unsupported/Eigen/src/Splines/Spline.h - mirror - Git at Google

 // 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 parameter vectors. */
     typedef typename SplineTraits<Spline>::ParameterVectorType ParameterVectorType;

     /** \brief The data type used to store non-zero basis functions. */
     typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType;

     /** \brief The data type used to store the values of the basis function derivatives. */
     typedef typename SplineTraits<Spline>::BasisDerivativeType BasisDerivativeType;

     /** \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(Dimension,(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);

     /**
      * \copydoc Spline::basisFunctionDerivatives
      * \param degree The degree of the underlying spline
      * \param knots The underlying spline's knot vector.
      **/
     static BasisDerivativeType BasisFunctionDerivatives(
       const Scalar u, const DenseIndex order, const DenseIndex degree, const KnotVectorType& knots);

   private:
     KnotVectorType m_knots; /*!< Knot vector. */
     ControlPointVectorType  m_ctrls; /*!< Control points. */

     template <typename DerivativeType>
     static void BasisFunctionDerivativesImpl(
       const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
       const DenseIndex order,
       const DenseIndex p,
       const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& U,
       DerivativeType& N_);
   };

   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 _Scalar, int _Dim, int _Degree>
   template <typename DerivativeType>
   void Spline<_Scalar, _Dim, _Degree>::BasisFunctionDerivativesImpl(
     const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
     const DenseIndex order,
     const DenseIndex p,
     const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& U,
     DerivativeType& N_)
   {
     typedef Spline<_Scalar, _Dim, _Degree> SplineType;
     enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };

     typedef typename SplineTraits<SplineType>::Scalar Scalar;
     typedef typename SplineTraits<SplineType>::BasisVectorType BasisVectorType;

     const DenseIndex span = SplineType::Span(u, p, 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 (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<_Scalar, _Dim, _Degree> >::BasisDerivativeType der;
     BasisFunctionDerivativesImpl(u, order, degree(), knots(), 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<_Scalar, _Dim, _Degree>, DerivativeOrder >::BasisDerivativeType der;
     BasisFunctionDerivativesImpl(u, order, degree(), knots(), der);
     return der;
   }

   template <typename _Scalar, int _Dim, int _Degree>
   typename SplineTraits<Spline<_Scalar, _Dim, _Degree> >::BasisDerivativeType
   Spline<_Scalar, _Dim, _Degree>::BasisFunctionDerivatives(
     const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
     const DenseIndex order,
     const DenseIndex degree,
     const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& knots)
   {
     typename SplineTraits<Spline>::BasisDerivativeType der;
     BasisFunctionDerivativesImpl(u, order, degree, knots, der);
     return der;
   }
 }

 #endif // EIGEN_SPLINE_H
	// 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 parameter vectors. */
	typedef typename SplineTraits<Spline>::ParameterVectorType ParameterVectorType;

	/** \brief The data type used to store non-zero basis functions. */
	typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType;

	/** \brief The data type used to store the values of the basis function derivatives. */
	typedef typename SplineTraits<Spline>::BasisDerivativeType BasisDerivativeType;

	/** \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(Dimension,(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);

	/**
	* \copydoc Spline::basisFunctionDerivatives
	* \param degree The degree of the underlying spline
	* \param knots The underlying spline's knot vector.
	**/
	static BasisDerivativeType BasisFunctionDerivatives(
	const Scalar u, const DenseIndex order, const DenseIndex degree, const KnotVectorType& knots);

	private:
	KnotVectorType m_knots; /!< Knot vector. /
	ControlPointVectorType m_ctrls; /!< Control points. /

	template <typename DerivativeType>
	static void BasisFunctionDerivativesImpl(
	const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
	const DenseIndex order,
	const DenseIndex p,
	const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& U,
	DerivativeType& N_);
	};

	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 _Scalar, int _Dim, int _Degree>
	template <typename DerivativeType>
	void Spline<_Scalar, _Dim, _Degree>::BasisFunctionDerivativesImpl(
	const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
	const DenseIndex order,
	const DenseIndex p,
	const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& U,
	DerivativeType& N_)
	{
	typedef Spline<_Scalar, _Dim, _Degree> SplineType;
	enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };

	typedef typename SplineTraits<SplineType>::Scalar Scalar;
	typedef typename SplineTraits<SplineType>::BasisVectorType BasisVectorType;

	const DenseIndex span = SplineType::Span(u, p, 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 (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<_Scalar, _Dim, _Degree> >::BasisDerivativeType der;
	BasisFunctionDerivativesImpl(u, order, degree(), knots(), 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<_Scalar, _Dim, _Degree>, DerivativeOrder >::BasisDerivativeType der;
	BasisFunctionDerivativesImpl(u, order, degree(), knots(), der);
	return der;
	}

	template <typename _Scalar, int _Dim, int _Degree>
	typename SplineTraits<Spline<_Scalar, _Dim, _Degree> >::BasisDerivativeType
	Spline<_Scalar, _Dim, _Degree>::BasisFunctionDerivatives(
	const typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
	const DenseIndex order,
	const DenseIndex degree,
	const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& knots)
	{
	typename SplineTraits<Spline>::BasisDerivativeType der;
	BasisFunctionDerivativesImpl(u, order, degree, knots, der);
	return der;
	}
	}

	#endif // EIGEN_SPLINE_H