text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra. Eigen itself is part of the KDE project.
  //
  // Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
  // Copyright (C) 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/.
  
  // no include guard, we'll include this twice from All.h from Eigen2Support, and it's internal anyway
  
  namespace Eigen { 
  
  /** \geometry_module \ingroup Geometry_Module
    *
    * \class ParametrizedLine
    *
    * \brief A parametrized line
    *
    * A parametrized line is defined by an origin point \f$ \mathbf{o} \f$ and a unit
    * direction vector \f$ \mathbf{d} \f$ such that the line corresponds to
    * the set \f$ l(t) = \mathbf{o} + t \mathbf{d} \f$, \f$ l \in \mathbf{R} \f$.
    *
    * \param _Scalar the scalar type, i.e., the type of the coefficients
    * \param _AmbientDim the dimension of the ambient space, can be a compile time value or Dynamic.
    */
  template <typename _Scalar, int _AmbientDim>
  class ParametrizedLine
  {
  public:
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim)
    enum { AmbientDimAtCompileTime = _AmbientDim };
    typedef _Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef Matrix<Scalar,AmbientDimAtCompileTime,1> VectorType;
  
    /** Default constructor without initialization */
    inline explicit ParametrizedLine() {}
  
    /** Constructs a dynamic-size line with \a _dim the dimension
      * of the ambient space */
    inline explicit ParametrizedLine(int _dim) : m_origin(_dim), m_direction(_dim) {}
  
    /** Initializes a parametrized line of direction \a direction and origin \a origin.
      * \warning the vector direction is assumed to be normalized.
      */
    ParametrizedLine(const VectorType& origin, const VectorType& direction)
      : m_origin(origin), m_direction(direction) {}
  
    explicit ParametrizedLine(const Hyperplane<_Scalar, _AmbientDim>& hyperplane);
  
    /** Constructs a parametrized line going from \a p0 to \a p1. */
    static inline ParametrizedLine Through(const VectorType& p0, const VectorType& p1)
    { return ParametrizedLine(p0, (p1-p0).normalized()); }
  
    ~ParametrizedLine() {}
  
    /** \returns the dimension in which the line holds */
    inline int dim() const { return m_direction.size(); }
  
    const VectorType& origin() const { return m_origin; }
    VectorType& origin() { return m_origin; }
  
    const VectorType& direction() const { return m_direction; }
    VectorType& direction() { return m_direction; }
  
    /** \returns the squared distance of a point \a p to its projection onto the line \c *this.
      * \sa distance()
      */
    RealScalar squaredDistance(const VectorType& p) const
    {
      VectorType diff = p-origin();
      return (diff - diff.eigen2_dot(direction())* direction()).squaredNorm();
    }
    /** \returns the distance of a point \a p to its projection onto the line \c *this.
      * \sa squaredDistance()
      */
    RealScalar distance(const VectorType& p) const { return ei_sqrt(squaredDistance(p)); }
  
    /** \returns the projection of a point \a p onto the line \c *this. */
    VectorType projection(const VectorType& p) const
    { return origin() + (p-origin()).eigen2_dot(direction()) * direction(); }
  
    Scalar intersection(const Hyperplane<_Scalar, _AmbientDim>& hyperplane);
  
    /** \returns \c *this with scalar type casted to \a NewScalarType
      *
      * Note that if \a NewScalarType is equal to the current scalar type of \c *this
      * then this function smartly returns a const reference to \c *this.
      */
    template<typename NewScalarType>
    inline typename internal::cast_return_type<ParametrizedLine,
             ParametrizedLine<NewScalarType,AmbientDimAtCompileTime> >::type cast() const
    {
      return typename internal::cast_return_type<ParametrizedLine,
                      ParametrizedLine<NewScalarType,AmbientDimAtCompileTime> >::type(*this);
    }
  
    /** Copy constructor with scalar type conversion */
    template<typename OtherScalarType>
    inline explicit ParametrizedLine(const ParametrizedLine<OtherScalarType,AmbientDimAtCompileTime>& other)
    {
      m_origin = other.origin().template cast<Scalar>();
      m_direction = other.direction().template cast<Scalar>();
    }
  
    /** \returns \c true if \c *this is approximately equal to \a other, within the precision
      * determined by \a prec.
      *
      * \sa MatrixBase::isApprox() */
    bool isApprox(const ParametrizedLine& other, typename NumTraits<Scalar>::Real prec = precision<Scalar>()) const
    { return m_origin.isApprox(other.m_origin, prec) && m_direction.isApprox(other.m_direction, prec); }
  
  protected:
  
    VectorType m_origin, m_direction;
  };
  
  /** Constructs a parametrized line from a 2D hyperplane
    *
    * \warning the ambient space must have dimension 2 such that the hyperplane actually describes a line
    */
  template <typename _Scalar, int _AmbientDim>
  inline ParametrizedLine<_Scalar, _AmbientDim>::ParametrizedLine(const Hyperplane<_Scalar, _AmbientDim>& hyperplane)
  {
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2)
    direction() = hyperplane.normal().unitOrthogonal();
    origin() = -hyperplane.normal()*hyperplane.offset();
  }
  
  /** \returns the parameter value of the intersection between \c *this and the given hyperplane
    */
  template <typename _Scalar, int _AmbientDim>
  inline _Scalar ParametrizedLine<_Scalar, _AmbientDim>::intersection(const Hyperplane<_Scalar, _AmbientDim>& hyperplane)
  {
    return -(hyperplane.offset()+origin().eigen2_dot(hyperplane.normal()))
            /(direction().eigen2_dot(hyperplane.normal()));
  }
  
  } // end namespace Eigen
  