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Diffstat (limited to 'unsupported/Eigen/src/Splines/Spline.h')
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diff --git a/unsupported/Eigen/src/Splines/Spline.h b/unsupported/Eigen/src/Splines/Spline.h new file mode 100644 index 000000000..bb4b73e5a --- /dev/null +++ b/unsupported/Eigen/src/Splines/Spline.h @@ -0,0 +1,407 @@ +// This file is part of Eigen, a lightweight C++ template library +// for linear algebra. +// +// Copyright (C) 20010-2011 Hauke Heibel <hauke.heibel@gmail.com> +// +// Eigen is free software; you can redistribute it and/or +// modify it under the terms of the GNU Lesser General Public +// License as published by the Free Software Foundation; either +// version 3 of the License, or (at your option) any later version. +// +// Alternatively, you can redistribute it and/or +// modify it under the terms of the GNU General Public License as +// published by the Free Software Foundation; either version 2 of +// the License, or (at your option) any later version. +// +// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY +// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS +// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the +// GNU General Public License for more details. +// +// You should have received a copy of the GNU Lesser General Public +// License and a copy of the GNU General Public License along with +// Eigen. If not, see <http://www.gnu.org/licenses/>. + +#ifndef EIGEN_SPLINE_H +#define EIGEN_SPLINE_H + +#include "SplineFwd.h" + +namespace Eigen +{ + /** + * \class Spline class + * \brief A class representing N-D spline curves. + * \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; + enum { Dimension = _Dim }; + enum { Degree = _Degree }; + + typedef typename SplineTraits<Spline>::PointType PointType; + typedef typename SplineTraits<Spline>::KnotVectorType KnotVectorType; + typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType; + typedef typename SplineTraits<Spline>::ControlPointVectorType ControlPointVectorType; + + /** + * \brief Creates a spline from a knot vector and control points. + **/ + template <typename OtherVectorType, typename OtherArrayType> + Spline(const OtherVectorType& knots, const OtherArrayType& ctrls) : m_knots(knots), m_ctrls(ctrls) {} + + template <int OtherDegree> + Spline(const Spline<Scalar, Dimension, OtherDegree>& spline) : + m_knots(spline.knots()), m_ctrls(spline.ctrls()) {} + + /* Const access methods for knots and control points. */ + const KnotVectorType& knots() const { return m_knots; } + const ControlPointVectorType& ctrls() const { return m_ctrls; } + + /* Spline evaluation. */ + PointType operator()(Scalar u) const; + + /* Evaluation of spline derivatives of up-to given order. + * The returned matrix has dimensions Dim-by-(Order+1) containing + * the 0-th order up-to Order-th order derivatives. + */ + typename SplineTraits<Spline>::DerivativeType + derivatives(Scalar u, DenseIndex order) const; + + /** + * Evaluation of spline derivatives of up-to given order. + * The function performs identically to derivatives(Scalar, int) but + * does not require any heap allocations. + * \sa derivatives(Scalar, int) + **/ + template <int DerivativeOrder> + typename SplineTraits<Spline,DerivativeOrder>::DerivativeType + derivatives(Scalar u, DenseIndex order = DerivativeOrder) const; + + /* Non-zero spline basis functions. */ + typename SplineTraits<Spline>::BasisVectorType + basisFunctions(Scalar u) const; + + /* Non-zero spline basis function derivatives up to given order. + * The order is different from the spline order - it is the order + * up to which derivatives will be computed. + * \sa basisFunctions(Scalar) + */ + typename SplineTraits<Spline>::BasisDerivativeType + basisFunctionDerivatives(Scalar u, DenseIndex order) const; + + /** + * Computes non-zero basis function derivatives up to the given derivative order. + * As opposed to basisFunctionDerivatives(Scalar, int) this function does not perform + * any heap allocations. + * \sa basisFunctionDerivatives(Scalar, int) + **/ + template <int DerivativeOrder> + typename SplineTraits<Spline,DerivativeOrder>::BasisDerivativeType + basisFunctionDerivatives(Scalar u, DenseIndex order = DerivativeOrder) const; + + /** + * \brief The current spline degree. It's a function of knot size and number + * of controls and thus does not require a dedicated member. + */ + DenseIndex degree() const; + + /** Computes the span within the knot vector in which u falls. */ + DenseIndex span(Scalar u) const; + + static DenseIndex Span(typename SplineTraits<Spline>::Scalar u, DenseIndex degree, const typename SplineTraits<Spline>::KnotVectorType& knots); + 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()); + } + + /** + * \brief A functor for the computation of a spline point. + * \sa Piegl & Tiller, "The NURBS Book", A4.1 (p. 124) + **/ + 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>::Scalar Scalar; + + typedef typename SplineTraits<SplineType>::BasisVectorType BasisVectorType; + 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(); + } + } + + /** + * \brief A functor for the computation of a spline point. + * \sa Piegl & Tiller, "The NURBS Book", A4.1 (p. 124) + **/ + 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; + } + + /** + * \brief A functor for the computation of a spline's non-zero basis functions. + * \sa Piegl & Tiller, "The NURBS Book", A2.2 (p. 70) + **/ + 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; + typedef typename SplineTraits<SplineType>::ControlPointVectorType ControlPointVectorType; + + 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 |