Eigen/src/Geometry/OrthoMethods.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2006-2008 Benoit Jacob <jacob.benoit.1@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_ORTHOMETHODS_H
#define EIGEN_ORTHOMETHODS_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

namespace internal {

// Vector3 version (default)
template <typename Derived, typename OtherDerived, int Size>
struct cross_impl {
  using Scalar = typename ScalarBinaryOpTraits<typename internal::traits<Derived>::Scalar,
                                               typename internal::traits<OtherDerived>::Scalar>::ReturnType;
  using return_type = Matrix<Scalar, MatrixBase<Derived>::RowsAtCompileTime, MatrixBase<Derived>::ColsAtCompileTime>;

  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE return_type run(const MatrixBase<Derived>& first,
                                                               const MatrixBase<OtherDerived>& second) {
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(Derived, 3)
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived, 3)

    // Note that there is no need for an expression here since the compiler
    // optimizes such a small temporary very well (even within a complex expression)
    typename internal::nested_eval<Derived, 2>::type lhs(first.derived());
    typename internal::nested_eval<OtherDerived, 2>::type rhs(second.derived());
    return return_type(numext::conj(lhs.coeff(1) * rhs.coeff(2) - lhs.coeff(2) * rhs.coeff(1)),
                       numext::conj(lhs.coeff(2) * rhs.coeff(0) - lhs.coeff(0) * rhs.coeff(2)),
                       numext::conj(lhs.coeff(0) * rhs.coeff(1) - lhs.coeff(1) * rhs.coeff(0)));
  }
};

// Vector2 version
template <typename Derived, typename OtherDerived>
struct cross_impl<Derived, OtherDerived, 2> {
  using Scalar = typename ScalarBinaryOpTraits<typename internal::traits<Derived>::Scalar,
                                               typename internal::traits<OtherDerived>::Scalar>::ReturnType;
  using return_type = Scalar;

  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE return_type run(const MatrixBase<Derived>& first,
                                                               const MatrixBase<OtherDerived>& second) {
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(Derived, 2)
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived, 2)
    typename internal::nested_eval<Derived, 2>::type lhs(first.derived());
    typename internal::nested_eval<OtherDerived, 2>::type rhs(second.derived());
    return numext::conj(lhs.coeff(0) * rhs.coeff(1) - lhs.coeff(1) * rhs.coeff(0));
  }
};

}  // end namespace internal

/** \geometry_module \ingroup Geometry_Module
 *
 * \returns the cross product of \c *this and \a other. This is either a scalar for size-2 vectors or a size-3 vector
 * for size-3 vectors.
 *
 * This method is implemented for two different cases: between vectors of fixed size 2 and between vectors of fixed
 * size 3.
 *
 * For vectors of size 3, the output is simply the traditional cross product.
 *
 * For vectors of size 2, the output is a scalar.
 * Given vectors \f$ v = \begin{bmatrix} v_1 & v_2 \end{bmatrix} \f$ and \f$ w = \begin{bmatrix} w_1 & w_2 \end{bmatrix}
 * \f$, the result is simply \f$ v\times w = \overline{v_1 w_2 - v_2 w_1} = \text{conj}\left|\begin{smallmatrix} v_1 &
 * w_1 \\ v_2 & w_2 \end{smallmatrix}\right| \f$; or, to put it differently, it is the third coordinate of the cross
 * product of \f$ \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix} \f$ and \f$ \begin{bmatrix} w_1 & w_2 & w_3
 * \end{bmatrix} \f$. For real-valued inputs, the result can be interpreted as the signed area of a parallelogram
 * spanned by the two vectors.
 *
 * \note With complex numbers, the cross product is implemented as
 * \f[ (\mathbf{a}+i\mathbf{b}) \times (\mathbf{c}+i\mathbf{d}) = (\mathbf{a} \times \mathbf{c} - \mathbf{b} \times
 * \mathbf{d}) - i(\mathbf{a} \times \mathbf{d} + \mathbf{b} \times \mathbf{c}).\f]
 * This definition preserves the orthogonality condition that \f$\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v}) =
 * \mathbf{v} \cdot (\mathbf{u} \times \mathbf{v}) = 0\f$.
 *
 * \sa MatrixBase::cross3()
 */
template <typename Derived>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename internal::cross_impl<Derived, OtherDerived>::return_type
MatrixBase<Derived>::cross(const MatrixBase<OtherDerived>& other) const {
  return internal::cross_impl<Derived, OtherDerived>::run(*this, other);
}

namespace internal {

template <int Arch, typename VectorLhs, typename VectorRhs, typename Scalar = typename VectorLhs::Scalar,
          bool Vectorizable =
              bool((int(evaluator<VectorLhs>::Flags) & int(evaluator<VectorRhs>::Flags)) & PacketAccessBit)>
struct cross3_impl {
  EIGEN_DEVICE_FUNC static inline typename internal::plain_matrix_type<VectorLhs>::type run(const VectorLhs& lhs,
                                                                                            const VectorRhs& rhs) {
    return typename internal::plain_matrix_type<VectorLhs>::type(
        numext::conj(lhs.coeff(1) * rhs.coeff(2) - lhs.coeff(2) * rhs.coeff(1)),
        numext::conj(lhs.coeff(2) * rhs.coeff(0) - lhs.coeff(0) * rhs.coeff(2)),
        numext::conj(lhs.coeff(0) * rhs.coeff(1) - lhs.coeff(1) * rhs.coeff(0)), 0);
  }
};

}  // namespace internal

/** \geometry_module \ingroup Geometry_Module
 *
 * \returns the cross product of \c *this and \a other using only the x, y, and z coefficients
 *
 * The size of \c *this and \a other must be four. This function is especially useful
 * when using 4D vectors instead of 3D ones to get advantage of SSE/AltiVec vectorization.
 *
 * \sa MatrixBase::cross()
 */
template <typename Derived>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC inline typename MatrixBase<Derived>::PlainObject MatrixBase<Derived>::cross3(
    const MatrixBase<OtherDerived>& other) const {
  EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(Derived, 4)
  EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived, 4)

  using DerivedNested = typename internal::nested_eval<Derived, 2>::type;
  using OtherDerivedNested = typename internal::nested_eval<OtherDerived, 2>::type;
  DerivedNested lhs(derived());
  OtherDerivedNested rhs(other.derived());

  return internal::cross3_impl<Architecture::Target, internal::remove_all_t<DerivedNested>,
                               internal::remove_all_t<OtherDerivedNested>>::run(lhs, rhs);
}

/** \geometry_module \ingroup Geometry_Module
 *
 * \returns a matrix expression of the cross product of each column or row
 * of the referenced expression with the \a other vector.
 *
 * The referenced matrix must have one dimension equal to 3.
 * The result matrix has the same dimensions as the referenced one.
 *
 * \sa MatrixBase::cross() */
template <typename ExpressionType, int Direction>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC const typename VectorwiseOp<ExpressionType, Direction>::CrossReturnType
VectorwiseOp<ExpressionType, Direction>::cross(const MatrixBase<OtherDerived>& other) const {
  EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived, 3)
  EIGEN_STATIC_ASSERT(
      (std::is_same<Scalar, typename OtherDerived::Scalar>::value),
      YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)

  typename internal::nested_eval<ExpressionType, 2>::type mat(_expression());
  typename internal::nested_eval<OtherDerived, 2>::type vec(other.derived());

  CrossReturnType res(_expression().rows(), _expression().cols());
  EIGEN_IF_CONSTEXPR (Direction == Vertical) {
    eigen_assert(CrossReturnType::RowsAtCompileTime == 3 && "the matrix must have exactly 3 rows");
    res.row(0) = (mat.row(1) * vec.coeff(2) - mat.row(2) * vec.coeff(1)).conjugate();
    res.row(1) = (mat.row(2) * vec.coeff(0) - mat.row(0) * vec.coeff(2)).conjugate();
    res.row(2) = (mat.row(0) * vec.coeff(1) - mat.row(1) * vec.coeff(0)).conjugate();
  } else {
    eigen_assert(CrossReturnType::ColsAtCompileTime == 3 && "the matrix must have exactly 3 columns");
    res.col(0) = (mat.col(1) * vec.coeff(2) - mat.col(2) * vec.coeff(1)).conjugate();
    res.col(1) = (mat.col(2) * vec.coeff(0) - mat.col(0) * vec.coeff(2)).conjugate();
    res.col(2) = (mat.col(0) * vec.coeff(1) - mat.col(1) * vec.coeff(0)).conjugate();
  }
  return res;
}

namespace internal {

template <typename Derived, int Size = Derived::SizeAtCompileTime>
struct unitOrthogonal_selector {
  using VectorType = typename plain_matrix_type<Derived>::type;
  using Scalar = typename traits<Derived>::Scalar;
  using RealScalar = typename NumTraits<Scalar>::Real;
  using Vector2 = Matrix<Scalar, 2, 1>;
  EIGEN_DEVICE_FUNC static inline VectorType run(const Derived& src) {
    VectorType perp = VectorType::Zero(src.size());
    Index maxi = 0;
    Index sndi = 0;
    src.cwiseAbs().maxCoeff(&maxi);
    if (maxi == 0) sndi = 1;
    RealScalar invnm = RealScalar(1) / (Vector2() << src.coeff(sndi), src.coeff(maxi)).finished().norm();
    perp.coeffRef(maxi) = -numext::conj(src.coeff(sndi)) * invnm;
    perp.coeffRef(sndi) = numext::conj(src.coeff(maxi)) * invnm;

    return perp;
  }
};

template <typename Derived>
struct unitOrthogonal_selector<Derived, 3> {
  using VectorType = typename plain_matrix_type<Derived>::type;
  using Scalar = typename traits<Derived>::Scalar;
  using RealScalar = typename NumTraits<Scalar>::Real;
  EIGEN_DEVICE_FUNC static inline VectorType run(const Derived& src) {
    VectorType perp;
    /* Let us compute the crossed product of *this with a vector
     * that is not too close to being collinear to *this.
     */

    /* unless the x and y coords are both close to zero, we can
     * simply take ( -y, x, 0 ) and normalize it.
     */
    if ((!isMuchSmallerThan(src.x(), src.z())) || (!isMuchSmallerThan(src.y(), src.z()))) {
      RealScalar invnm = RealScalar(1) / src.template head<2>().norm();
      perp.coeffRef(0) = -numext::conj(src.y()) * invnm;
      perp.coeffRef(1) = numext::conj(src.x()) * invnm;
      perp.coeffRef(2) = 0;
    }
    /* if both x and y are close to zero, then the vector is close
     * to the z-axis, so it's far from collinear to the x-axis for instance.
     * So we take the crossed product with (1,0,0) and normalize it.
     */
    else {
      RealScalar invnm = RealScalar(1) / src.template tail<2>().norm();
      perp.coeffRef(0) = 0;
      perp.coeffRef(1) = -numext::conj(src.z()) * invnm;
      perp.coeffRef(2) = numext::conj(src.y()) * invnm;
    }

    return perp;
  }
};

template <typename Derived>
struct unitOrthogonal_selector<Derived, 2> {
  using VectorType = typename plain_matrix_type<Derived>::type;
  EIGEN_DEVICE_FUNC static inline VectorType run(const Derived& src) {
    return VectorType(-numext::conj(src.y()), numext::conj(src.x())).normalized();
  }
};

}  // end namespace internal

/** \geometry_module \ingroup Geometry_Module
 *
 * \returns a unit vector which is orthogonal to \c *this
 *
 * The size of \c *this must be at least 2. If the size is exactly 2 at compile time,
 * then the returned vector is a counter-clockwise rotation of \c *this, i.e., (-y,x).normalized().
 * A vector whose size is only known at runtime takes the generic code path even when its size is 2,
 * and then returns the clockwise rotation (y,-x).normalized() if |y| > |x|.
 *
 * \sa cross()
 */
template <typename Derived>
EIGEN_DEVICE_FUNC typename MatrixBase<Derived>::PlainObject MatrixBase<Derived>::unitOrthogonal() const {
  EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
  return internal::unitOrthogonal_selector<Derived>::run(derived());
}

}  // end namespace Eigen

#endif  // EIGEN_ORTHOMETHODS_H