Eigen::Spline< Scalar_, Dim_, Degree_ > Class Template Reference

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).

Public Types
enum	{ Dimension }
enum	{ Degree }
typedef SplineTraits< Spline >::BasisDerivativeType	BasisDerivativeType
	The data type used to store the values of the basis function derivatives.
typedef SplineTraits< Spline >::BasisVectorType	BasisVectorType
	The data type used to store non-zero basis functions.
typedef SplineTraits< Spline >::ControlPointVectorType	ControlPointVectorType
	The data type representing the spline's control points.
typedef SplineTraits< Spline >::KnotVectorType	KnotVectorType
	The data type used to store knot vectors.
typedef SplineTraits< Spline >::ParameterVectorType	ParameterVectorType
	The data type used to store parameter vectors.
typedef SplineTraits< Spline >::PointType	PointType
	The point type the spline is representing.
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.
DenseIndex	degree () const
	Returns the spline degree.
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.
PointType	operator() (Scalar u) const
	Returns the spline value at a given site \(u\). 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.
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 span within the provided knot vector in which u is falling.

Member Typedef Documentation

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() [1/2]

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() [2/2]

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()

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.

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.

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.

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

• Spline.h