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/.
// SPDX-License-Identifier: MPL-2.0
 
#ifndef EIGEN_SPLINE_H
#define EIGEN_SPLINE_H
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
#include "SplineFwd.h"
 
namespace Eigen {
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>::ParameterVectorType ParameterVectorType;

  typedef typename SplineTraits<Spline>::BasisVectorType BasisVectorType;

  typedef typename SplineTraits<Spline>::BasisDerivativeType BasisDerivativeType;

  typedef typename SplineTraits<Spline>::ControlPointVectorType ControlPointVectorType;

  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();
  }

  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 KnotVectorType& knots() const { return m_knots; }

  const ControlPointVectorType& ctrls() const { return m_ctrls; }

  PointType operator()(Scalar u) const;

  typename SplineTraits<Spline>::DerivativeType derivatives(Scalar u, DenseIndex order) const;

  template <int DerivativeOrder>
  typename SplineTraits<Spline, DerivativeOrder>::DerivativeType derivatives(Scalar u,
                                                                             DenseIndex order = DerivativeOrder) const;

  typename SplineTraits<Spline>::BasisVectorType basisFunctions(Scalar u) const;

  typename SplineTraits<Spline>::BasisDerivativeType basisFunctionDerivatives(Scalar u, DenseIndex order) const;

  template <int DerivativeOrder>
  typename SplineTraits<Spline, DerivativeOrder>::BasisDerivativeType basisFunctionDerivatives(
      Scalar u, DenseIndex order = DerivativeOrder) const;

  DenseIndex degree() const;

  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);

  static BasisDerivativeType BasisFunctionDerivatives(const Scalar u, const DenseIndex order, const DenseIndex degree,
                                                      const KnotVectorType& knots);
 
 private:
  KnotVectorType m_knots;         
  ControlPointVectorType m_ctrls; 
 
  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) {
  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 {
  EIGEN_IF_CONSTEXPR (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 };
 
  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);
 
  Scalar saved, temp;  // FIXME: These were double instead of Scalar. Was there a reason for that?
 
  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) {
      Scalar 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;
}
}  // namespace Eigen
 
#endif  // EIGEN_SPLINE_H