// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 20010-2011 Hauke Heibel <hauke.heibel@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_SPLINE_H
#define EIGEN_SPLINE_H
#include "SplineFwd.h"
namespace Eigen
{
/**
* \ingroup Splines_Module
* \class Spline
* \brief 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
* \f{align*}
* C(u) & = \sum_{i=0}^{n}N_{i,p}(u)P_i
* \f}
*
* \tparam _Scalar The underlying data type (typically float or double)
* \tparam _Dim The curve dimension (e.g. 2 or 3)
* \tparam _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).
**/
template <typename _Scalar, int _Dim, int _Degree>
class Spline
{
public:
typedef _Scalar Scalar; /*!< The spline curve's scalar type. */
enum { Dimension = _Dim /*!< The spline curve's dimension. */ };
enum { Degree = _Degree /*!< The spline curve's degree. */ };
/** \brief The point type the spline is representing. */
typedef typename SplineTraits<Spline>::PointType PointType;
/** \brief The data type used to store knot vectors. */
typedef typename SplineTraits<Spline>::KnotVectorType KnotVectorType;
/** \brief The data type used to store non-zero basis functions. */
typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType;
/** \brief The data type representing the spline's control points. */
typedef typename SplineTraits<Spline>::ControlPointVectorType ControlPointVectorType;
/**
* \brief Creates a (constant) zero spline.
* For Splines with dynamic degree, the resulting degree will be 0.
**/
Spline()
: m_knots(1, (Degree==Dynamic ? 2 : 2*Degree+2))
, m_ctrls(ControlPointVectorType::Zero(2,(Degree==Dynamic ? 1 : Degree+1)))
{
// in theory this code can go to the initializer list but it will get pretty
// much unreadable ...
enum { MinDegree = (Degree==Dynamic ? 0 : Degree) };
m_knots.template segment<MinDegree+1>(0) = Array<Scalar,1,MinDegree+1>::Zero();
m_knots.template segment<MinDegree+1>(MinDegree+1) = Array<Scalar,1,MinDegree+1>::Ones();
}
/**
* \brief Creates a spline from a knot vector and control points.
* \param knots The spline's knot vector.
* \param ctrls The spline's control point vector.
**/
template <typename OtherVectorType, typename OtherArrayType>
Spline(const OtherVectorType& knots, const OtherArrayType& ctrls) : m_knots(knots), m_ctrls(ctrls) {}
/**
* \brief Copy constructor for splines.
* \param spline The input spline.
**/
template <int OtherDegree>
Spline(const Spline<Scalar, Dimension, OtherDegree>& spline) :
m_knots(spline.knots()), m_ctrls(spline.ctrls()) {}
/**
* \brief Returns the knots of the underlying spline.
**/
const KnotVectorType& knots() const { return m_knots; }
/**
* \brief Returns the knots of the underlying spline.
**/
const ControlPointVectorType& ctrls() const { return m_ctrls; }
/**
* \brief Returns the spline value at a given site \f$u\f$.
*
* The function returns
* \f{align*}
* C(u) & = \sum_{i=0}^{n}N_{i,p}P_i
* \f}
*
* \param u Parameter \f$u \in [0;1]\f$ at which the spline is evaluated.
* \return The spline value at the given location \f$u\f$.
**/
PointType operator()(Scalar u) const;
/**
* \brief Evaluation of spline derivatives of up-to given order.
*
* The function returns
* \f{align*}
* \frac{d^i}{du^i}C(u) & = \sum_{i=0}^{n} \frac{d^i}{du^i} N_{i,p}(u)P_i
* \f}
* for i ranging between 0 and order.
*
* \param u Parameter \f$u \in [0;1]\f$ at which the spline derivative is evaluated.
* \param order The order up to which the derivatives are computed.
**/
typename SplineTraits<Spline>::DerivativeType
derivatives(Scalar u, DenseIndex order) const;
/**
* \copydoc Spline::derivatives
* Using the template version of this function is more efficieent since
* temporary objects are allocated on the stack whenever this is possible.
**/
template <int DerivativeOrder>
typename SplineTraits<Spline,DerivativeOrder>::DerivativeType
derivatives(Scalar u, DenseIndex order = DerivativeOrder) const;
/**
* \brief Computes the non-zero basis functions at the given site.
*
* Splines have local support and a point from their image is defined
* by exactly \f$p+1\f$ control points \f$P_i\f$ where \f$p\f$ is the
* spline degree.
*
* This function computes the \f$p+1\f$ non-zero basis function values
* for a given parameter value \f$u\f$. It returns
* \f{align*}{
* N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
* \f}
*
* \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis functions
* are computed.
**/
typename SplineTraits<Spline>::BasisVectorType
basisFunctions(Scalar u) const;
/**
* \brief Computes the non-zero spline basis function derivatives up to given order.
*
* The function computes
* \f{align*}{
* \frac{d^i}{du^i} N_{i,p}(u), \hdots, \frac{d^i}{du^i} N_{i+p+1,p}(u)
* \f}
* with i ranging from 0 up to the specified order.
*
* \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis function
* derivatives are computed.
* \param order The order up to which the basis function derivatives are computes.
**/
typename SplineTraits<Spline>::BasisDerivativeType
basisFunctionDerivatives(Scalar u, DenseIndex order) const;
/**
* \copydoc Spline::basisFunctionDerivatives
* Using the template version of this function is more efficieent since
* temporary objects are allocated on the stack whenever this is possible.
**/
template <int DerivativeOrder>
typename SplineTraits<Spline,DerivativeOrder>::BasisDerivativeType
basisFunctionDerivatives(Scalar u, DenseIndex order = DerivativeOrder) const;
/**
* \brief Returns the spline degree.
**/
DenseIndex degree() const;
/**
* \brief Returns the span within the knot vector in which u is falling.
* \param u The site for which the span is determined.
**/
DenseIndex span(Scalar u) const;
/**
* \brief Computes the spang within the provided knot vector in which u is falling.
**/
static DenseIndex Span(typename SplineTraits<Spline>::Scalar u, DenseIndex degree, const typename SplineTraits<Spline>::KnotVectorType& knots);
/**
* \brief Returns the spline's non-zero basis functions.
*
* The function computes and returns
* \f{align*}{
* N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
* \f}
*
* \param u The site at which the basis functions are computed.
* \param degree The degree of the underlying spline.
* \param knots The underlying spline's knot vector.
**/
static BasisVectorType BasisFunctions(Scalar u, DenseIndex degree, const KnotVectorType& knots);
private:
KnotVectorType m_knots; /*!< Knot vector. */
ControlPointVectorType m_ctrls; /*!< Control points. */
};
template <typename _Scalar, int _Dim, int _Degree>
DenseIndex Spline<_Scalar, _Dim, _Degree>::Span(
typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::Scalar u,
DenseIndex degree,
const typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::KnotVectorType& knots)
{
// Piegl & Tiller, "The NURBS Book", A2.1 (p. 68)
if (u <= knots(0)) return degree;
const Scalar* pos = std::upper_bound(knots.data()+degree-1, knots.data()+knots.size()-degree-1, u);
return static_cast<DenseIndex>( std::distance(knots.data(), pos) - 1 );
}
template <typename _Scalar, int _Dim, int _Degree>
typename Spline<_Scalar, _Dim, _Degree>::BasisVectorType
Spline<_Scalar, _Dim, _Degree>::BasisFunctions(
typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
DenseIndex degree,
const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& knots)
{
typedef typename Spline<_Scalar, _Dim, _Degree>::BasisVectorType BasisVectorType;
const DenseIndex p = degree;
const DenseIndex i = Spline::Span(u, degree, knots);
const KnotVectorType& U = knots;
BasisVectorType left(p+1); left(0) = Scalar(0);
BasisVectorType right(p+1); right(0) = Scalar(0);
VectorBlock<BasisVectorType,Degree>(left,1,p) = u - VectorBlock<const KnotVectorType,Degree>(U,i+1-p,p).reverse();
VectorBlock<BasisVectorType,Degree>(right,1,p) = VectorBlock<const KnotVectorType,Degree>(U,i+1,p) - u;
BasisVectorType N(1,p+1);
N(0) = Scalar(1);
for (DenseIndex j=1; j<=p; ++j)
{
Scalar saved = Scalar(0);
for (DenseIndex r=0; r<j; r++)
{
const Scalar tmp = N(r)/(right(r+1)+left(j-r));
N[r] = saved + right(r+1)*tmp;
saved = left(j-r)*tmp;
}
N(j) = saved;
}
return N;
}
template <typename _Scalar, int _Dim, int _Degree>
DenseIndex Spline<_Scalar, _Dim, _Degree>::degree() const
{
if (_Degree == Dynamic)
return m_knots.size() - m_ctrls.cols() - 1;
else
return _Degree;
}
template <typename _Scalar, int _Dim, int _Degree>
DenseIndex Spline<_Scalar, _Dim, _Degree>::span(Scalar u) const
{
return Spline::Span(u, degree(), knots());
}
template <typename _Scalar, int _Dim, int _Degree>
typename Spline<_Scalar, _Dim, _Degree>::PointType Spline<_Scalar, _Dim, _Degree>::operator()(Scalar u) const
{
enum { Order = SplineTraits<Spline>::OrderAtCompileTime };
const DenseIndex span = this->span(u);
const DenseIndex p = degree();
const BasisVectorType basis_funcs = basisFunctions(u);
const Replicate<BasisVectorType,Dimension,1> ctrl_weights(basis_funcs);
const Block<const ControlPointVectorType,Dimension,Order> ctrl_pts(ctrls(),0,span-p,Dimension,p+1);
return (ctrl_weights * ctrl_pts).rowwise().sum();
}
/* --------------------------------------------------------------------------------------------- */
template <typename SplineType, typename DerivativeType>
void derivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& der)
{
enum { Dimension = SplineTraits<SplineType>::Dimension };
enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };
enum { DerivativeOrder = DerivativeType::ColsAtCompileTime };
typedef typename SplineTraits<SplineType>::ControlPointVectorType ControlPointVectorType;
typedef typename SplineTraits<SplineType,DerivativeOrder>::BasisDerivativeType BasisDerivativeType;
typedef typename BasisDerivativeType::ConstRowXpr BasisDerivativeRowXpr;
const DenseIndex p = spline.degree();
const DenseIndex span = spline.span(u);
const DenseIndex n = (std::min)(p, order);
der.resize(Dimension,n+1);
// Retrieve the basis function derivatives up to the desired order...
const BasisDerivativeType basis_func_ders = spline.template basisFunctionDerivatives<DerivativeOrder>(u, n+1);
// ... and perform the linear combinations of the control points.
for (DenseIndex der_order=0; der_order<n+1; ++der_order)
{
const Replicate<BasisDerivativeRowXpr,Dimension,1> ctrl_weights( basis_func_ders.row(der_order) );
const Block<const ControlPointVectorType,Dimension,Order> ctrl_pts(spline.ctrls(),0,span-p,Dimension,p+1);
der.col(der_order) = (ctrl_weights * ctrl_pts).rowwise().sum();
}
}
template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::DerivativeType
Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const
{
typename SplineTraits< Spline >::DerivativeType res;
derivativesImpl(*this, u, order, res);
return res;
}
template <typename _Scalar, int _Dim, int _Degree>
template <int DerivativeOrder>
typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::DerivativeType
Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const
{
typename SplineTraits< Spline, DerivativeOrder >::DerivativeType res;
derivativesImpl(*this, u, order, res);
return res;
}
template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::BasisVectorType
Spline<_Scalar, _Dim, _Degree>::basisFunctions(Scalar u) const
{
return Spline::BasisFunctions(u, degree(), knots());
}
/* --------------------------------------------------------------------------------------------- */
template <typename SplineType, typename DerivativeType>
void basisFunctionDerivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& N_)
{
enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };
typedef typename SplineTraits<SplineType>::Scalar Scalar;
typedef typename SplineTraits<SplineType>::BasisVectorType BasisVectorType;
typedef typename SplineTraits<SplineType>::KnotVectorType KnotVectorType;
const KnotVectorType& U = spline.knots();
const DenseIndex p = spline.degree();
const DenseIndex span = spline.span(u);
const DenseIndex n = (std::min)(p, order);
N_.resize(n+1, p+1);
BasisVectorType left = BasisVectorType::Zero(p+1);
BasisVectorType right = BasisVectorType::Zero(p+1);
Matrix<Scalar,Order,Order> ndu(p+1,p+1);
double saved, temp;
ndu(0,0) = 1.0;
DenseIndex j;
for (j=1; j<=p; ++j)
{
left[j] = u-U[span+1-j];
right[j] = U[span+j]-u;
saved = 0.0;
for (DenseIndex r=0; r<j; ++r)
{
/* Lower triangle */
ndu(j,r) = right[r+1]+left[j-r];
temp = ndu(r,j-1)/ndu(j,r);
/* Upper triangle */
ndu(r,j) = static_cast<Scalar>(saved+right[r+1] * temp);
saved = left[j-r] * temp;
}
ndu(j,j) = static_cast<Scalar>(saved);
}
for (j = p; j>=0; --j)
N_(0,j) = ndu(j,p);
// Compute the derivatives
DerivativeType a(n+1,p+1);
DenseIndex r=0;
for (; r<=p; ++r)
{
DenseIndex s1,s2;
s1 = 0; s2 = 1; // alternate rows in array a
a(0,0) = 1.0;
// Compute the k-th derivative
for (DenseIndex k=1; k<=static_cast<DenseIndex>(n); ++k)
{
double d = 0.0;
DenseIndex rk,pk,j1,j2;
rk = r-k; pk = p-k;
if (r>=k)
{
a(s2,0) = a(s1,0)/ndu(pk+1,rk);
d = a(s2,0)*ndu(rk,pk);
}
if (rk>=-1) j1 = 1;
else j1 = -rk;
if (r-1 <= pk) j2 = k-1;
else j2 = p-r;
for (j=j1; j<=j2; ++j)
{
a(s2,j) = (a(s1,j)-a(s1,j-1))/ndu(pk+1,rk+j);
d += a(s2,j)*ndu(rk+j,pk);
}
if (r<=pk)
{
a(s2,k) = -a(s1,k-1)/ndu(pk+1,r);
d += a(s2,k)*ndu(r,pk);
}
N_(k,r) = static_cast<Scalar>(d);
j = s1; s1 = s2; s2 = j; // Switch rows
}
}
/* Multiply through by the correct factors */
/* (Eq. [2.9]) */
r = p;
for (DenseIndex k=1; k<=static_cast<DenseIndex>(n); ++k)
{
for (DenseIndex j=p; j>=0; --j) N_(k,j) *= r;
r *= p-k;
}
}
template <typename _Scalar, int _Dim, int _Degree>
typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::BasisDerivativeType
Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const
{
typename SplineTraits< Spline >::BasisDerivativeType der;
basisFunctionDerivativesImpl(*this, u, order, der);
return der;
}
template <typename _Scalar, int _Dim, int _Degree>
template <int DerivativeOrder>
typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::BasisDerivativeType
Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const
{
typename SplineTraits< Spline, DerivativeOrder >::BasisDerivativeType der;
basisFunctionDerivativesImpl(*this, u, order, der);
return der;
}
}
#endif // EIGEN_SPLINE_H