#include <aikido/common/BSpline.hpp>

Public Types
enum	{ Dimension = _Dim }

enum	{ Degree = _Degree }

typedef _Scalar	Scalar

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

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

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

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

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

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

Public Member Functions
	BSpline ()
	Creates a (constant) zero spline. More...

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

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

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

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

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

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

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

template<int DerivativeOrder>
SplineTraits< BSpline, DerivativeOrder >::DerivativeType	derivatives (Scalar u, Eigen::DenseIndex order=DerivativeOrder) const

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

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

template<int DerivativeOrder>
SplineTraits< BSpline, DerivativeOrder >::BasisDerivativeType	basisFunctionDerivatives (Scalar u, Eigen::DenseIndex order=DerivativeOrder) const

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

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

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

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

static BasisDerivativeType	BasisFunctionDerivatives (const Scalar u, const Eigen::DenseIndex order, const Eigen::DenseIndex degree, const KnotVectorType &knots)

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

Private Attributes
KnotVectorType	m_knots

ControlPointVectorType	m_ctrls

Member Typedef Documentation

◆ BasisDerivativeType

template<typename _Scalar , int _Dim, int _Degree>

typedef SplineTraits<BSpline>::BasisDerivativeType aikido::common::BSpline< _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<BSpline>::BasisVectorType aikido::common::BSpline< _Scalar, _Dim, _Degree >::BasisVectorType

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

◆ ControlPointVectorType

template<typename _Scalar , int _Dim, int _Degree>

typedef SplineTraits<BSpline>::ControlPointVectorType aikido::common::BSpline< _Scalar, _Dim, _Degree >::ControlPointVectorType

The data type representing the spline's control points.

◆ KnotVectorType

template<typename _Scalar , int _Dim, int _Degree>

typedef SplineTraits<BSpline>::KnotVectorType aikido::common::BSpline< _Scalar, _Dim, _Degree >::KnotVectorType

The data type used to store knot vectors.

◆ ParameterVectorType

template<typename _Scalar , int _Dim, int _Degree>

typedef SplineTraits<BSpline>::ParameterVectorType aikido::common::BSpline< _Scalar, _Dim, _Degree >::ParameterVectorType

The data type used to store parameter vectors.

◆ PointType

template<typename _Scalar , int _Dim, int _Degree>

typedef SplineTraits<BSpline>::PointType aikido::common::BSpline< _Scalar, _Dim, _Degree >::PointType

The point type the spline is representing.

◆ Scalar

template<typename _Scalar , int _Dim, int _Degree>

typedef _Scalar aikido::common::BSpline< _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

◆ BSpline() [1/3]

template<typename _Scalar , int _Dim, int _Degree>

aikido::common::BSpline< _Scalar, _Dim, _Degree >::BSpline ( )

inline

Creates a (constant) zero spline.

For Splines with dynamic degree, the resulting degree will be 0.

◆ BSpline() [2/3]

template<typename _Scalar , int _Dim, int _Degree>

template<typename OtherVectorType , typename OtherArrayType >

aikido::common::BSpline< _Scalar, _Dim, _Degree >::BSpline	(	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.

◆ BSpline() [3/3]

template<typename _Scalar , int _Dim, int _Degree>

template<int OtherDegree>

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

static

Parameters

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

Using the template version of this function is more efficieent since temporary objects are allocated on the stack whenever this is possible.

◆ BasisFunctionDerivativesImpl()

template<typename _Scalar , int _Dim, int _Degree>

template<typename DerivativeType >

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

staticprivate

◆ basisFunctions()

template<typename _Scalar , int _Dim, int _Degree>

SplineTraits< BSpline< _Scalar, _Dim, _Degree > >::BasisVectorType aikido::common::BSpline< _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>

BSpline< _Scalar, _Dim, _Degree >::BasisVectorType aikido::common::BSpline< _Scalar, _Dim, _Degree >::BasisFunctions	(	Scalar	u,
		Eigen::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.

◆ ctrls() [1/2]

template<typename _Scalar , int _Dim, int _Degree>

ControlPointVectorType& aikido::common::BSpline< _Scalar, _Dim, _Degree >::ctrls ( )

inline

Returns the ctrls of the underlying spline.

◆ ctrls() [2/2]

template<typename _Scalar , int _Dim, int _Degree>

const ControlPointVectorType& aikido::common::BSpline< _Scalar, _Dim, _Degree >::ctrls ( ) const

inline

Returns the ctrls of the underlying spline.

◆ degree()

template<typename _Scalar , int _Dim, int _Degree>

Eigen::DenseIndex aikido::common::BSpline< _Scalar, _Dim, _Degree >::degree

Returns the spline degree.

◆ derivatives() [1/2]

template<typename _Scalar , int _Dim, int _Degree>

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

Using the template version of this function is more efficieent since temporary objects are allocated on the stack whenever this is possible.

◆ knots()

template<typename _Scalar , int _Dim, int _Degree>

const KnotVectorType& aikido::common::BSpline< _Scalar, _Dim, _Degree >::knots ( ) const

inline

Returns the knots of the underlying spline.

◆ operator()()

template<typename _Scalar , int _Dim, int _Degree>

BSpline< _Scalar, _Dim, _Degree >::PointType aikido::common::BSpline< _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 .

◆ span()

template<typename _Scalar , int _Dim, int _Degree>

Eigen::DenseIndex aikido::common::BSpline< _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.

◆ Span()

template<typename _Scalar , int _Dim, int _Degree>

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

static

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

Member Data Documentation

◆ m_ctrls

template<typename _Scalar , int _Dim, int _Degree>

ControlPointVectorType aikido::common::BSpline< _Scalar, _Dim, _Degree >::m_ctrls

private

Control points.

◆ m_knots

template<typename _Scalar , int _Dim, int _Degree>

KnotVectorType aikido::common::BSpline< _Scalar, _Dim, _Degree >::m_knots

private

Knot vector.

Public Types

Public Member Functions

Static Public Member Functions

Static Private Member Functions

Private Attributes

Member Typedef Documentation

◆ BasisDerivativeType

◆ BasisVectorType

◆ ControlPointVectorType

◆ KnotVectorType

◆ ParameterVectorType

◆ PointType

◆ Scalar

Member Enumeration Documentation

◆ anonymous enum

◆ anonymous enum

Constructor & Destructor Documentation

◆ BSpline() [1/3]

◆ BSpline() [2/3]

◆ BSpline() [3/3]

Member Function Documentation

◆ BasisFunctionDerivatives()

◆ basisFunctionDerivatives() [1/2]

◆ basisFunctionDerivatives() [2/2]

◆ BasisFunctionDerivativesImpl()

◆ basisFunctions()

◆ BasisFunctions()

◆ ctrls() [1/2]

◆ ctrls() [2/2]

◆ degree()

◆ derivatives() [1/2]

◆ derivatives() [2/2]

◆ knots()

◆ operator()()

◆ span()

◆ Span()

Member Data Documentation

◆ m_ctrls

◆ m_knots