Public Types | Public Member Functions | Static Public Member Functions
Spline< _Scalar, _Dim, _Degree > Class Template Reference

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

List of all members.

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.
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.
SplineTraits< Spline >
::BasisVectorType 
basisFunctions (Scalar u) const
 Computes the non-zero basis functions at the given site.
const ControlPointVectorTypectrls () 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.
template<int DerivativeOrder>
SplineTraits< Spline,
DerivativeOrder >
::DerivativeType 
derivatives (Scalar u, DenseIndex order=DerivativeOrder) const
 Evaluation of spline derivatives of up-to given order.
const KnotVectorTypeknots () const
 Returns the knots of the underlying spline.
PointType operator() (Scalar u) const
 Returns the spline value at a given site $u$.
DenseIndex span (Scalar u) const
 Returns the span within the knot vector in which u is falling.
template<typename OtherVectorType , typename OtherArrayType >
 Spline (const OtherVectorType &knots, const OtherArrayType &ctrls)
 Creates a spline from a knot vector and control points.
template<int OtherDegree>
 Spline (const Spline< Scalar, Dimension, OtherDegree > &spline)
 Copy constructor for splines.

Static Public Member Functions

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

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:
_ScalarThe underlying data type (typically float or double)
_DimThe curve dimension (e.g. 2 or 3)
_DegreePer 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

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:
knotsThe spline's knot vector.
ctrlsThe spline's control point vector.
Spline ( const Spline< Scalar, Dimension, OtherDegree > &  spline)
inline

Copy constructor for splines.

Parameters:
splineThe 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:
uParameter $u \in [0;1]$ at which the non-zero basis function derivatives are computed.
orderThe 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:
uParameter $u \in [0;1]$ at which the non-zero basis function derivatives are computed.
orderThe 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:
uParameter $u \in [0;1]$ at which the non-zero basis functions are computed.

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

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:
uThe site at which the basis functions are computed.
degreeThe degree of the underlying spline.
knotsThe underlying spline's knot vector.

References 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:
uParameter $u \in [0;1]$ at which the spline derivative is evaluated.
orderThe 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:
uParameter $u \in [0;1]$ at which the spline derivative is evaluated.
orderThe 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*}

Parameters:
uParameter $u \in [0;1]$ at which the spline is evaluated.
Returns:
The spline value at the given location $u$.
DenseIndex span ( Scalar  u) const

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

Parameters:
uThe site for which the span is determined.

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


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