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 > ::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 > ::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 knots 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 . 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 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.

Member Typedef Documentation

typedef _Scalar Scalar

The spline curve's scalar type.

Member Enumeration Documentation

anonymous enum

Enumerator
Dimension	The spline curve's dimension.

anonymous enum

Enumerator
Degree	The spline curve's degree.

Constructor & Destructor Documentation

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 ( const Spline< Scalar, Dimension, OtherDegree > & spline )

inline

Copy constructor for splines.

Parameters

spline The input spline.

Member Function Documentation

SplineTraits< Spline< _Scalar, _Dim, _Degree >, DerivativeOrder >::BasisDerivativeType 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.

SplineTraits<Spline,DerivativeOrder>::BasisDerivativeType 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.

SplineTraits< Spline< _Scalar, _Dim, _Degree > >::BasisVectorType 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.

References Spline< _Scalar, _Dim, _Degree >::BasisFunctions(), Spline< _Scalar, _Dim, _Degree >::degree(), and Spline< _Scalar, _Dim, _Degree >::knots().

Referenced by Spline< _Scalar, _Dim, _Degree >::operator()().

Spline< _Scalar, _Dim, _Degree >::BasisVectorType 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.

References Spline< _Scalar, _Dim, _Degree >::degree(), Spline< _Scalar, _Dim, _Degree >::knots(), and Spline< _Scalar, _Dim, _Degree >::Span().

Referenced by Spline< _Scalar, _Dim, _Degree >::basisFunctions().

SplineTraits< Spline< _Scalar, _Dim, _Degree >, DerivativeOrder >::DerivativeType 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.

SplineTraits<Spline,DerivativeOrder>::DerivativeType 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.

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

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

The function returns

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