A class representing multi-dimensional spline curves. More...

#include <Spline.h>

Public Types
enum	{ Dimension = _Dim }

enum	{ Degree = _Degree }

typedef SplineTraits< Spline >::BasisDerivativeType	BasisDerivativeType
	The data type used to store the values of the basis function derivatives. More...

typedef SplineTraits< Spline >::BasisVectorType	BasisVectorType
	The data type used to store non-zero basis functions. More...

typedef SplineTraits< Spline >::ControlPointVectorType	ControlPointVectorType
	The data type representing the spline's control points. More...

typedef SplineTraits< Spline >::KnotVectorType	KnotVectorType
	The data type used to store knot vectors. More...

typedef SplineTraits< Spline >::ParameterVectorType	ParameterVectorType
	The data type used to store parameter vectors. More...

typedef SplineTraits< Spline >::PointType	PointType
	The point type the spline is representing. More...

typedef _Scalar	Scalar

Public Member Functions
SplineTraits< Spline >::BasisDerivativeType	basisFunctionDerivatives (Scalar u, DenseIndex order) const
	Computes the non-zero spline basis function derivatives up to given order. More...

template<int DerivativeOrder>
SplineTraits< Spline, DerivativeOrder >::BasisDerivativeType	basisFunctionDerivatives (Scalar u, DenseIndex order=DerivativeOrder) const
	Computes the non-zero spline basis function derivatives up to given order. More...

SplineTraits< Spline >::BasisVectorType	basisFunctions (Scalar u) const
	Computes the non-zero basis functions at the given site. More...

const ControlPointVectorType &	ctrls () const
	Returns the ctrls of the underlying spline. More...

DenseIndex	degree () const
	Returns the spline degree. More...

SplineTraits< Spline >::DerivativeType	derivatives (Scalar u, DenseIndex order) const
	Evaluation of spline derivatives of up-to given order. More...

template<int DerivativeOrder>
SplineTraits< Spline, DerivativeOrder >::DerivativeType	derivatives (Scalar u, DenseIndex order=DerivativeOrder) const
	Evaluation of spline derivatives of up-to given order. More...

const KnotVectorType &	knots () const
	Returns the knots of the underlying spline. More...

PointType	operator() (Scalar u) const
	Returns the spline value at a given site . More...

DenseIndex	span (Scalar u) const
	Returns the span within the knot vector in which u is falling. More...

	Spline ()
	Creates a (constant) zero spline. For Splines with dynamic degree, the resulting degree will be 0. More...

template<typename OtherVectorType , typename OtherArrayType >
	Spline (const OtherVectorType &knots, const OtherArrayType &ctrls)
	Creates a spline from a knot vector and control points. More...

template<int OtherDegree>
	Spline (const Spline< Scalar, Dimension, OtherDegree > &spline)
	Copy constructor for splines. More...

Static Public Member Functions
static BasisDerivativeType	BasisFunctionDerivatives (const Scalar u, const DenseIndex order, const DenseIndex degree, const KnotVectorType &knots)
	Computes the non-zero spline basis function derivatives up to given order. More...

static BasisVectorType	BasisFunctions (Scalar u, DenseIndex degree, const KnotVectorType &knots)
	Returns the spline's non-zero basis functions. More...

static DenseIndex	Span (typename SplineTraits< Spline >::Scalar u, DenseIndex degree, const typename SplineTraits< Spline >::KnotVectorType &knots)
	Computes the spang within the provided knot vector in which u is falling. More...

Static Private Member Functions
template<typename DerivativeType >
static void	BasisFunctionDerivativesImpl (const typename Spline< _Scalar, _Dim, _Degree >::Scalar u, const DenseIndex order, const DenseIndex p, const typename Spline< _Scalar, _Dim, _Degree >::KnotVectorType &U, DerivativeType &N_)

Private Attributes
ControlPointVectorType	m_ctrls

KnotVectorType	m_knots

Detailed Description

template<typename _Scalar, int _Dim, int _Degree>
class Eigen::Spline< _Scalar, _Dim, _Degree >

A class representing multi-dimensional spline curves.

The class represents B-splines with non-uniform knot vectors. Each control point of the B-spline is associated with a basis function

$\begin{align*} C(u) & = \sum_{i=0}^{n}N_{i,p}(u)P_i \end{align*}$

Template Parameters

_Scalar	The underlying data type (typically float or double)
_Dim	The curve dimension (e.g. 2 or 3)
_Degree	Per default set to Dynamic; could be set to the actual desired degree for optimization purposes (would result in stack allocation of several temporary variables).

Definition at line 35 of file Spline.h.

Member Typedef Documentation

template<typename _Scalar, int _Dim, int _Degree>

typedef SplineTraits<Spline>::BasisDerivativeType Eigen::Spline< _Scalar, _Dim, _Degree >::BasisDerivativeType

The data type used to store the values of the basis function derivatives.

Definition at line 55 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

typedef SplineTraits<Spline>::BasisVectorType Eigen::Spline< _Scalar, _Dim, _Degree >::BasisVectorType

The data type used to store non-zero basis functions.

Definition at line 52 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

typedef SplineTraits<Spline>::ControlPointVectorType Eigen::Spline< _Scalar, _Dim, _Degree >::ControlPointVectorType

The data type representing the spline's control points.

Definition at line 58 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

typedef SplineTraits<Spline>::KnotVectorType Eigen::Spline< _Scalar, _Dim, _Degree >::KnotVectorType

The data type used to store knot vectors.

Definition at line 46 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

typedef SplineTraits<Spline>::ParameterVectorType Eigen::Spline< _Scalar, _Dim, _Degree >::ParameterVectorType

The data type used to store parameter vectors.

Definition at line 49 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

typedef SplineTraits<Spline>::PointType Eigen::Spline< _Scalar, _Dim, _Degree >::PointType

The point type the spline is representing.

Definition at line 43 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

typedef _Scalar Eigen::Spline< _Scalar, _Dim, _Degree >::Scalar

The spline curve's scalar type.

Definition at line 38 of file Spline.h.

Member Enumeration Documentation

template<typename _Scalar, int _Dim, int _Degree>

anonymous enum

Enumerator
Dimension	The spline curve's dimension.

Definition at line 39 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

anonymous enum

Enumerator
Degree	The spline curve's degree.

Definition at line 40 of file Spline.h.

Constructor & Destructor Documentation

template<typename _Scalar, int _Dim, int _Degree>

Eigen::Spline< _Scalar, _Dim, _Degree >::Spline ( )

inline

Creates a (constant) zero spline. For Splines with dynamic degree, the resulting degree will be 0.

Definition at line 64 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

template<typename OtherVectorType , typename OtherArrayType >

Eigen::Spline< _Scalar, _Dim, _Degree >::Spline	(	const OtherVectorType &	knots,
		const OtherArrayType &	ctrls
	)

inline

Creates a spline from a knot vector and control points.

Parameters

knots	The spline's knot vector.
ctrls	The spline's control point vector.

Definition at line 81 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

template<int OtherDegree>

Eigen::Spline< _Scalar, _Dim, _Degree >::Spline ( const Spline< Scalar, Dimension, OtherDegree > & spline )

inline

Copy constructor for splines.

Parameters

spline The input spline.

Definition at line 88 of file Spline.h.

Member Function Documentation

template<typename _Scalar , int _Dim, int _Degree>

SplineTraits< Spline< _Scalar, _Dim, _Degree >, DerivativeOrder >::BasisDerivativeType Eigen::Spline< _Scalar, _Dim, _Degree >::basisFunctionDerivatives	(	Scalar	u,
		DenseIndex	order
	)		const

Computes the non-zero spline basis function derivatives up to given order.

The function computes

$\begin{align*} \frac{d^i}{du^i} N_{i,p}(u), \hdots, \frac{d^i}{du^i} N_{i+p+1,p}(u) \end{align*}$

with i ranging from 0 up to the specified order.

Parameters

u	Parameter $u \in [0;1]$ at which the non-zero basis function derivatives are computed.
order	The order up to which the basis function derivatives are computes.

Definition at line 481 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

template<int DerivativeOrder>

SplineTraits<Spline,DerivativeOrder>::BasisDerivativeType Eigen::Spline< _Scalar, _Dim, _Degree >::basisFunctionDerivatives	(	Scalar	u,
		DenseIndex	order = `DerivativeOrder`
	)		const

Computes the non-zero spline basis function derivatives up to given order.

The function computes

$\begin{align*} \frac{d^i}{du^i} N_{i,p}(u), \hdots, \frac{d^i}{du^i} N_{i+p+1,p}(u) \end{align*}$

with i ranging from 0 up to the specified order.

Parameters

u	Parameter $u \in [0;1]$ at which the non-zero basis function derivatives are computed.
order	The order up to which the basis function derivatives are computes. Using the template version of this function is more efficieent since temporary objects are allocated on the stack whenever this is possible.

template<typename _Scalar, int _Dim, int _Degree>

SplineTraits< Spline< _Scalar, _Dim, _Degree > >::BasisDerivativeType Eigen::Spline< _Scalar, _Dim, _Degree >::BasisFunctionDerivatives	(	const Scalar	u,
		const DenseIndex	order,
		const DenseIndex	degree,
		const KnotVectorType &	knots
	)

static

Computes the non-zero spline basis function derivatives up to given order.

The function computes

$\begin{align*} \frac{d^i}{du^i} N_{i,p}(u), \hdots, \frac{d^i}{du^i} N_{i+p+1,p}(u) \end{align*}$

with i ranging from 0 up to the specified order.

Parameters

u	Parameter $u \in [0;1]$ at which the non-zero basis function derivatives are computed.
order	The order up to which the basis function derivatives are computes.
degree	The degree of the underlying spline
knots	The underlying spline's knot vector.

Definition at line 500 of file Spline.h.

template<typename _Scalar , int _Dim, int _Degree>

template<typename DerivativeType >

void Eigen::Spline< _Scalar, _Dim, _Degree >::BasisFunctionDerivativesImpl	(	const typename Spline< _Scalar, _Dim, _Degree >::Scalar	u,
		const DenseIndex	order,
		const DenseIndex	p,
		const typename Spline< _Scalar, _Dim, _Degree >::KnotVectorType &	U,
		DerivativeType &	N_
	)

staticprivate

Definition at line 373 of file Spline.h.

template<typename _Scalar , int _Dim, int _Degree>

SplineTraits< Spline< _Scalar, _Dim, _Degree > >::BasisVectorType Eigen::Spline< _Scalar, _Dim, _Degree >::basisFunctions ( Scalar u ) const

Computes the non-zero basis functions at the given site.

Splines have local support and a point from their image is defined by exactly $p+1$ control points $P_i$ where $p$ is the spline degree.

This function computes the $p+1$ non-zero basis function values for a given parameter value $u$ . It returns

$\begin{align*} N_{i,p}(u), \hdots, N_{i+p+1,p}(u) \end{align*}$

Parameters

u	Parameter $u \in [0;1]$ at which the non-zero basis functions are computed.

Definition at line 363 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

Spline< _Scalar, _Dim, _Degree >::BasisVectorType Eigen::Spline< _Scalar, _Dim, _Degree >::BasisFunctions	(	Scalar	u,
		DenseIndex	degree,
		const KnotVectorType &	knots
	)

static

Returns the spline's non-zero basis functions.

The function computes and returns

$\begin{align*} N_{i,p}(u), \hdots, N_{i+p+1,p}(u) \end{align*}$

Parameters

u	The site at which the basis functions are computed.
degree	The degree of the underlying spline.
knots	The underlying spline's knot vector.

Definition at line 247 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

const ControlPointVectorType& Eigen::Spline< _Scalar, _Dim, _Degree >::ctrls ( ) const

inline

Returns the ctrls of the underlying spline.

Definition at line 99 of file Spline.h.

template<typename _Scalar , int _Dim, int _Degree>

DenseIndex Eigen::Spline< _Scalar, _Dim, _Degree >::degree ( ) const

Returns the spline degree.

Definition at line 282 of file Spline.h.

template<typename _Scalar , int _Dim, int _Degree>

SplineTraits< Spline< _Scalar, _Dim, _Degree >, DerivativeOrder >::DerivativeType Eigen::Spline< _Scalar, _Dim, _Degree >::derivatives	(	Scalar	u,
		DenseIndex	order
	)		const

Evaluation of spline derivatives of up-to given order.

The function returns

$\begin{align*} \frac{d^i}{du^i}C(u) & = \sum_{i=0}^{n} \frac{d^i}{du^i} N_{i,p}(u)P_i \end{align*}$

for i ranging between 0 and order.

Parameters

u	Parameter $u \in [0;1]$ at which the spline derivative is evaluated.
order	The order up to which the derivatives are computed.

Definition at line 344 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

template<int DerivativeOrder>

SplineTraits<Spline,DerivativeOrder>::DerivativeType Eigen::Spline< _Scalar, _Dim, _Degree >::derivatives	(	Scalar	u,
		DenseIndex	order = `DerivativeOrder`
	)		const

Evaluation of spline derivatives of up-to given order.

The function returns

$\begin{align*} \frac{d^i}{du^i}C(u) & = \sum_{i=0}^{n} \frac{d^i}{du^i} N_{i,p}(u)P_i \end{align*}$

for i ranging between 0 and order.

Parameters

u	Parameter $u \in [0;1]$ at which the spline derivative is evaluated.
order	The order up to which the derivatives are computed. Using the template version of this function is more efficieent since temporary objects are allocated on the stack whenever this is possible.

template<typename _Scalar, int _Dim, int _Degree>

const KnotVectorType& Eigen::Spline< _Scalar, _Dim, _Degree >::knots ( ) const

inline

Returns the knots of the underlying spline.

Definition at line 94 of file Spline.h.

template<typename _Scalar , int _Dim, int _Degree>

Spline< _Scalar, _Dim, _Degree >::PointType Eigen::Spline< _Scalar, _Dim, _Degree >::operator() ( Scalar u ) const

Returns the spline value at a given site $u$ .

The function returns

$\begin{align*} C(u) & = \sum_{i=0}^{n}N_{i,p}P_i \end{align*}$

Parameters

u	Parameter $u \in [0;1]$ at which the spline is evaluated.

Returns: The spline value at the given location .

Definition at line 297 of file Spline.h.

template<typename _Scalar , int _Dim, int _Degree>

DenseIndex Eigen::Spline< _Scalar, _Dim, _Degree >::span ( Scalar u ) const

Returns the span within the knot vector in which u is falling.

Parameters

u	The site for which the span is determined.

Definition at line 291 of file Spline.h.

template<typename _Scalar , int _Dim, int _Degree>

DenseIndex Eigen::Spline< _Scalar, _Dim, _Degree >::Span	(	typename SplineTraits< Spline< _Scalar, _Dim, _Degree > >::Scalar	u,
		DenseIndex	degree,
		const typename SplineTraits< Spline< _Scalar, _Dim, _Degree > >::KnotVectorType &	knots
	)

static

Computes the spang within the provided knot vector in which u is falling.

Definition at line 234 of file Spline.h.

Member Data Documentation

template<typename _Scalar, int _Dim, int _Degree>

ControlPointVectorType Eigen::Spline< _Scalar, _Dim, _Degree >::m_ctrls

private

Control points.

Definition at line 222 of file Spline.h.

template<typename _Scalar, int _Dim, int _Degree>

KnotVectorType Eigen::Spline< _Scalar, _Dim, _Degree >::m_knots

private

Knot vector.

Definition at line 221 of file Spline.h.

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

Spline.h

Public Types

Public Member Functions

Static Public Member Functions

Static Private Member Functions

Private Attributes

Detailed Description

template<typename _Scalar, int _Dim, int _Degree> class Eigen::Spline< _Scalar, _Dim, _Degree >

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<typename _Scalar, int _Dim, int _Degree>
class Eigen::Spline< _Scalar, _Dim, _Degree >