Eigen/src/Core/SkewSymmetricMatrix3.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2007-2009 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/.

#ifndef EIGEN_SKEWSYMMETRICMATRIX3_H
#define EIGEN_SKEWSYMMETRICMATRIX3_H

#include "./InternalHeaderCheck.h"

namespace Eigen {

/** \class SkewSymmetricBase
 * \ingroup Core_Module
 *
 * \brief Base class for skew symmetric matrices and expressions
 *
 * This is the base class that is inherited by SkewSymmetricMatrix3 and related expression
 * types, which internally use a three vector for storing the entries. SkewSymmetric
 * types always represent square three times three matrices.
 *
 * This implementations follows class DiagonalMatrix
 *
 * \tparam Derived is the derived type, a SkewSymmetricMatrix3 or SkewSymmetricWrapper.
 *
 * \sa class SkewSymmetricMatrix3, class SkewSymmetricWrapper
 */
template<typename Derived>
class SkewSymmetricBase : public EigenBase<Derived>
{
  public:
    typedef typename internal::traits<Derived>::SkewSymmetricVectorType SkewSymmetricVectorType;
    typedef typename SkewSymmetricVectorType::Scalar Scalar;
    typedef typename SkewSymmetricVectorType::RealScalar RealScalar;
    typedef typename internal::traits<Derived>::StorageKind StorageKind;
    typedef typename internal::traits<Derived>::StorageIndex StorageIndex;

    enum {
      RowsAtCompileTime = SkewSymmetricVectorType::SizeAtCompileTime,
      ColsAtCompileTime = SkewSymmetricVectorType::SizeAtCompileTime,
      MaxRowsAtCompileTime = SkewSymmetricVectorType::MaxSizeAtCompileTime,
      MaxColsAtCompileTime = SkewSymmetricVectorType::MaxSizeAtCompileTime,
      IsVectorAtCompileTime = 0,
      Flags = NoPreferredStorageOrderBit
    };

    typedef Matrix<Scalar, RowsAtCompileTime, ColsAtCompileTime, 0, MaxRowsAtCompileTime, MaxColsAtCompileTime> DenseMatrixType;
    typedef DenseMatrixType DenseType;
    typedef SkewSymmetricMatrix3<Scalar> PlainObject;

    /** \returns a reference to the derived object. */
    EIGEN_DEVICE_FUNC
    inline const Derived& derived() const { return *static_cast<const Derived*>(this); }
    /** \returns a const reference to the derived object. */
    EIGEN_DEVICE_FUNC
    inline Derived& derived() { return *static_cast<Derived*>(this); }

    /**
     * Constructs a dense matrix from \c *this. Note, this directly returns a dense matrix type,
     * not an expression.
     * \returns A dense matrix, with its entries set from the the derived object. */
    EIGEN_DEVICE_FUNC
    DenseMatrixType toDenseMatrix() const { return derived(); }

    /** Determinant vanishes */
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR
    inline Scalar determinant() const { return 0; }

    /** A.transpose() = -A */
    EIGEN_DEVICE_FUNC
    PlainObject transpose() const { return (-vector()).asSkewSymmetric(); }

    /** \returns the exponential of this matrix using Rodrigues’ formula */
    EIGEN_DEVICE_FUNC
    DenseMatrixType exponential() const {
      DenseMatrixType retVal = DenseMatrixType::Identity();
      const SkewSymmetricVectorType& v = vector();
      if (v.isZero()) {
        return retVal;
      }
      const Scalar norm2 = v.squaredNorm();
      const Scalar norm = numext::sqrt(norm2);
      retVal += ((((1 - numext::cos(norm))/norm2)*derived())*derived()) + (numext::sin(norm)/norm)*derived().toDenseMatrix();
      return retVal;
    }

    /** \returns a reference to the derived object's vector of coefficients. */
    EIGEN_DEVICE_FUNC
    inline const SkewSymmetricVectorType& vector() const { return derived().vector(); }
    /** \returns a const reference to the derived object's vector of coefficients. */
    EIGEN_DEVICE_FUNC
    inline SkewSymmetricVectorType& vector() { return derived().vector(); }

    /** \returns the number of rows. */
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR 
    inline Index rows() const { return 3; }
    /** \returns the number of columns. */
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR 
    inline Index cols() const { return 3; }

    /** \returns the matrix product of \c *this by the dense matrix, \a matrix */
    template<typename MatrixDerived>
    EIGEN_DEVICE_FUNC
    Product<Derived,MatrixDerived,LazyProduct>
    operator*(const MatrixBase<MatrixDerived> &matrix) const
    {
      return Product<Derived, MatrixDerived, LazyProduct>(derived(), matrix.derived());
    }

    /** \returns the matrix product of \c *this by the skew symmetric matrix, \a matrix */
    template<typename MatrixDerived>
    EIGEN_DEVICE_FUNC
    Product<Derived,MatrixDerived,LazyProduct>
    operator*(const SkewSymmetricBase<MatrixDerived> &matrix) const
    {
      return Product<Derived, MatrixDerived, LazyProduct>(derived(), matrix.derived());
    }

    template <typename OtherDerived>
    using SkewSymmetricProductReturnType = SkewSymmetricWrapper<const EIGEN_CWISE_BINARY_RETURN_TYPE(
        SkewSymmetricVectorType, typename OtherDerived::SkewSymmetricVectorType, product)>;

    /** \returns the wedge product of \c *this by the skew symmetric matrix \a other
     *  A wedge B = AB - BA */
    template <typename OtherDerived>
    EIGEN_DEVICE_FUNC SkewSymmetricProductReturnType<OtherDerived> wedge(
        const SkewSymmetricBase<OtherDerived>& other) const {
      return vector().cross(other.vector()).asSkewSymmetric();
    }

    using SkewSymmetricScaleReturnType =
        SkewSymmetricWrapper<const EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(SkewSymmetricVectorType, Scalar, product)>;

    /** \returns the product of \c *this by the scalar \a scalar */
    EIGEN_DEVICE_FUNC
    inline SkewSymmetricScaleReturnType operator*(const Scalar& scalar) const {
      return (vector() * scalar).asSkewSymmetric();
    }

    using ScaleSkewSymmetricReturnType =
        SkewSymmetricWrapper<const EIGEN_SCALAR_BINARYOP_EXPR_RETURN_TYPE(Scalar, SkewSymmetricVectorType, product)>;

    /** \returns the product of a scalar and the skew symmetric matrix \a other */
    EIGEN_DEVICE_FUNC
    friend inline ScaleSkewSymmetricReturnType operator*(const Scalar& scalar, const SkewSymmetricBase& other) {
      return (scalar * other.vector()).asSkewSymmetric();
    }

    template <typename OtherDerived>
    using SkewSymmetricSumReturnType = SkewSymmetricWrapper<const EIGEN_CWISE_BINARY_RETURN_TYPE(
        SkewSymmetricVectorType, typename OtherDerived::SkewSymmetricVectorType, sum)>;

    /** \returns the sum of \c *this and the skew symmetric matrix \a other */
    template <typename OtherDerived>
    EIGEN_DEVICE_FUNC inline SkewSymmetricSumReturnType<OtherDerived> operator+(
        const SkewSymmetricBase<OtherDerived>& other) const {
      return (vector() + other.vector()).asSkewSymmetric();
    }

    template <typename OtherDerived>
    using SkewSymmetricDifferenceReturnType = SkewSymmetricWrapper<const EIGEN_CWISE_BINARY_RETURN_TYPE(
        SkewSymmetricVectorType, typename OtherDerived::SkewSymmetricVectorType, difference)>;

    /** \returns the difference of \c *this and the skew symmetric matrix \a other */
    template <typename OtherDerived>
    EIGEN_DEVICE_FUNC inline SkewSymmetricDifferenceReturnType<OtherDerived> operator-(
        const SkewSymmetricBase<OtherDerived>& other) const {
      return (vector() - other.vector()).asSkewSymmetric();
    }
};

/** \class SkewSymmetricMatrix3
 * \ingroup Core_Module
 *
 * \brief Represents a 3x3 skew symmetric matrix with its storage
 *
 * \tparam Scalar_ the type of coefficients
 *
 * \sa class SkewSymmetricBase, class SkewSymmetricWrapper
 */

namespace internal {
template<typename Scalar_>
struct traits<SkewSymmetricMatrix3<Scalar_> >
 : traits<Matrix<Scalar_,3,3,0,3,3> >
{
  typedef Matrix<Scalar_,3,1,0,3,1> SkewSymmetricVectorType;
  typedef SkewSymmetricShape StorageKind;
  enum {
    Flags = LvalueBit | NoPreferredStorageOrderBit | NestByRefBit
  };
};
}
template<typename Scalar_>
class SkewSymmetricMatrix3
  : public SkewSymmetricBase<SkewSymmetricMatrix3<Scalar_> >
{
  public:
    #ifndef EIGEN_PARSED_BY_DOXYGEN
    typedef typename internal::traits<SkewSymmetricMatrix3>::SkewSymmetricVectorType SkewSymmetricVectorType;
    typedef const SkewSymmetricMatrix3& Nested;
    typedef Scalar_ Scalar;
    typedef typename internal::traits<SkewSymmetricMatrix3>::StorageKind StorageKind;
    typedef typename internal::traits<SkewSymmetricMatrix3>::StorageIndex StorageIndex;
    #endif

  protected:

    SkewSymmetricVectorType m_vector;

  public:

    /** const version of vector(). */
    EIGEN_DEVICE_FUNC
    inline const SkewSymmetricVectorType& vector() const { return m_vector; }
    /** \returns a reference to the stored vector of coefficients. */
    EIGEN_DEVICE_FUNC
    inline SkewSymmetricVectorType& vector() { return m_vector; }

    /** Default constructor without initialization */
    EIGEN_DEVICE_FUNC
    inline SkewSymmetricMatrix3() {}

    /** Constructor from three scalars */
    EIGEN_DEVICE_FUNC
    inline SkewSymmetricMatrix3(const Scalar& x, const Scalar& y, const Scalar& z) : m_vector(x,y,z) {}

    /** \brief Constructs a SkewSymmetricMatrix3 from an r-value vector type */
    EIGEN_DEVICE_FUNC
    explicit inline SkewSymmetricMatrix3(SkewSymmetricVectorType&& vec) : m_vector(std::move(vec)) {}

    /** generic constructor from expression of the coefficients */
    template<typename OtherDerived>
    EIGEN_DEVICE_FUNC
        explicit inline SkewSymmetricMatrix3(const MatrixBase<OtherDerived>& other) : m_vector(other)
    {}

    /** Copy constructor. */
    template<typename OtherDerived>
    EIGEN_DEVICE_FUNC
    inline SkewSymmetricMatrix3(const SkewSymmetricBase<OtherDerived>& other) : m_vector(other.vector()) {}

    #ifndef EIGEN_PARSED_BY_DOXYGEN
    /** copy constructor. prevent a default copy constructor from hiding the other templated constructor */
    inline SkewSymmetricMatrix3(const SkewSymmetricMatrix3& other) : m_vector(other.vector()) {}
    #endif

    /** Copy operator. */
    template<typename OtherDerived>
    EIGEN_DEVICE_FUNC
    SkewSymmetricMatrix3& operator=(const SkewSymmetricBase<OtherDerived>& other)
    {
      m_vector = other.vector();
      return *this;
    }

    #ifndef EIGEN_PARSED_BY_DOXYGEN
    /** This is a special case of the templated operator=. Its purpose is to
      * prevent a default operator= from hiding the templated operator=.
      */
    EIGEN_DEVICE_FUNC
    SkewSymmetricMatrix3& operator=(const SkewSymmetricMatrix3& other)
    {
      m_vector = other.vector();
      return *this;
    }
    #endif

    typedef SkewSymmetricWrapper<const CwiseNullaryOp<internal::scalar_constant_op<Scalar>, SkewSymmetricVectorType>>
        InitializeReturnType;

    /** Initializes a skew symmetric matrix with coefficients set to zero */
    EIGEN_DEVICE_FUNC
    static InitializeReturnType Zero() { return SkewSymmetricVectorType::Zero().asSkewSymmetric(); }

    /** Sets all coefficients to zero. */
    EIGEN_DEVICE_FUNC
    inline void setZero() { m_vector.setZero(); }
};

/** \class SkewSymmetricWrapper
  * \ingroup Core_Module
  *
  * \brief Expression of a skew symmetric matrix
  *
  * \tparam SkewSymmetricVectorType_ the type of the vector of coefficients
  *
  * This class is an expression of a skew symmetric matrix, but not storing its own vector of coefficients,
  * instead wrapping an existing vector expression. It is the return type of MatrixBase::asSkewSymmetric()
  * and most of the time this is the only way that it is used.
  *
  * \sa class SkewSymmetricMatrix3, class SkewSymmetricBase, MatrixBase::asSkewSymmetric()
  */

namespace internal {
template<typename SkewSymmetricVectorType_>
struct traits<SkewSymmetricWrapper<SkewSymmetricVectorType_> >
{
  typedef SkewSymmetricVectorType_ SkewSymmetricVectorType;
  typedef typename SkewSymmetricVectorType::Scalar Scalar;
  typedef typename SkewSymmetricVectorType::StorageIndex StorageIndex;
  typedef SkewSymmetricShape StorageKind;
  typedef typename traits<SkewSymmetricVectorType>::XprKind XprKind;
  enum {
    RowsAtCompileTime = SkewSymmetricVectorType::SizeAtCompileTime,
    ColsAtCompileTime = SkewSymmetricVectorType::SizeAtCompileTime,
    MaxRowsAtCompileTime = SkewSymmetricVectorType::MaxSizeAtCompileTime,
    MaxColsAtCompileTime = SkewSymmetricVectorType::MaxSizeAtCompileTime,
    Flags =  (traits<SkewSymmetricVectorType>::Flags & LvalueBit) | NoPreferredStorageOrderBit
  };
};
}

template<typename SkewSymmetricVectorType_>
class SkewSymmetricWrapper
  : public SkewSymmetricBase<SkewSymmetricWrapper<SkewSymmetricVectorType_> >, internal::no_assignment_operator
{
  public:
    #ifndef EIGEN_PARSED_BY_DOXYGEN
    typedef SkewSymmetricVectorType_ SkewSymmetricVectorType;
    typedef SkewSymmetricWrapper Nested;
    #endif

    /** Constructor from expression of coefficients to wrap. */
    EIGEN_DEVICE_FUNC
    explicit inline SkewSymmetricWrapper(SkewSymmetricVectorType& a_vector) : m_vector(a_vector) {}

    /** \returns a const reference to the wrapped expression of coefficients. */
    EIGEN_DEVICE_FUNC
    const SkewSymmetricVectorType& vector() const { return m_vector; }

  protected:
    typename SkewSymmetricVectorType::Nested m_vector;
};

/** \returns a pseudo-expression of a skew symmetric matrix with *this as vector of coefficients
  *
  * \only_for_vectors
  *
  * \sa class SkewSymmetricWrapper, class SkewSymmetricMatrix3, vector(), isSkewSymmetric()
  **/
template<typename Derived>
EIGEN_DEVICE_FUNC inline const SkewSymmetricWrapper<const Derived>
MatrixBase<Derived>::asSkewSymmetric() const
{
  return SkewSymmetricWrapper<const Derived>(derived());
}

/** \returns true if *this is approximately equal to a skew symmetric matrix,
  *          within the precision given by \a prec.
  */
template<typename Derived>
bool MatrixBase<Derived>::isSkewSymmetric(const RealScalar& prec) const
{
  if(cols() != rows()) return false;
  return (this->transpose() + *this).isZero(prec);
}

/** \returns the matrix product of \c *this by the skew symmetric matrix \skew.
 */
template<typename Derived>
template<typename SkewDerived>
EIGEN_DEVICE_FUNC inline const Product<Derived, SkewDerived, LazyProduct>
MatrixBase<Derived>::operator*(const SkewSymmetricBase<SkewDerived> &skew) const
{
  return Product<Derived, SkewDerived, LazyProduct>(derived(), skew.derived());
}

namespace internal {

template<> struct storage_kind_to_shape<SkewSymmetricShape> { typedef SkewSymmetricShape Shape; };

struct SkewSymmetric2Dense {};

template<> struct AssignmentKind<DenseShape,SkewSymmetricShape> { typedef SkewSymmetric2Dense Kind; };

// SkewSymmetric matrix to Dense assignment
template< typename DstXprType, typename SrcXprType, typename Functor>
struct Assignment<DstXprType, SrcXprType, Functor, SkewSymmetric2Dense>
{
  static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename SrcXprType::Scalar> &/*func*/)
  {
    if((dst.rows()!=3) || (dst.cols()!=3)) {
      dst.resize(3, 3);
    }
    dst.diagonal().setZero();
    const typename SrcXprType::SkewSymmetricVectorType v = src.vector();
    dst(0, 1) = -v(2);
    dst(1, 0) = v(2);
    dst(0, 2) = v(1);
    dst(2, 0) = -v(1);
    dst(1, 2) = -v(0);
    dst(2, 1) = v(0);
  }
  
  static void run(DstXprType &dst, const SrcXprType &src, const internal::add_assign_op<typename DstXprType::Scalar,typename SrcXprType::Scalar> &/*func*/)
  { dst.vector() += src.vector(); }
  
  static void run(DstXprType &dst, const SrcXprType &src, const internal::sub_assign_op<typename DstXprType::Scalar,typename SrcXprType::Scalar> &/*func*/)
  { dst.vector() -= src.vector(); }
};

} // namespace internal

}  // end namespace Eigen

#endif // EIGEN_SKEWSYMMETRICMATRIX3_H