text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.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/.
  
  #ifndef EIGEN_PARAMETRIZEDLINE_H
  #define EIGEN_PARAMETRIZEDLINE_H
  
  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$, \f$ t \in \mathbf{R} \f$.
    *
    * \tparam _Scalar the scalar type, i.e., the type of the coefficients
    * \tparam _AmbientDim the dimension of the ambient space, can be a compile time value or Dynamic.
    */
  template <typename _Scalar, int _AmbientDim, int _Options>
  class ParametrizedLine
  {
  public:
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_AmbientDim)
    enum {
      AmbientDimAtCompileTime = _AmbientDim,
      Options = _Options
    };
    typedef _Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
    typedef Matrix<Scalar,AmbientDimAtCompileTime,1,Options> VectorType;
  
    /** Default constructor without initialization */
    EIGEN_DEVICE_FUNC inline ParametrizedLine() {}
    
    template<int OtherOptions>
    EIGEN_DEVICE_FUNC ParametrizedLine(const ParametrizedLine<Scalar,AmbientDimAtCompileTime,OtherOptions>& other)
     : m_origin(other.origin()), m_direction(other.direction())
    {}
  
    /** Constructs a dynamic-size line with \a _dim the dimension
      * of the ambient space */
    EIGEN_DEVICE_FUNC inline explicit ParametrizedLine(Index _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.
      */
    EIGEN_DEVICE_FUNC ParametrizedLine(const VectorType& o, const VectorType& d)
      : m_origin(o), m_direction(d) {}
  
    template <int OtherOptions>
    EIGEN_DEVICE_FUNC explicit ParametrizedLine(const Hyperplane<_Scalar, _AmbientDim, OtherOptions>& hyperplane);
  
    /** Constructs a parametrized line going from \a p0 to \a p1. */
    EIGEN_DEVICE_FUNC static inline ParametrizedLine Through(const VectorType& p0, const VectorType& p1)
    { return ParametrizedLine(p0, (p1-p0).normalized()); }
  
    EIGEN_DEVICE_FUNC ~ParametrizedLine() {}
  
    /** \returns the dimension in which the line holds */
    EIGEN_DEVICE_FUNC inline Index dim() const { return m_direction.size(); }
  
    EIGEN_DEVICE_FUNC const VectorType& origin() const { return m_origin; }
    EIGEN_DEVICE_FUNC VectorType& origin() { return m_origin; }
  
    EIGEN_DEVICE_FUNC const VectorType& direction() const { return m_direction; }
    EIGEN_DEVICE_FUNC VectorType& direction() { return m_direction; }
  
    /** \returns the squared distance of a point \a p to its projection onto the line \c *this.
      * \sa distance()
      */
    EIGEN_DEVICE_FUNC RealScalar squaredDistance(const VectorType& p) const
    {
      VectorType diff = p - origin();
      return (diff - direction().dot(diff) * direction()).squaredNorm();
    }
    /** \returns the distance of a point \a p to its projection onto the line \c *this.
      * \sa squaredDistance()
      */
    EIGEN_DEVICE_FUNC RealScalar distance(const VectorType& p) const { EIGEN_USING_STD_MATH(sqrt) return sqrt(squaredDistance(p)); }
  
    /** \returns the projection of a point \a p onto the line \c *this. */
    EIGEN_DEVICE_FUNC VectorType projection(const VectorType& p) const
    { return origin() + direction().dot(p-origin()) * direction(); }
  
    EIGEN_DEVICE_FUNC VectorType pointAt(const Scalar& t) const;
    
    template <int OtherOptions>
    EIGEN_DEVICE_FUNC Scalar intersectionParameter(const Hyperplane<_Scalar, _AmbientDim, OtherOptions>& hyperplane) const;
   
    template <int OtherOptions>
    EIGEN_DEVICE_FUNC Scalar intersection(const Hyperplane<_Scalar, _AmbientDim, OtherOptions>& hyperplane) const;
    
    template <int OtherOptions>
    EIGEN_DEVICE_FUNC VectorType intersectionPoint(const Hyperplane<_Scalar, _AmbientDim, OtherOptions>& hyperplane) const;
  
    /** \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>
    EIGEN_DEVICE_FUNC inline typename internal::cast_return_type<ParametrizedLine,
             ParametrizedLine<NewScalarType,AmbientDimAtCompileTime,Options> >::type cast() const
    {
      return typename internal::cast_return_type<ParametrizedLine,
                      ParametrizedLine<NewScalarType,AmbientDimAtCompileTime,Options> >::type(*this);
    }
  
    /** Copy constructor with scalar type conversion */
    template<typename OtherScalarType,int OtherOptions>
    EIGEN_DEVICE_FUNC inline explicit ParametrizedLine(const ParametrizedLine<OtherScalarType,AmbientDimAtCompileTime,OtherOptions>& 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() */
    EIGEN_DEVICE_FUNC bool isApprox(const ParametrizedLine& other, const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) 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, int _Options>
  template <int OtherOptions>
  EIGEN_DEVICE_FUNC inline ParametrizedLine<_Scalar, _AmbientDim,_Options>::ParametrizedLine(const Hyperplane<_Scalar, _AmbientDim,OtherOptions>& hyperplane)
  {
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2)
    direction() = hyperplane.normal().unitOrthogonal();
    origin() = -hyperplane.normal()*hyperplane.offset();
  }
  
  /** \returns the point at \a t along this line
    */
  template <typename _Scalar, int _AmbientDim, int _Options>
  EIGEN_DEVICE_FUNC inline typename ParametrizedLine<_Scalar, _AmbientDim,_Options>::VectorType
  ParametrizedLine<_Scalar, _AmbientDim,_Options>::pointAt(const _Scalar& t) const
  {
    return origin() + (direction()*t); 
  }
  
  /** \returns the parameter value of the intersection between \c *this and the given \a hyperplane
    */
  template <typename _Scalar, int _AmbientDim, int _Options>
  template <int OtherOptions>
  EIGEN_DEVICE_FUNC inline _Scalar ParametrizedLine<_Scalar, _AmbientDim,_Options>::intersectionParameter(const Hyperplane<_Scalar, _AmbientDim, OtherOptions>& hyperplane) const
  {
    return -(hyperplane.offset()+hyperplane.normal().dot(origin()))
            / hyperplane.normal().dot(direction());
  }
  
  
  /** \deprecated use intersectionParameter()
    * \returns the parameter value of the intersection between \c *this and the given \a hyperplane
    */
  template <typename _Scalar, int _AmbientDim, int _Options>
  template <int OtherOptions>
  EIGEN_DEVICE_FUNC inline _Scalar ParametrizedLine<_Scalar, _AmbientDim,_Options>::intersection(const Hyperplane<_Scalar, _AmbientDim, OtherOptions>& hyperplane) const
  {
    return intersectionParameter(hyperplane);
  }
  
  /** \returns the point of the intersection between \c *this and the given hyperplane
    */
  template <typename _Scalar, int _AmbientDim, int _Options>
  template <int OtherOptions>
  EIGEN_DEVICE_FUNC inline typename ParametrizedLine<_Scalar, _AmbientDim,_Options>::VectorType
  ParametrizedLine<_Scalar, _AmbientDim,_Options>::intersectionPoint(const Hyperplane<_Scalar, _AmbientDim, OtherOptions>& hyperplane) const
  {
    return pointAt(intersectionParameter(hyperplane));
  }
  
  } // end namespace Eigen
  
  #endif // EIGEN_PARAMETRIZEDLINE_H
  