Squashed 'third_party/eigen/' content from commit 61d72f6
Change-Id: Iccc90fa0b55ab44037f018046d2fcffd90d9d025
git-subtree-dir: third_party/eigen
git-subtree-split: 61d72f6383cfa842868c53e30e087b0258177257
diff --git a/unsupported/Eigen/src/Splines/Spline.h b/unsupported/Eigen/src/Splines/Spline.h
new file mode 100644
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+++ b/unsupported/Eigen/src/Splines/Spline.h
@@ -0,0 +1,474 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 20010-2011 Hauke Heibel <hauke.heibel@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_SPLINE_H
+#define EIGEN_SPLINE_H
+
+#include "SplineFwd.h"
+
+namespace Eigen
+{
+ /**
+ * \ingroup Splines_Module
+ * \class Spline
+ * \brief A class representing multi-dimensional spline curves.
+ *
+ * The class represents B-splines with non-uniform knot vectors. Each control
+ * point of the B-spline is associated with a basis function
+ * \f{align*}
+ * C(u) & = \sum_{i=0}^{n}N_{i,p}(u)P_i
+ * \f}
+ *
+ * \tparam _Scalar The underlying data type (typically float or double)
+ * \tparam _Dim The curve dimension (e.g. 2 or 3)
+ * \tparam _Degree Per default set to Dynamic; could be set to the actual desired
+ * degree for optimization purposes (would result in stack allocation
+ * of several temporary variables).
+ **/
+ template <typename _Scalar, int _Dim, int _Degree>
+ class Spline
+ {
+ public:
+ typedef _Scalar Scalar; /*!< The spline curve's scalar type. */
+ enum { Dimension = _Dim /*!< The spline curve's dimension. */ };
+ enum { Degree = _Degree /*!< The spline curve's degree. */ };
+
+ /** \brief The point type the spline is representing. */
+ typedef typename SplineTraits<Spline>::PointType PointType;
+
+ /** \brief The data type used to store knot vectors. */
+ typedef typename SplineTraits<Spline>::KnotVectorType KnotVectorType;
+
+ /** \brief The data type used to store non-zero basis functions. */
+ typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType;
+
+ /** \brief The data type representing the spline's control points. */
+ typedef typename SplineTraits<Spline>::ControlPointVectorType ControlPointVectorType;
+
+ /**
+ * \brief Creates a (constant) zero spline.
+ * For Splines with dynamic degree, the resulting degree will be 0.
+ **/
+ Spline()
+ : m_knots(1, (Degree==Dynamic ? 2 : 2*Degree+2))
+ , m_ctrls(ControlPointVectorType::Zero(2,(Degree==Dynamic ? 1 : Degree+1)))
+ {
+ // in theory this code can go to the initializer list but it will get pretty
+ // much unreadable ...
+ enum { MinDegree = (Degree==Dynamic ? 0 : Degree) };
+ m_knots.template segment<MinDegree+1>(0) = Array<Scalar,1,MinDegree+1>::Zero();
+ m_knots.template segment<MinDegree+1>(MinDegree+1) = Array<Scalar,1,MinDegree+1>::Ones();
+ }
+
+ /**
+ * \brief Creates a spline from a knot vector and control points.
+ * \param knots The spline's knot vector.
+ * \param ctrls The spline's control point vector.
+ **/
+ template <typename OtherVectorType, typename OtherArrayType>
+ Spline(const OtherVectorType& knots, const OtherArrayType& ctrls) : m_knots(knots), m_ctrls(ctrls) {}
+
+ /**
+ * \brief Copy constructor for splines.
+ * \param spline The input spline.
+ **/
+ template <int OtherDegree>
+ Spline(const Spline<Scalar, Dimension, OtherDegree>& spline) :
+ m_knots(spline.knots()), m_ctrls(spline.ctrls()) {}
+
+ /**
+ * \brief Returns the knots of the underlying spline.
+ **/
+ const KnotVectorType& knots() const { return m_knots; }
+
+ /**
+ * \brief Returns the knots of the underlying spline.
+ **/
+ const ControlPointVectorType& ctrls() const { return m_ctrls; }
+
+ /**
+ * \brief Returns the spline value at a given site \f$u\f$.
+ *
+ * The function returns
+ * \f{align*}
+ * C(u) & = \sum_{i=0}^{n}N_{i,p}P_i
+ * \f}
+ *
+ * \param u Parameter \f$u \in [0;1]\f$ at which the spline is evaluated.
+ * \return The spline value at the given location \f$u\f$.
+ **/
+ PointType operator()(Scalar u) const;
+
+ /**
+ * \brief Evaluation of spline derivatives of up-to given order.
+ *
+ * The function returns
+ * \f{align*}
+ * \frac{d^i}{du^i}C(u) & = \sum_{i=0}^{n} \frac{d^i}{du^i} N_{i,p}(u)P_i
+ * \f}
+ * for i ranging between 0 and order.
+ *
+ * \param u Parameter \f$u \in [0;1]\f$ at which the spline derivative is evaluated.
+ * \param order The order up to which the derivatives are computed.
+ **/
+ typename SplineTraits<Spline>::DerivativeType
+ derivatives(Scalar u, DenseIndex order) const;
+
+ /**
+ * \copydoc Spline::derivatives
+ * Using the template version of this function is more efficieent since
+ * temporary objects are allocated on the stack whenever this is possible.
+ **/
+ template <int DerivativeOrder>
+ typename SplineTraits<Spline,DerivativeOrder>::DerivativeType
+ derivatives(Scalar u, DenseIndex order = DerivativeOrder) const;
+
+ /**
+ * \brief Computes the non-zero basis functions at the given site.
+ *
+ * Splines have local support and a point from their image is defined
+ * by exactly \f$p+1\f$ control points \f$P_i\f$ where \f$p\f$ is the
+ * spline degree.
+ *
+ * This function computes the \f$p+1\f$ non-zero basis function values
+ * for a given parameter value \f$u\f$. It returns
+ * \f{align*}{
+ * N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
+ * \f}
+ *
+ * \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis functions
+ * are computed.
+ **/
+ typename SplineTraits<Spline>::BasisVectorType
+ basisFunctions(Scalar u) const;
+
+ /**
+ * \brief Computes the non-zero spline basis function derivatives up to given order.
+ *
+ * The function computes
+ * \f{align*}{
+ * \frac{d^i}{du^i} N_{i,p}(u), \hdots, \frac{d^i}{du^i} N_{i+p+1,p}(u)
+ * \f}
+ * with i ranging from 0 up to the specified order.
+ *
+ * \param u Parameter \f$u \in [0;1]\f$ at which the non-zero basis function
+ * derivatives are computed.
+ * \param order The order up to which the basis function derivatives are computes.
+ **/
+ typename SplineTraits<Spline>::BasisDerivativeType
+ basisFunctionDerivatives(Scalar u, DenseIndex order) const;
+
+ /**
+ * \copydoc Spline::basisFunctionDerivatives
+ * Using the template version of this function is more efficieent since
+ * temporary objects are allocated on the stack whenever this is possible.
+ **/
+ template <int DerivativeOrder>
+ typename SplineTraits<Spline,DerivativeOrder>::BasisDerivativeType
+ basisFunctionDerivatives(Scalar u, DenseIndex order = DerivativeOrder) const;
+
+ /**
+ * \brief Returns the spline degree.
+ **/
+ DenseIndex degree() const;
+
+ /**
+ * \brief Returns the span within the knot vector in which u is falling.
+ * \param u The site for which the span is determined.
+ **/
+ DenseIndex span(Scalar u) const;
+
+ /**
+ * \brief Computes the spang within the provided knot vector in which u is falling.
+ **/
+ static DenseIndex Span(typename SplineTraits<Spline>::Scalar u, DenseIndex degree, const typename SplineTraits<Spline>::KnotVectorType& knots);
+
+ /**
+ * \brief Returns the spline's non-zero basis functions.
+ *
+ * The function computes and returns
+ * \f{align*}{
+ * N_{i,p}(u), \hdots, N_{i+p+1,p}(u)
+ * \f}
+ *
+ * \param u The site at which the basis functions are computed.
+ * \param degree The degree of the underlying spline.
+ * \param knots The underlying spline's knot vector.
+ **/
+ static BasisVectorType BasisFunctions(Scalar u, DenseIndex degree, const KnotVectorType& knots);
+
+
+ private:
+ KnotVectorType m_knots; /*!< Knot vector. */
+ ControlPointVectorType m_ctrls; /*!< Control points. */
+ };
+
+ template <typename _Scalar, int _Dim, int _Degree>
+ DenseIndex Spline<_Scalar, _Dim, _Degree>::Span(
+ typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::Scalar u,
+ DenseIndex degree,
+ const typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::KnotVectorType& knots)
+ {
+ // Piegl & Tiller, "The NURBS Book", A2.1 (p. 68)
+ if (u <= knots(0)) return degree;
+ const Scalar* pos = std::upper_bound(knots.data()+degree-1, knots.data()+knots.size()-degree-1, u);
+ return static_cast<DenseIndex>( std::distance(knots.data(), pos) - 1 );
+ }
+
+ template <typename _Scalar, int _Dim, int _Degree>
+ typename Spline<_Scalar, _Dim, _Degree>::BasisVectorType
+ Spline<_Scalar, _Dim, _Degree>::BasisFunctions(
+ typename Spline<_Scalar, _Dim, _Degree>::Scalar u,
+ DenseIndex degree,
+ const typename Spline<_Scalar, _Dim, _Degree>::KnotVectorType& knots)
+ {
+ typedef typename Spline<_Scalar, _Dim, _Degree>::BasisVectorType BasisVectorType;
+
+ const DenseIndex p = degree;
+ const DenseIndex i = Spline::Span(u, degree, knots);
+
+ const KnotVectorType& U = knots;
+
+ BasisVectorType left(p+1); left(0) = Scalar(0);
+ BasisVectorType right(p+1); right(0) = Scalar(0);
+
+ VectorBlock<BasisVectorType,Degree>(left,1,p) = u - VectorBlock<const KnotVectorType,Degree>(U,i+1-p,p).reverse();
+ VectorBlock<BasisVectorType,Degree>(right,1,p) = VectorBlock<const KnotVectorType,Degree>(U,i+1,p) - u;
+
+ BasisVectorType N(1,p+1);
+ N(0) = Scalar(1);
+ for (DenseIndex j=1; j<=p; ++j)
+ {
+ Scalar saved = Scalar(0);
+ for (DenseIndex r=0; r<j; r++)
+ {
+ const Scalar tmp = N(r)/(right(r+1)+left(j-r));
+ N[r] = saved + right(r+1)*tmp;
+ saved = left(j-r)*tmp;
+ }
+ N(j) = saved;
+ }
+ return N;
+ }
+
+ template <typename _Scalar, int _Dim, int _Degree>
+ DenseIndex Spline<_Scalar, _Dim, _Degree>::degree() const
+ {
+ if (_Degree == Dynamic)
+ return m_knots.size() - m_ctrls.cols() - 1;
+ else
+ return _Degree;
+ }
+
+ template <typename _Scalar, int _Dim, int _Degree>
+ DenseIndex Spline<_Scalar, _Dim, _Degree>::span(Scalar u) const
+ {
+ return Spline::Span(u, degree(), knots());
+ }
+
+ template <typename _Scalar, int _Dim, int _Degree>
+ typename Spline<_Scalar, _Dim, _Degree>::PointType Spline<_Scalar, _Dim, _Degree>::operator()(Scalar u) const
+ {
+ enum { Order = SplineTraits<Spline>::OrderAtCompileTime };
+
+ const DenseIndex span = this->span(u);
+ const DenseIndex p = degree();
+ const BasisVectorType basis_funcs = basisFunctions(u);
+
+ const Replicate<BasisVectorType,Dimension,1> ctrl_weights(basis_funcs);
+ const Block<const ControlPointVectorType,Dimension,Order> ctrl_pts(ctrls(),0,span-p,Dimension,p+1);
+ return (ctrl_weights * ctrl_pts).rowwise().sum();
+ }
+
+ /* --------------------------------------------------------------------------------------------- */
+
+ template <typename SplineType, typename DerivativeType>
+ void derivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& der)
+ {
+ enum { Dimension = SplineTraits<SplineType>::Dimension };
+ enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };
+ enum { DerivativeOrder = DerivativeType::ColsAtCompileTime };
+
+ typedef typename SplineTraits<SplineType>::ControlPointVectorType ControlPointVectorType;
+ typedef typename SplineTraits<SplineType,DerivativeOrder>::BasisDerivativeType BasisDerivativeType;
+ typedef typename BasisDerivativeType::ConstRowXpr BasisDerivativeRowXpr;
+
+ const DenseIndex p = spline.degree();
+ const DenseIndex span = spline.span(u);
+
+ const DenseIndex n = (std::min)(p, order);
+
+ der.resize(Dimension,n+1);
+
+ // Retrieve the basis function derivatives up to the desired order...
+ const BasisDerivativeType basis_func_ders = spline.template basisFunctionDerivatives<DerivativeOrder>(u, n+1);
+
+ // ... and perform the linear combinations of the control points.
+ for (DenseIndex der_order=0; der_order<n+1; ++der_order)
+ {
+ const Replicate<BasisDerivativeRowXpr,Dimension,1> ctrl_weights( basis_func_ders.row(der_order) );
+ const Block<const ControlPointVectorType,Dimension,Order> ctrl_pts(spline.ctrls(),0,span-p,Dimension,p+1);
+ der.col(der_order) = (ctrl_weights * ctrl_pts).rowwise().sum();
+ }
+ }
+
+ template <typename _Scalar, int _Dim, int _Degree>
+ typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::DerivativeType
+ Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const
+ {
+ typename SplineTraits< Spline >::DerivativeType res;
+ derivativesImpl(*this, u, order, res);
+ return res;
+ }
+
+ template <typename _Scalar, int _Dim, int _Degree>
+ template <int DerivativeOrder>
+ typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::DerivativeType
+ Spline<_Scalar, _Dim, _Degree>::derivatives(Scalar u, DenseIndex order) const
+ {
+ typename SplineTraits< Spline, DerivativeOrder >::DerivativeType res;
+ derivativesImpl(*this, u, order, res);
+ return res;
+ }
+
+ template <typename _Scalar, int _Dim, int _Degree>
+ typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::BasisVectorType
+ Spline<_Scalar, _Dim, _Degree>::basisFunctions(Scalar u) const
+ {
+ return Spline::BasisFunctions(u, degree(), knots());
+ }
+
+ /* --------------------------------------------------------------------------------------------- */
+
+ template <typename SplineType, typename DerivativeType>
+ void basisFunctionDerivativesImpl(const SplineType& spline, typename SplineType::Scalar u, DenseIndex order, DerivativeType& N_)
+ {
+ enum { Order = SplineTraits<SplineType>::OrderAtCompileTime };
+
+ typedef typename SplineTraits<SplineType>::Scalar Scalar;
+ typedef typename SplineTraits<SplineType>::BasisVectorType BasisVectorType;
+ typedef typename SplineTraits<SplineType>::KnotVectorType KnotVectorType;
+
+ const KnotVectorType& U = spline.knots();
+
+ const DenseIndex p = spline.degree();
+ const DenseIndex span = spline.span(u);
+
+ const DenseIndex n = (std::min)(p, order);
+
+ N_.resize(n+1, p+1);
+
+ BasisVectorType left = BasisVectorType::Zero(p+1);
+ BasisVectorType right = BasisVectorType::Zero(p+1);
+
+ Matrix<Scalar,Order,Order> ndu(p+1,p+1);
+
+ double saved, temp;
+
+ ndu(0,0) = 1.0;
+
+ DenseIndex j;
+ for (j=1; j<=p; ++j)
+ {
+ left[j] = u-U[span+1-j];
+ right[j] = U[span+j]-u;
+ saved = 0.0;
+
+ for (DenseIndex r=0; r<j; ++r)
+ {
+ /* Lower triangle */
+ ndu(j,r) = right[r+1]+left[j-r];
+ temp = ndu(r,j-1)/ndu(j,r);
+ /* Upper triangle */
+ ndu(r,j) = static_cast<Scalar>(saved+right[r+1] * temp);
+ saved = left[j-r] * temp;
+ }
+
+ ndu(j,j) = static_cast<Scalar>(saved);
+ }
+
+ for (j = p; j>=0; --j)
+ N_(0,j) = ndu(j,p);
+
+ // Compute the derivatives
+ DerivativeType a(n+1,p+1);
+ DenseIndex r=0;
+ for (; r<=p; ++r)
+ {
+ DenseIndex s1,s2;
+ s1 = 0; s2 = 1; // alternate rows in array a
+ a(0,0) = 1.0;
+
+ // Compute the k-th derivative
+ for (DenseIndex k=1; k<=static_cast<DenseIndex>(n); ++k)
+ {
+ double d = 0.0;
+ DenseIndex rk,pk,j1,j2;
+ rk = r-k; pk = p-k;
+
+ if (r>=k)
+ {
+ a(s2,0) = a(s1,0)/ndu(pk+1,rk);
+ d = a(s2,0)*ndu(rk,pk);
+ }
+
+ if (rk>=-1) j1 = 1;
+ else j1 = -rk;
+
+ if (r-1 <= pk) j2 = k-1;
+ else j2 = p-r;
+
+ for (j=j1; j<=j2; ++j)
+ {
+ a(s2,j) = (a(s1,j)-a(s1,j-1))/ndu(pk+1,rk+j);
+ d += a(s2,j)*ndu(rk+j,pk);
+ }
+
+ if (r<=pk)
+ {
+ a(s2,k) = -a(s1,k-1)/ndu(pk+1,r);
+ d += a(s2,k)*ndu(r,pk);
+ }
+
+ N_(k,r) = static_cast<Scalar>(d);
+ j = s1; s1 = s2; s2 = j; // Switch rows
+ }
+ }
+
+ /* Multiply through by the correct factors */
+ /* (Eq. [2.9]) */
+ r = p;
+ for (DenseIndex k=1; k<=static_cast<DenseIndex>(n); ++k)
+ {
+ for (DenseIndex j=p; j>=0; --j) N_(k,j) *= r;
+ r *= p-k;
+ }
+ }
+
+ template <typename _Scalar, int _Dim, int _Degree>
+ typename SplineTraits< Spline<_Scalar, _Dim, _Degree> >::BasisDerivativeType
+ Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const
+ {
+ typename SplineTraits< Spline >::BasisDerivativeType der;
+ basisFunctionDerivativesImpl(*this, u, order, der);
+ return der;
+ }
+
+ template <typename _Scalar, int _Dim, int _Degree>
+ template <int DerivativeOrder>
+ typename SplineTraits< Spline<_Scalar, _Dim, _Degree>, DerivativeOrder >::BasisDerivativeType
+ Spline<_Scalar, _Dim, _Degree>::basisFunctionDerivatives(Scalar u, DenseIndex order) const
+ {
+ typename SplineTraits< Spline, DerivativeOrder >::BasisDerivativeType der;
+ basisFunctionDerivativesImpl(*this, u, order, der);
+ return der;
+ }
+}
+
+#endif // EIGEN_SPLINE_H