree d and below at (u,v) More...

fmatvec::Vec3	normal (double u, double v) const
	Computes the normal of the surface at (u,v) More...

void	globalInterp (const fmatvec::GeneralMatrix< fmatvec::Vec3 > &Q, int DegU, int DegV)

void	globalInterp (const fmatvec::GeneralMatrix< fmatvec::Vec3 > &Q, const fmatvec::VecV &uk, const fmatvec::VecV &vk, int DegU, int DegV)

void	globalInterpClosedU (const fmatvec::GeneralMatrix< fmatvec::Vec3 > &Q, int DegU, int DegV)

void	globalInterpClosedU (const fmatvec::GeneralMatrix< fmatvec::Vec3 > &Q, const fmatvec::VecV &uk, const fmatvec::VecV &vk, int DegU, int DegV)

void	globalInterpH (const fmatvec::GeneralMatrix< fmatvec::Vec4 > &Q, int DegU, int DegV, Method method=chordLength)

void	globalInterpClosedUH (const fmatvec::GeneralMatrix< fmatvec::Vec4 > &Q, int DegU, int DegV, Method method=chordLength)

Protected Member Functions
void	findSpan (double u, double v, int spanU, int spanV) const
	Finds the multiplicity of a knot in the U knot. More...

int	findSpanU (double u) const
	finds the span in the U direction More...

int	findSpanV (double v) const
	finds the span in the V direction More...

int	findMultU (int r) const

int	findMultV (int r) const

Protected Attributes
fmatvec::VecV	U
	the U knot vector

fmatvec::VecV	V
	the V knot vector

fmatvec::GeneralMatrix< fmatvec::Vec4 >	P
	The matrix of control points.

int	degU {0}
	the degree of the surface in U

int	degV {0}
	the degree of the surface in V

fmatvec::Mat	InverseU
	The matrix for the computation of the control points in U-direction.

fmatvec::Mat	InverseV
	The matrix for the computation of the control points in V-direction.

Related Functions
(Note that these are not member functions.)
int	surfMeshParams (const GeneralMatrix< fmatvec::Vec3 > &Q, VecV &uk, VecV &vl)
	Computes the parameters for global surface interpolation. More...

Detailed Description

A class to represent a NURBS surface.

The NURBS surface is composed of points in homogenous space. It can have any degree in both the u and the v direction.

Member Function Documentation

◆ deriveAt()

void MBSim::NurbsSurface::deriveAt	(	double	u,
		double	v,
		int	d,
		fmatvec::GeneralMatrix< fmatvec::Vec3 > &	skl
	)		const

Find the non-zero basis functions in the U and V direction.

Parameters

u	the u parametric value
v	the v parametric value
spanU	the span of u
spanV	the span of v
Nu	the vector containing the U non-zero basis functions
Nv	the vector containing the V non-zero basis functions

Author: Philippe Lavoie

Date: 24 January, 1997

Finds the non-zero basis function in the U direction

Parameters

u	the u parametric value
span	the span of u
N	the vector containing the basis functions

Author: Philippe Lavoie

Date: 24 January, 1997 ‍/ template <class T, int nD> void NurbsSurface<T,nD>::basisFunsU(T u, int span, Vector<T>& N) const { //Vector<T> left(degU+1), right(degU+1) ; vector<T> left(degU+1) ; vector<T> right(degU+1) ; T temp,saved ;

N.resize(degU+1) ;

N(0] = 1.0 ; for(int j=1; j<= degU ; j++){ left[j] = u-U[span+1-j] ; right[j] = U[span+j]-u ; saved = 0.0 ; for(int r=0 ; r<j; r++){ temp = N[r]/(right[r+1]+left[j-r]) ; N[r] = saved+right[r+1]*temp ; saved = left[j-r]*temp ; } N[j] = saved ; }

}

/*!

Finds the non-zero basis function in the V direction

Parameters

v	the v parametric value
span	the span of v
N	the vector containing the basis functions

Author: Philippe Lavoie

Date: 24 January, 1997 ‍/ template <class T, int nD> void NurbsSurface<T,nD>::basisFunsV(T v, int span, Vector<T>& N) const { //Vector<T> left(degV+1), right(degV+1) ; vector<T> left(degV+1) ; vector<T> right(degV+1) ; T temp,saved ;

N.resize(degV+1) ;

N[0] = 1.0 ; for(int j=1; j<= degV ; j++){ left[j] = v-V[span+1-j] ; right[j] = V[span+j]-v ; saved = 0.0 ; for(int r=0 ; r<j; r++){ temp = N[r]/(right[r+1]+left[j-r]) ; N[r] = saved+right[r+1]*temp ; saved = left[j-r]*temp ; } N[j] = saved ; }

} /*!

Computes the point and the derivatives of degree d and below at (u,v)

Computes the matrix of derivatives at u,v . The value of skl(k,l) represents the derivative of the surface S(u,v) with respect to u, k times and to v, l times.

Parameters

u	the u parametric value
v	the v parametric value
d	the derivative is computed up to and including to this value
skl	the matrix containing the derivatives

Author: Philippe Lavoie

Date: 24 January, 1997

◆ deriveAtH()

void MBSim::NurbsSurface::deriveAtH	(	double	u,
		double	v,
		int	d,
		fmatvec::GeneralMatrix< fmatvec::Vec4 > &	skl
	)		const

computes the point and the derivatives of degree d and below at (u,v)

Computes the matrix of derivatives at u,v . The value of skl(k,l) represents the derivative of the surface S(u,v) with respect to u k times and to $v$ $l$ times.

Parameters

u	the u parametric value
v	the v parametric value
d	the derivative is computed up to and including to this value
skl	the matrix containing the derivatives

Author: Philippe Lavoie

Date: 24 January, 1997

◆ findSpan()

void MBSim::NurbsSurface::findSpan	(	double	u,
		double	v,
		int	spanU,
		int	spanV
	)		const

protected

Finds the multiplicity of a knot in the U knot.

Parameters

r	the knot to observe

Returns: the multiplicity of the knot

Warning: r must be a valid U knot index

Author: Philippe Lavoie

Date: 24 January, 1997

finds the multiplicity of a knot in the V knot

Parameters

r	the knot to observe

Returns: the multiplicity of the V knot

Warning: r must be a valid knot index

Author: Philippe Lavoie

Date: 24 January, 1997

finds the span in the U and V direction

Finds the span in the U and V direction. The spanU is the index k for which the parameter u is valid in the [u_k,u_{k+1}] range. The spanV is the index k for which the parameter v is valid in the [v_k,v_{k+1}] range.

Parameters

u	find the U span for this parametric value
v	find the V span for this parametric value
spanU	the U span
spanV	the V span

Author: Philippe Lavoie

Date: 24 January, 1997

◆ findSpanU()

int MBSim::NurbsSurface::findSpanU ( double u ) const

protected

finds the span in the U direction

Finds the span in the U direction. The span is the index k for which the parameter u is valid in the [u_k,u_{k+1}] range.

Parameters

u	--> find the span for this parametric value

Returns: the span for u

Author: Philippe Lavoie

Date: 24 January, 1997

\modified 20 January, 1999 (Alejandro Frangi)

◆ findSpanV()

int MBSim::NurbsSurface::findSpanV ( double v ) const

protected

finds the span in the V direction

Finds the span in the V direction. The span is the index k for which the parameter v is valid in the [v_k,v_{k+1}] range.

Parameters

v	find the span for this parametric value

Returns: the span for v

Author: Philippe Lavoie

Date: 24 January, 1997 \modified 20 January, 1999 (Alejandro Frangi)

◆ normal()

Vec3 MBSim::NurbsSurface::normal	(	double	u,
		double	v
	)		const

Computes the normal of the surface at (u,v)

Parameters

u	the u parametric value
v	the v parametric value

Returns: the normal at (u,v) .

Author: Philippe Lavoie

Date: 24 January, 1997

◆ operator()()

fmatvec::HPoint< 3 > MBSim::NurbsSurface::operator()	(	double	u,
		double	v
	)		const

Returns the point on the surface at u,v.

Returns the point on the surface at u,v

Parameters

u	the u parametric value
v	the v parametric value

Returns: The homogenous point at u,v

Author: Philippe Lavoie

Date: 24 January, 1997

◆ operator=()

NurbsSurface & MBSim::NurbsSurface::operator= ( const NurbsSurface & nS )

virtual

NurbsSurface assignment.

Parameters

nS	the NURBS surface to copy

Author: Philippe Lavoie

Date: 24 January, 1997

◆ pointAt()

fmatvec::Point< 3 > MBSim::NurbsSurface::pointAt	(	double	u,
		double	v
	)		const

rojects the point in the normal space

Projects the point in the normal space.

◆ resize()

void MBSim::NurbsSurface::resize	(	int	Pu,
		int	Pv,
		int	DegU,
		int	DegV
	)

Resize the surface.

Resize the surface. Proper values must be assigned once this function has been called since the resize operator is destructive.

Parameters

Pu	the number of control points in the U direction
Pv	the number of control points in the V direction
DegU	the degree of the surface in the U direction
DegV	the degree of the surface in the V direction

Author: Philippe Lavoie

Date: 24 January, 1997

Friends And Related Function Documentation

◆ surfMeshParams()

int surfMeshParams	(	const GeneralMatrix< fmatvec::Vec3 > &	Q,
		VecV &	uk,
		VecV &	vl
	)

Computes the parameters for global surface interpolation. For more information, see A9.3 on p377 on the NURBS book.

Parameters

Q	the matrix of 3D points
uk	the knot coefficients in the U direction
vl	the knot coefficients in the V direction

Returns: 0 if an error occurs, 1 otherwise

Warning

Author: Philippe Lavoie

Date: 24 January, 1997

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

/mbsim-env/mbsim/kernel/mbsim/numerics/nurbs/nurbs_surface.h
/mbsim-env/mbsim/kernel/mbsim/numerics/nurbs/nurbs_surface.cc

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Related Functions

Detailed Description

Member Function Documentation

◆ deriveAt()

◆ deriveAtH()

◆ findSpan()

◆ findSpanU()

◆ findSpanV()

◆ normal()

◆ operator()()

◆ operator=()

◆ pointAt()

◆ resize()

Friends And Related Function Documentation

◆ surfMeshParams()