[136] | 1 | // This file is part of Eigen, a lightweight C++ template library
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| 2 | // for linear algebra.
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| 3 | //
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| 4 | // Copyright (C) 20010-2011 Hauke Heibel <hauke.heibel@gmail.com>
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| 5 | //
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| 6 | // This Source Code Form is subject to the terms of the Mozilla
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| 7 | // Public License v. 2.0. If a copy of the MPL was not distributed
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| 8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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| 9 |
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| 10 | #ifndef EIGEN_SPLINE_H
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| 11 | #define EIGEN_SPLINE_H
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| 12 |
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| 13 | #include "SplineFwd.h"
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| 14 |
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| 15 | namespace Eigen
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| 16 | {
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| 17 | /**
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| 18 | * \ingroup Splines_Module
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| 19 | * \class Spline
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| 20 | * \brief A class representing multi-dimensional spline curves.
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| 21 | *
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| 22 | * The class represents B-splines with non-uniform knot vectors. Each control
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| 23 | * point of the B-spline is associated with a basis function
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| 24 | * \f{align*}
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| 25 | * C(u) & = \sum_{i=0}^{n}N_{i,p}(u)P_i
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| 26 | * \f}
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| 27 | *
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| 28 | * \tparam _Scalar The underlying data type (typically float or double)
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| 29 | * \tparam _Dim The curve dimension (e.g. 2 or 3)
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| 30 | * \tparam _Degree Per default set to Dynamic; could be set to the actual desired
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| 31 | * degree for optimization purposes (would result in stack allocation
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| 32 | * of several temporary variables).
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| 33 | **/
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| 34 | template <typename _Scalar, int _Dim, int _Degree>
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| 35 | class Spline
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| 36 | {
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| 37 | public:
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| 38 | typedef _Scalar Scalar; /*!< The spline curve's scalar type. */
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| 39 | enum { Dimension = _Dim /*!< The spline curve's dimension. */ };
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| 40 | enum { Degree = _Degree /*!< The spline curve's degree. */ };
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| 41 |
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| 42 | /** \brief The point type the spline is representing. */
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| 43 | typedef typename SplineTraits<Spline>::PointType PointType;
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| 44 |
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| 45 | /** \brief The data type used to store knot vectors. */
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| 46 | typedef typename SplineTraits<Spline>::KnotVectorType KnotVectorType;
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| 47 |
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| 48 | /** \brief The data type used to store non-zero basis functions. */
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| 49 | typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType;
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| 50 |
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| 51 | /** \brief The data type representing the spline's control points. */
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| 52 | typedef typename SplineTraits<Spline>::ControlPointVectorType ControlPointVectorType;
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| 53 |
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| 54 | /**
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| 55 | * \brief Creates a (constant) zero spline.
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| 56 | * For Splines with dynamic degree, the resulting degree will be 0.
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| 57 | **/
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| 58 | Spline()
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| 59 | : m_knots(1, (Degree==Dynamic ? 2 : 2*Degree+2))
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| 60 | , m_ctrls(ControlPointVectorType::Zero(2,(Degree==Dynamic ? 1 : Degree+1)))
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| 61 | {
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| 62 | // in theory this code can go to the initializer list but it will get pretty
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| 63 | // much unreadable ...
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| 64 | enum { MinDegree = (Degree==Dynamic ? 0 : Degree) };
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| 65 | m_knots.template segment<MinDegree+1>(0) = Array<Scalar,1,MinDegree+1>::Zero();
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| 66 | m_knots.template segment<MinDegree+1>(MinDegree+1) = Array<Scalar,1,MinDegree+1>::Ones();
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| 67 | }
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| 68 |
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| 69 | /**
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| 70 | * \brief Creates a spline from a knot vector and control points.
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| 71 | * \param knots The spline's knot vector.
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| 72 | * \param ctrls The spline's control point vector.
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| 73 | **/
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| 74 | template <typename OtherVectorType, typename OtherArrayType>
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| 75 | Spline(const OtherVectorType& knots, const OtherArrayType& ctrls) : m_knots(knots), m_ctrls(ctrls) {}
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| 76 |
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| 77 | /**
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| 78 | * \brief Copy constructor for splines.
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| 79 | * \param spline The input spline.
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| 80 | **/
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| 81 | template <int OtherDegree>
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| 82 | Spline(const Spline<Scalar, Dimension, OtherDegree>& spline) :
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| 83 | m_knots(spline.knots()), m_ctrls(spline.ctrls()) {}
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| 84 |
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| 85 | /**
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| 86 | * \brief Returns the knots of the underlying spline.
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| 87 | **/
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| 88 | const KnotVectorType& knots() const { return m_knots; }
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| 89 |
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| 90 | /**
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| 91 | * \brief Returns the knots of the underlying spline.
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| 92 | **/
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| 93 | const ControlPointVectorType& ctrls() const { return m_ctrls; }
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| 94 |
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| 95 | /**
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| 96 | * \brief Returns the spline value at a given site \f$u\f$.
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| 97 | *
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| 98 | * The function returns
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| 99 | * \f{align*}
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| 100 | * C(u) & = \sum_{i=0}^{n}N_{i,p}P_i
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| 101 | * \f}
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| 102 | *
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| 103 | * \param u Parameter \f$u \in [0;1]\f$ at which the spline is evaluated.
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| 104 | * \return The spline value at the given location \f$u\f$.
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| 105 | **/
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| 106 | PointType operator()(Scalar u) const;
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| 107 |
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| 108 | /**
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| 109 | * \brief Evaluation of spline derivatives of up-to given order.
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| 110 | *
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| 111 | * The function returns
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| 112 | * \f{align*}
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| 113 | * \frac{d^i}{du^i}C(u) & = \sum_{i=0}^{n} \frac{d^i}{du^i} N_{i,p}(u)P_i
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| 114 | * \f}
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| 115 | * for i ranging between 0 and order.
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| 116 | *
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| 117 | * \param u Parameter \f$u \in [0;1]\f$ at which the spline derivative is evaluated.
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| 118 | * \param order The order up to which the derivatives are computed.
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| 119 | **/
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| 120 | typename SplineTraits<Spline>::DerivativeType
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| 121 | derivatives(Scalar u, DenseIndex order) const;
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| 122 |
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| 123 | /**
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| 124 | * \copydoc Spline::derivatives
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| 125 | * Using the template version of this function is more efficieent since
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| 126 | * temporary objects are allocated on the stack whenever this is possible.
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| 127 | **/
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| 128 | template <int DerivativeOrder>
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| 129 | typename SplineTraits<Spline,DerivativeOrder>::DerivativeType
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| 130 | derivatives(Scalar u, DenseIndex order = DerivativeOrder) const;
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| 131 |
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| 132 | /**
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| 133 | * \brief Computes the non-zero basis functions at the given site.
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| 134 | *
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| 135 | * Splines have local support and a point from their image is defined
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| 136 | * by exactly \f$p+1\f$ control points \f$P_i\f$ where \f$p\f$ is the
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| 137 | * spline degree.
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| 138 | *
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| 139 | * This function computes the \f$p+1\f$ non-zero basis function values
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| 140 | * for a given parameter value \f$u\f$. It returns
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| 141 | * \f{align*}{
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| 142 | * N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
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| 143 | * \f}
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| 144 | *
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| 145 | * \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis functions
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| 146 | * are computed.
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| 147 | **/
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| 148 | typename SplineTraits<Spline>::BasisVectorType
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| 149 | basisFunctions(Scalar u) const;
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| 150 |
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| 151 | /**
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| 152 | * \brief Computes the non-zero spline basis function derivatives up to given order.
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| 153 | *
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| 154 | * The function computes
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| 155 | * \f{align*}{
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| 156 | * \frac{d^i}{du^i} N_{i,p}(u), \hdots, \frac{d^i}{du^i} N_{i+p+1,p}(u)
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| 157 | * \f}
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| 158 | * with i ranging from 0 up to the specified order.
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| 159 | *
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| 160 | * \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis function
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| 161 | * derivatives are computed.
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| 162 | * \param order The order up to which the basis function derivatives are computes.
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| 163 | **/
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| 164 | typename SplineTraits<Spline>::BasisDerivativeType
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| 165 | basisFunctionDerivatives(Scalar u, DenseIndex order) const;
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| 166 |
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| 167 | /**
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| 168 | * \copydoc Spline::basisFunctionDerivatives
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| 169 | * Using the template version of this function is more efficieent since
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| 170 | * temporary objects are allocated on the stack whenever this is possible.
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| 171 | **/
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| 172 | template <int DerivativeOrder>
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| 173 | typename SplineTraits<Spline,DerivativeOrder>::BasisDerivativeType
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| 174 | basisFunctionDerivatives(Scalar u, DenseIndex order = DerivativeOrder) const;
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| 175 |
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| 176 | /**
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| 177 | * \brief Returns the spline degree.
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| 178 | **/
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| 179 | DenseIndex degree() const;
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| 180 |
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| 181 | /**
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| 182 | * \brief Returns the span within the knot vector in which u is falling.
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| 183 | * \param u The site for which the span is determined.
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| 184 | **/
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| 185 | DenseIndex span(Scalar u) const;
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| 186 |
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| 187 | /**
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| 188 | * \brief Computes the spang within the provided knot vector in which u is falling.
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| 189 | **/
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| 190 | static DenseIndex Span(typename SplineTraits<Spline>::Scalar u, DenseIndex degree, const typename SplineTraits<Spline>::KnotVectorType& knots);
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| 191 |
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| 192 | /**
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| 193 | * \brief Returns the spline's non-zero basis functions.
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| 194 | *
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| 195 | * The function computes and returns
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| 196 | * \f{align*}{
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| 197 | * N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
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| 198 | * \f}
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| 199 | *
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| 200 | * \param u The site at which the basis functions are computed.
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| 201 | * \param degree The degree of the underlying spline.
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| 202 | * \param knots The underlying spline's knot vector.
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| 203 | **/
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| 204 | static BasisVectorType BasisFunctions(Scalar u, DenseIndex degree, const KnotVectorType& knots);
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| 205 |
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| 206 |
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| 207 | private:
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| 208 | KnotVectorType m_knots; /*!< Knot vector. */
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| 209 | ControlPointVectorType m_ctrls; /*!< Control points. */
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| 210 | };
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| 211 |
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| 212 | template <typename _Scalar, int _Dim, int _Degree>
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| 213 | DenseIndex Spline<_Scalar, _Dim, _Degree>::Span(
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| 214 | typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::Scalar u,
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| 215 | DenseIndex degree,
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| 216 | const typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::KnotVectorType& knots)
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| 217 | {
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| 218 | // Piegl & Tiller, "The NURBS Book", A2.1 (p. 68)
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| 219 | if (u <= knots(0)) return degree;
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| 220 | const Scalar* pos = std::upper_bound(knots.data()+degree-1, knots.data()+knots.size()-degree-1, u);
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| 221 | return static_cast<DenseIndex>( std::distance(knots.data(), pos) - 1 );
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| 222 | }
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| 223 |
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| 224 | template <typename _Scalar, int _Dim, int _Degree>
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| 225 | typename Spline<_Scalar, _Dim, _Degree>::BasisVectorType
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| 226 | Spline<_Scalar, _Dim, _Degree>::BasisFunctions(
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| 227 | typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
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| 228 | DenseIndex degree,
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| 229 | const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& knots)
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| 230 | {
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| 231 | typedef typename Spline<_Scalar, _Dim, _Degree>::BasisVectorType BasisVectorType;
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| 232 |
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| 233 | const DenseIndex p = degree;
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| 234 | const DenseIndex i = Spline::Span(u, degree, knots);
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| 235 |
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| 236 | const KnotVectorType& U = knots;
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| 237 |
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| 238 | BasisVectorType left(p+1); left(0) = Scalar(0);
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| 239 | BasisVectorType right(p+1); right(0) = Scalar(0);
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| 240 |
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| 241 | VectorBlock<BasisVectorType,Degree>(left,1,p) = u - VectorBlock<const KnotVectorType,Degree>(U,i+1-p,p).reverse();
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| 242 | VectorBlock<BasisVectorType,Degree>(right,1,p) = VectorBlock<const KnotVectorType,Degree>(U,i+1,p) - u;
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| 243 |
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| 244 | BasisVectorType N(1,p+1);
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| 245 | N(0) = Scalar(1);
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| 246 | for (DenseIndex j=1; j<=p; ++j)
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| 247 | {
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| 248 | Scalar saved = Scalar(0);
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| 249 | for (DenseIndex r=0; r<j; r++)
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| 250 | {
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| 251 | const Scalar tmp = N(r)/(right(r+1)+left(j-r));
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| 252 | N[r] = saved + right(r+1)*tmp;
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| 253 | saved = left(j-r)*tmp;
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| 254 | }
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| 255 | N(j) = saved;
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| 256 | }
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| 257 | return N;
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| 258 | }
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| 259 |
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| 260 | template <typename _Scalar, int _Dim, int _Degree>
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| 261 | DenseIndex Spline<_Scalar, _Dim, _Degree>::degree() const
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| 262 | {
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| 263 | if (_Degree == Dynamic)
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| 264 | return m_knots.size() - m_ctrls.cols() - 1;
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| 265 | else
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| 266 | return _Degree;
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| 267 | }
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| 268 |
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| 269 | template <typename _Scalar, int _Dim, int _Degree>
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| 270 | DenseIndex Spline<_Scalar, _Dim, _Degree>::span(Scalar u) const
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| 271 | {
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| 272 | return Spline::Span(u, degree(), knots());
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| 273 | }
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| 274 |
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| 275 | template <typename _Scalar, int _Dim, int _Degree>
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| 276 | typename Spline<_Scalar, _Dim, _Degree>::PointType Spline<_Scalar, _Dim, _Degree>::operator()(Scalar u) const
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| 277 | {
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| 278 | enum { Order = SplineTraits<Spline>::OrderAtCompileTime };
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| 279 |
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| 280 | const DenseIndex span = this->span(u);
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| 281 | const DenseIndex p = degree();
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| 282 | const BasisVectorType basis_funcs = basisFunctions(u);
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| 283 |
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| 284 | const Replicate<BasisVectorType,Dimension,1> ctrl_weights(basis_funcs);
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| 285 | const Block<const ControlPointVectorType,Dimension,Order> ctrl_pts(ctrls(),0,span-p,Dimension,p+1);
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| 286 | return (ctrl_weights * ctrl_pts).rowwise().sum();
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| 287 | }
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| 288 |
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| 289 | /* --------------------------------------------------------------------------------------------- */
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| 290 |
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| 291 | template <typename SplineType, typename DerivativeType>
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| 292 | void derivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& der)
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| 293 | {
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| 294 | enum { Dimension = SplineTraits<SplineType>::Dimension };
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| 295 | enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };
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| 296 | enum { DerivativeOrder = DerivativeType::ColsAtCompileTime };
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| 297 |
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| 298 | typedef typename SplineTraits<SplineType>::ControlPointVectorType ControlPointVectorType;
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| 299 | typedef typename SplineTraits<SplineType,DerivativeOrder>::BasisDerivativeType BasisDerivativeType;
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| 300 | typedef typename BasisDerivativeType::ConstRowXpr BasisDerivativeRowXpr;
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| 301 |
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| 302 | const DenseIndex p = spline.degree();
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| 303 | const DenseIndex span = spline.span(u);
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| 304 |
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| 305 | const DenseIndex n = (std::min)(p, order);
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| 306 |
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| 307 | der.resize(Dimension,n+1);
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| 308 |
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| 309 | // Retrieve the basis function derivatives up to the desired order...
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| 310 | const BasisDerivativeType basis_func_ders = spline.template basisFunctionDerivatives<DerivativeOrder>(u, n+1);
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| 311 |
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| 312 | // ... and perform the linear combinations of the control points.
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| 313 | for (DenseIndex der_order=0; der_order<n+1; ++der_order)
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| 314 | {
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| 315 | const Replicate<BasisDerivativeRowXpr,Dimension,1> ctrl_weights( basis_func_ders.row(der_order) );
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| 316 | const Block<const ControlPointVectorType,Dimension,Order> ctrl_pts(spline.ctrls(),0,span-p,Dimension,p+1);
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| 317 | der.col(der_order) = (ctrl_weights * ctrl_pts).rowwise().sum();
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| 318 | }
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| 319 | }
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| 320 |
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| 321 | template <typename _Scalar, int _Dim, int _Degree>
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| 322 | typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::DerivativeType
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| 323 | Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const
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| 324 | {
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| 325 | typename SplineTraits< Spline >::DerivativeType res;
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| 326 | derivativesImpl(*this, u, order, res);
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| 327 | return res;
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| 328 | }
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| 329 |
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| 330 | template <typename _Scalar, int _Dim, int _Degree>
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| 331 | template <int DerivativeOrder>
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| 332 | typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::DerivativeType
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| 333 | Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const
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| 334 | {
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| 335 | typename SplineTraits< Spline, DerivativeOrder >::DerivativeType res;
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| 336 | derivativesImpl(*this, u, order, res);
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| 337 | return res;
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| 338 | }
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| 339 |
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| 340 | template <typename _Scalar, int _Dim, int _Degree>
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| 341 | typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::BasisVectorType
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| 342 | Spline<_Scalar, _Dim, _Degree>::basisFunctions(Scalar u) const
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| 343 | {
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| 344 | return Spline::BasisFunctions(u, degree(), knots());
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| 345 | }
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| 346 |
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| 347 | /* --------------------------------------------------------------------------------------------- */
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| 348 |
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| 349 | template <typename SplineType, typename DerivativeType>
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| 350 | void basisFunctionDerivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& N_)
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| 351 | {
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| 352 | enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };
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| 353 |
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| 354 | typedef typename SplineTraits<SplineType>::Scalar Scalar;
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| 355 | typedef typename SplineTraits<SplineType>::BasisVectorType BasisVectorType;
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| 356 | typedef typename SplineTraits<SplineType>::KnotVectorType KnotVectorType;
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| 357 |
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| 358 | const KnotVectorType& U = spline.knots();
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| 359 |
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| 360 | const DenseIndex p = spline.degree();
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| 361 | const DenseIndex span = spline.span(u);
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| 362 |
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| 363 | const DenseIndex n = (std::min)(p, order);
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| 364 |
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| 365 | N_.resize(n+1, p+1);
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| 366 |
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| 367 | BasisVectorType left = BasisVectorType::Zero(p+1);
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| 368 | BasisVectorType right = BasisVectorType::Zero(p+1);
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| 369 |
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| 370 | Matrix<Scalar,Order,Order> ndu(p+1,p+1);
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| 371 |
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| 372 | double saved, temp;
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| 373 |
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| 374 | ndu(0,0) = 1.0;
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| 375 |
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| 376 | DenseIndex j;
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| 377 | for (j=1; j<=p; ++j)
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| 378 | {
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| 379 | left[j] = u-U[span+1-j];
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| 380 | right[j] = U[span+j]-u;
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| 381 | saved = 0.0;
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| 382 |
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| 383 | for (DenseIndex r=0; r<j; ++r)
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| 384 | {
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| 385 | /* Lower triangle */
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| 386 | ndu(j,r) = right[r+1]+left[j-r];
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| 387 | temp = ndu(r,j-1)/ndu(j,r);
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| 388 | /* Upper triangle */
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| 389 | ndu(r,j) = static_cast<Scalar>(saved+right[r+1] * temp);
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| 390 | saved = left[j-r] * temp;
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| 391 | }
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| 392 |
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| 393 | ndu(j,j) = static_cast<Scalar>(saved);
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| 394 | }
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| 395 |
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| 396 | for (j = p; j>=0; --j)
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| 397 | N_(0,j) = ndu(j,p);
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| 398 |
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| 399 | // Compute the derivatives
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| 400 | DerivativeType a(n+1,p+1);
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| 401 | DenseIndex r=0;
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| 402 | for (; r<=p; ++r)
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| 403 | {
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| 404 | DenseIndex s1,s2;
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| 405 | s1 = 0; s2 = 1; // alternate rows in array a
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| 406 | a(0,0) = 1.0;
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| 407 |
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| 408 | // Compute the k-th derivative
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| 409 | for (DenseIndex k=1; k<=static_cast<DenseIndex>(n); ++k)
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| 410 | {
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| 411 | double d = 0.0;
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| 412 | DenseIndex rk,pk,j1,j2;
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| 413 | rk = r-k; pk = p-k;
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| 414 |
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| 415 | if (r>=k)
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| 416 | {
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| 417 | a(s2,0) = a(s1,0)/ndu(pk+1,rk);
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| 418 | d = a(s2,0)*ndu(rk,pk);
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| 419 | }
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| 420 |
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| 421 | if (rk>=-1) j1 = 1;
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| 422 | else j1 = -rk;
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| 423 |
|
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| 424 | if (r-1 <= pk) j2 = k-1;
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| 425 | else j2 = p-r;
|
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| 426 |
|
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| 427 | for (j=j1; j<=j2; ++j)
|
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| 428 | {
|
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| 429 | a(s2,j) = (a(s1,j)-a(s1,j-1))/ndu(pk+1,rk+j);
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| 430 | d += a(s2,j)*ndu(rk+j,pk);
|
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| 431 | }
|
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| 432 |
|
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| 433 | if (r<=pk)
|
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| 434 | {
|
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| 435 | a(s2,k) = -a(s1,k-1)/ndu(pk+1,r);
|
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| 436 | d += a(s2,k)*ndu(r,pk);
|
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| 437 | }
|
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| 438 |
|
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| 439 | N_(k,r) = static_cast<Scalar>(d);
|
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| 440 | j = s1; s1 = s2; s2 = j; // Switch rows
|
---|
| 441 | }
|
---|
| 442 | }
|
---|
| 443 |
|
---|
| 444 | /* Multiply through by the correct factors */
|
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| 445 | /* (Eq. [2.9]) */
|
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| 446 | r = p;
|
---|
| 447 | for (DenseIndex k=1; k<=static_cast<DenseIndex>(n); ++k)
|
---|
| 448 | {
|
---|
| 449 | for (DenseIndex j=p; j>=0; --j) N_(k,j) *= r;
|
---|
| 450 | r *= p-k;
|
---|
| 451 | }
|
---|
| 452 | }
|
---|
| 453 |
|
---|
| 454 | template <typename _Scalar, int _Dim, int _Degree>
|
---|
| 455 | typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::BasisDerivativeType
|
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| 456 | Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const
|
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| 457 | {
|
---|
| 458 | typename SplineTraits< Spline >::BasisDerivativeType der;
|
---|
| 459 | basisFunctionDerivativesImpl(*this, u, order, der);
|
---|
| 460 | return der;
|
---|
| 461 | }
|
---|
| 462 |
|
---|
| 463 | template <typename _Scalar, int _Dim, int _Degree>
|
---|
| 464 | template <int DerivativeOrder>
|
---|
| 465 | typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::BasisDerivativeType
|
---|
| 466 | Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const
|
---|
| 467 | {
|
---|
| 468 | typename SplineTraits< Spline, DerivativeOrder >::BasisDerivativeType der;
|
---|
| 469 | basisFunctionDerivativesImpl(*this, u, order, der);
|
---|
| 470 | return der;
|
---|
| 471 | }
|
---|
| 472 | }
|
---|
| 473 |
|
---|
| 474 | #endif // EIGEN_SPLINE_H
|
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