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

#include <Spline.h>

Public Types
enum	{ Dimension = _Dim }

enum	{ Degree = _Degree }

typedef _Scalar	Scalar

typedef SplineTraits< Spline >::PointType	PointType
	The point type the spline is representing. 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 >::BasisVectorType	BasisVectorType
	The data type used to store non-zero basis functions. More...

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

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

Public Member Functions
	Spline ()
	Creates a (constant) zero spline. 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...

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

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

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

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

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...

DenseIndex	degree () const
	Returns the spline degree.

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

Static Public Member Functions
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.

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

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...

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).

Member Typedef Documentation

§ BasisDerivativeType

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.

§ BasisVectorType

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.

§ ControlPointVectorType

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.

§ KnotVectorType

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.

§ ParameterVectorType

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.

§ PointType

template<typename _Scalar, int _Dim, int _Degree>

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

The point type the spline is representing.

§ Scalar

template<typename _Scalar, int _Dim, int _Degree>

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

The spline curve's scalar type.

Member Enumeration Documentation

§ anonymous enum

template<typename _Scalar, int _Dim, int _Degree>

anonymous enum

Enumerator
Dimension	The spline curve's dimension.

§ anonymous enum

template<typename _Scalar, int _Dim, int _Degree>

anonymous enum

Enumerator
Degree	The spline curve's degree.

Constructor & Destructor Documentation

§ Spline() [1/3]

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.

§ Spline() [2/3]

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.

§ Spline() [3/3]

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.

Member Function Documentation

§ basisFunctionDerivatives() [1/2]

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.

§ basisFunctionDerivatives() [2/2]

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.

§ BasisFunctionDerivatives()

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.

§ basisFunctions()

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.

§ BasisFunctions()

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.

§ derivatives() [1/2]

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.

§ derivatives() [2/2]

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.

§ operator()()

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 \(u\).

§ span()

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.

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

third-party-libs/eigen3/unsupported/Eigen/src/Splines/Spline.h

Public Types

Public Member Functions

Static Public Member Functions

Detailed Description

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

Member Typedef Documentation

§ BasisDerivativeType

§ BasisVectorType

§ ControlPointVectorType

§ KnotVectorType

§ ParameterVectorType

§ PointType

§ Scalar

Member Enumeration Documentation

§ anonymous enum

§ anonymous enum

Constructor & Destructor Documentation

§ Spline() [1/3]

§ Spline() [2/3]

§ Spline() [3/3]

Member Function Documentation

§ basisFunctionDerivatives() [1/2]

§ basisFunctionDerivatives() [2/2]

§ BasisFunctionDerivatives()

§ basisFunctions()

§ BasisFunctions()

§ derivatives() [1/2]

§ derivatives() [2/2]

§ operator()()

§ span()

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