Quaternion.h
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Created: Tue, Jul 1, 15:55

Quaternion.h
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	// This file is part of Eigen, a lightweight C++ template library
	// for linear algebra.
	//
	// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
	// Copyright (C) 2009 Mathieu Gautier <mathieu.gautier@cea.fr>
	//
	// 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_QUATERNION_H
	#define EIGEN_QUATERNION_H
	namespace Eigen {


	/***************************************************************************
	* Definition of QuaternionBase<Derived>
	* The implementation is at the end of the file
	***************************************************************************/

	namespace internal {
	template<typename Other,
	int OtherRows=Other::RowsAtCompileTime,
	int OtherCols=Other::ColsAtCompileTime>
	struct quaternionbase_assign_impl;
	}

	/** \geometry_module \ingroup Geometry_Module
	* \class QuaternionBase
	* \brief Base class for quaternion expressions
	* \tparam Derived derived type (CRTP)
	* \sa class Quaternion
	*/
	template<class Derived>
	class QuaternionBase : public RotationBase<Derived, 3>
	{
	public:
	typedef RotationBase<Derived, 3> Base;

	using Base::operator*;
	using Base::derived;

	typedef typename internal::traits<Derived>::Scalar Scalar;
	typedef typename NumTraits<Scalar>::Real RealScalar;
	typedef typename internal::traits<Derived>::Coefficients Coefficients;
	typedef typename Coefficients::CoeffReturnType CoeffReturnType;
	typedef typename internal::conditional<bool(internal::traits<Derived>::Flags&LvalueBit),
	Scalar&, CoeffReturnType>::type NonConstCoeffReturnType;


	enum {
	Flags = Eigen::internal::traits<Derived>::Flags
	};

	// typedef typename Matrix<Scalar,4,1> Coefficients;
	/** the type of a 3D vector */
	typedef Matrix<Scalar,3,1> Vector3;
	/** the equivalent rotation matrix type */
	typedef Matrix<Scalar,3,3> Matrix3;
	/** the equivalent angle-axis type */
	typedef AngleAxis<Scalar> AngleAxisType;



	/** \returns the \c x coefficient */
	EIGEN_DEVICE_FUNC inline CoeffReturnType x() const { return this->derived().coeffs().coeff(0); }
	/** \returns the \c y coefficient */
	EIGEN_DEVICE_FUNC inline CoeffReturnType y() const { return this->derived().coeffs().coeff(1); }
	/** \returns the \c z coefficient */
	EIGEN_DEVICE_FUNC inline CoeffReturnType z() const { return this->derived().coeffs().coeff(2); }
	/** \returns the \c w coefficient */
	EIGEN_DEVICE_FUNC inline CoeffReturnType w() const { return this->derived().coeffs().coeff(3); }

	/** \returns a reference to the \c x coefficient (if Derived is a non-const lvalue) */
	EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType x() { return this->derived().coeffs().x(); }
	/** \returns a reference to the \c y coefficient (if Derived is a non-const lvalue) */
	EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType y() { return this->derived().coeffs().y(); }
	/** \returns a reference to the \c z coefficient (if Derived is a non-const lvalue) */
	EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType z() { return this->derived().coeffs().z(); }
	/** \returns a reference to the \c w coefficient (if Derived is a non-const lvalue) */
	EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType w() { return this->derived().coeffs().w(); }

	/** \returns a read-only vector expression of the imaginary part (x,y,z) */
	EIGEN_DEVICE_FUNC inline const VectorBlock<const Coefficients,3> vec() const { return coeffs().template head<3>(); }

	/** \returns a vector expression of the imaginary part (x,y,z) */
	EIGEN_DEVICE_FUNC inline VectorBlock<Coefficients,3> vec() { return coeffs().template head<3>(); }

	/** \returns a read-only vector expression of the coefficients (x,y,z,w) */
	EIGEN_DEVICE_FUNC inline const typename internal::traits<Derived>::Coefficients& coeffs() const { return derived().coeffs(); }

	/** \returns a vector expression of the coefficients (x,y,z,w) */
	EIGEN_DEVICE_FUNC inline typename internal::traits<Derived>::Coefficients& coeffs() { return derived().coeffs(); }

	EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE QuaternionBase<Derived>& operator=(const QuaternionBase<Derived>& other);
	template<class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& operator=(const QuaternionBase<OtherDerived>& other);

	// disabled this copy operator as it is giving very strange compilation errors when compiling
	// test_stdvector with GCC 4.4.2. This looks like a GCC bug though, so feel free to re-enable it if it's
	// useful; however notice that we already have the templated operator= above and e.g. in MatrixBase
	// we didn't have to add, in addition to templated operator=, such a non-templated copy operator.
	// Derived& operator=(const QuaternionBase& other)
	// { return operator=<Derived>(other); }

	EIGEN_DEVICE_FUNC Derived& operator=(const AngleAxisType& aa);
	template<class OtherDerived> EIGEN_DEVICE_FUNC Derived& operator=(const MatrixBase<OtherDerived>& m);

	/** \returns a quaternion representing an identity rotation
	* \sa MatrixBase::Identity()
	*/
	EIGEN_DEVICE_FUNC static inline Quaternion<Scalar> Identity() { return Quaternion<Scalar>(Scalar(1), Scalar(0), Scalar(0), Scalar(0)); }

	/** \sa QuaternionBase::Identity(), MatrixBase::setIdentity()
	*/
	EIGEN_DEVICE_FUNC inline QuaternionBase& setIdentity() { coeffs() << Scalar(0), Scalar(0), Scalar(0), Scalar(1); return *this; }

	/** \returns the squared norm of the quaternion's coefficients
	* \sa QuaternionBase::norm(), MatrixBase::squaredNorm()
	*/
	EIGEN_DEVICE_FUNC inline Scalar squaredNorm() const { return coeffs().squaredNorm(); }

	/** \returns the norm of the quaternion's coefficients
	* \sa QuaternionBase::squaredNorm(), MatrixBase::norm()
	*/
	EIGEN_DEVICE_FUNC inline Scalar norm() const { return coeffs().norm(); }

	/** Normalizes the quaternion \c *this
	* \sa normalized(), MatrixBase::normalize() */
	EIGEN_DEVICE_FUNC inline void normalize() { coeffs().normalize(); }
	/** \returns a normalized copy of \c *this
	* \sa normalize(), MatrixBase::normalized() */
	EIGEN_DEVICE_FUNC inline Quaternion<Scalar> normalized() const { return Quaternion<Scalar>(coeffs().normalized()); }

	/** \returns the dot product of \c *this and \a other
	* Geometrically speaking, the dot product of two unit quaternions
	* corresponds to the cosine of half the angle between the two rotations.
	* \sa angularDistance()
	*/
	template<class OtherDerived> EIGEN_DEVICE_FUNC inline Scalar dot(const QuaternionBase<OtherDerived>& other) const { return coeffs().dot(other.coeffs()); }

	template<class OtherDerived> EIGEN_DEVICE_FUNC Scalar angularDistance(const QuaternionBase<OtherDerived>& other) const;

	/** \returns an equivalent 3x3 rotation matrix */
	EIGEN_DEVICE_FUNC inline Matrix3 toRotationMatrix() const;

	/** \returns the quaternion which transform \a a into \a b through a rotation */
	template<typename Derived1, typename Derived2>
	EIGEN_DEVICE_FUNC Derived& setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);

	template<class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion<Scalar> operator* (const QuaternionBase<OtherDerived>& q) const;
	template<class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& operator*= (const QuaternionBase<OtherDerived>& q);

	/** \returns the quaternion describing the inverse rotation */
	EIGEN_DEVICE_FUNC Quaternion<Scalar> inverse() const;

	/** \returns the conjugated quaternion */
	EIGEN_DEVICE_FUNC Quaternion<Scalar> conjugate() const;

	template<class OtherDerived> EIGEN_DEVICE_FUNC Quaternion<Scalar> slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const;

	/** \returns true if each coefficients of \c *this and \a other are all exactly equal.
	* \warning When using floating point scalar values you probably should rather use a
	* fuzzy comparison such as isApprox()
	* \sa isApprox(), operator!= */
	template<class OtherDerived>
	EIGEN_DEVICE_FUNC inline bool operator==(const QuaternionBase<OtherDerived>& other) const
	{ return coeffs() == other.coeffs(); }

	/** \returns true if at least one pair of coefficients of \c *this and \a other are not exactly equal to each other.
	* \warning When using floating point scalar values you probably should rather use a
	* fuzzy comparison such as isApprox()
	* \sa isApprox(), operator== */
	template<class OtherDerived>
	EIGEN_DEVICE_FUNC inline bool operator!=(const QuaternionBase<OtherDerived>& other) const
	{ return coeffs() != other.coeffs(); }

	/** \returns \c true if \c *this is approximately equal to \a other, within the precision
	* determined by \a prec.
	*
	* \sa MatrixBase::isApprox() */
	template<class OtherDerived>
	EIGEN_DEVICE_FUNC bool isApprox(const QuaternionBase<OtherDerived>& other, const RealScalar& prec = NumTraits<Scalar>::dummy_precision()) const
	{ return coeffs().isApprox(other.coeffs(), prec); }

	/** return the result vector of \a v through the rotation*/
	EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Vector3 _transformVector(const Vector3& v) const;

	#ifdef EIGEN_PARSED_BY_DOXYGEN
	/** \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<Derived,Quaternion<NewScalarType> >::type cast() const;

	#else

	template<typename NewScalarType>
	EIGEN_DEVICE_FUNC inline
	typename internal::enable_if<internal::is_same<Scalar,NewScalarType>::value,const Derived&>::type cast() const
	{
	return derived();
	}

	template<typename NewScalarType>
	EIGEN_DEVICE_FUNC inline
	typename internal::enable_if<!internal::is_same<Scalar,NewScalarType>::value,Quaternion<NewScalarType> >::type cast() const
	{
	return Quaternion<NewScalarType>(coeffs().template cast<NewScalarType>());
	}
	#endif

	#ifndef EIGEN_NO_IO
	friend std::ostream& operator<<(std::ostream& s, const QuaternionBase<Derived>& q) {
	s << q.x() << "i + " << q.y() << "j + " << q.z() << "k" << " + " << q.w();
	return s;
	}
	#endif

	#ifdef EIGEN_QUATERNIONBASE_PLUGIN
	# include EIGEN_QUATERNIONBASE_PLUGIN
	#endif
	protected:
	EIGEN_DEFAULT_COPY_CONSTRUCTOR(QuaternionBase)
	EIGEN_DEFAULT_EMPTY_CONSTRUCTOR_AND_DESTRUCTOR(QuaternionBase)
	};

	/***************************************************************************
	* Definition/implementation of Quaternion<Scalar>
	***************************************************************************/

	/** \geometry_module \ingroup Geometry_Module
	*
	* \class Quaternion
	*
	* \brief The quaternion class used to represent 3D orientations and rotations
	*
	* \tparam _Scalar the scalar type, i.e., the type of the coefficients
	* \tparam _Options controls the memory alignment of the coefficients. Can be \# AutoAlign or \# DontAlign. Default is AutoAlign.
	*
	* This class represents a quaternion \f$ w+xi+yj+zk \f$ that is a convenient representation of
	* orientations and rotations of objects in three dimensions. Compared to other representations
	* like Euler angles or 3x3 matrices, quaternions offer the following advantages:
	* \li \b compact storage (4 scalars)
	* \li \b efficient to compose (28 flops),
	* \li \b stable spherical interpolation
	*
	* The following two typedefs are provided for convenience:
	* \li \c Quaternionf for \c float
	* \li \c Quaterniond for \c double
	*
	* \warning Operations interpreting the quaternion as rotation have undefined behavior if the quaternion is not normalized.
	*
	* \sa class AngleAxis, class Transform
	*/

	namespace internal {
	template<typename _Scalar,int _Options>
	struct traits<Quaternion<_Scalar,_Options> >
	{
	typedef Quaternion<_Scalar,_Options> PlainObject;
	typedef _Scalar Scalar;
	typedef Matrix<_Scalar,4,1,_Options> Coefficients;
	enum{
	Alignment = internal::traits<Coefficients>::Alignment,
	Flags = LvalueBit
	};
	};
	}

	template<typename _Scalar, int _Options>
	class Quaternion : public QuaternionBase<Quaternion<_Scalar,_Options> >
	{
	public:
	typedef QuaternionBase<Quaternion<_Scalar,_Options> > Base;
	enum { NeedsAlignment = internal::traits<Quaternion>::Alignment>0 };

	typedef _Scalar Scalar;

	EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Quaternion)
	using Base::operator*=;

	typedef typename internal::traits<Quaternion>::Coefficients Coefficients;
	typedef typename Base::AngleAxisType AngleAxisType;

	/** Default constructor leaving the quaternion uninitialized. */
	EIGEN_DEVICE_FUNC inline Quaternion() {}

	/** Constructs and initializes the quaternion \f$ w+xi+yj+zk \f$ from
	* its four coefficients \a w, \a x, \a y and \a z.
	*
	* \warning Note the order of the arguments: the real \a w coefficient first,
	* while internally the coefficients are stored in the following order:
	* [\c x, \c y, \c z, \c w]
	*/
	EIGEN_DEVICE_FUNC inline Quaternion(const Scalar& w, const Scalar& x, const Scalar& y, const Scalar& z) : m_coeffs(x, y, z, w){}

	/** Constructs and initialize a quaternion from the array data */
	EIGEN_DEVICE_FUNC explicit inline Quaternion(const Scalar* data) : m_coeffs(data) {}

	/** Copy constructor */
	template<class Derived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion(const QuaternionBase<Derived>& other) { this->Base::operator=(other); }

	/** Constructs and initializes a quaternion from the angle-axis \a aa */
	EIGEN_DEVICE_FUNC explicit inline Quaternion(const AngleAxisType& aa) { *this = aa; }

	/** Constructs and initializes a quaternion from either:
	* - a rotation matrix expression,
	* - a 4D vector expression representing quaternion coefficients.
	*/
	template<typename Derived>
	EIGEN_DEVICE_FUNC explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; }

	/** Explicit copy constructor with scalar conversion */
	template<typename OtherScalar, int OtherOptions>
	EIGEN_DEVICE_FUNC explicit inline Quaternion(const Quaternion<OtherScalar, OtherOptions>& other)
	{ m_coeffs = other.coeffs().template cast<Scalar>(); }

	#if EIGEN_HAS_RVALUE_REFERENCES
	// We define a copy constructor, which means we don't get an implicit move constructor or assignment operator.
	/** Default move constructor */
	EIGEN_DEVICE_FUNC inline Quaternion(Quaternion&& other) EIGEN_NOEXCEPT_IF(std::is_nothrow_move_constructible<Scalar>::value)
	: m_coeffs(std::move(other.coeffs()))
	{}

	/** Default move assignment operator */
	EIGEN_DEVICE_FUNC Quaternion& operator=(Quaternion&& other) EIGEN_NOEXCEPT_IF(std::is_nothrow_move_assignable<Scalar>::value)
	{
	m_coeffs = std::move(other.coeffs());
	return *this;
	}
	#endif

	EIGEN_DEVICE_FUNC static Quaternion UnitRandom();

	template<typename Derived1, typename Derived2>
	EIGEN_DEVICE_FUNC static Quaternion FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);

	EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs;}
	EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs;}

	EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF(bool(NeedsAlignment))

	#ifdef EIGEN_QUATERNION_PLUGIN
	# include EIGEN_QUATERNION_PLUGIN
	#endif

	protected:
	Coefficients m_coeffs;

	#ifndef EIGEN_PARSED_BY_DOXYGEN
	static EIGEN_STRONG_INLINE void _check_template_params()
	{
	EIGEN_STATIC_ASSERT( (_Options & DontAlign) == _Options,
	INVALID_MATRIX_TEMPLATE_PARAMETERS)
	}
	#endif
	};

	/** \ingroup Geometry_Module
	* single precision quaternion type */
	typedef Quaternion<float> Quaternionf;
	/** \ingroup Geometry_Module
	* double precision quaternion type */
	typedef Quaternion<double> Quaterniond;

	/***************************************************************************
	* Specialization of Map<Quaternion<Scalar>>
	***************************************************************************/

	namespace internal {
	template<typename _Scalar, int _Options>
	struct traits<Map<Quaternion<_Scalar>, _Options> > : traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> >
	{
	typedef Map<Matrix<_Scalar,4,1>, _Options> Coefficients;
	};
	}

	namespace internal {
	template<typename _Scalar, int _Options>
	struct traits<Map<const Quaternion<_Scalar>, _Options> > : traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> >
	{
	typedef Map<const Matrix<_Scalar,4,1>, _Options> Coefficients;
	typedef traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> > TraitsBase;
	enum {
	Flags = TraitsBase::Flags & ~LvalueBit
	};
	};
	}

	/** \ingroup Geometry_Module
	* \brief Quaternion expression mapping a constant memory buffer
	*
	* \tparam _Scalar the type of the Quaternion coefficients
	* \tparam _Options see class Map
	*
	* This is a specialization of class Map for Quaternion. This class allows to view
	* a 4 scalar memory buffer as an Eigen's Quaternion object.
	*
	* \sa class Map, class Quaternion, class QuaternionBase
	*/
	template<typename _Scalar, int _Options>
	class Map<const Quaternion<_Scalar>, _Options >
	: public QuaternionBase<Map<const Quaternion<_Scalar>, _Options> >
	{
	public:
	typedef QuaternionBase<Map<const Quaternion<_Scalar>, _Options> > Base;

	typedef _Scalar Scalar;
	typedef typename internal::traits<Map>::Coefficients Coefficients;
	EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map)
	using Base::operator*=;

	/** Constructs a Mapped Quaternion object from the pointer \a coeffs
	*
	* The pointer \a coeffs must reference the four coefficients of Quaternion in the following order:
	* \code *coeffs == {x, y, z, w} \endcode
	*
	* If the template parameter _Options is set to #Aligned, then the pointer coeffs must be aligned. */
	EIGEN_DEVICE_FUNC explicit EIGEN_STRONG_INLINE Map(const Scalar* coeffs) : m_coeffs(coeffs) {}

	EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs;}

	protected:
	const Coefficients m_coeffs;
	};

	/** \ingroup Geometry_Module
	* \brief Expression of a quaternion from a memory buffer
	*
	* \tparam _Scalar the type of the Quaternion coefficients
	* \tparam _Options see class Map
	*
	* This is a specialization of class Map for Quaternion. This class allows to view
	* a 4 scalar memory buffer as an Eigen's Quaternion object.
	*
	* \sa class Map, class Quaternion, class QuaternionBase
	*/
	template<typename _Scalar, int _Options>
	class Map<Quaternion<_Scalar>, _Options >
	: public QuaternionBase<Map<Quaternion<_Scalar>, _Options> >
	{
	public:
	typedef QuaternionBase<Map<Quaternion<_Scalar>, _Options> > Base;

	typedef _Scalar Scalar;
	typedef typename internal::traits<Map>::Coefficients Coefficients;
	EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map)
	using Base::operator*=;

	/** Constructs a Mapped Quaternion object from the pointer \a coeffs
	*
	* The pointer \a coeffs must reference the four coefficients of Quaternion in the following order:
	* \code *coeffs == {x, y, z, w} \endcode
	*
	* If the template parameter _Options is set to #Aligned, then the pointer coeffs must be aligned. */
	EIGEN_DEVICE_FUNC explicit EIGEN_STRONG_INLINE Map(Scalar* coeffs) : m_coeffs(coeffs) {}

	EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; }
	EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; }

	protected:
	Coefficients m_coeffs;
	};

	/** \ingroup Geometry_Module
	* Map an unaligned array of single precision scalars as a quaternion */
	typedef Map<Quaternion<float>, 0> QuaternionMapf;
	/** \ingroup Geometry_Module
	* Map an unaligned array of double precision scalars as a quaternion */
	typedef Map<Quaternion<double>, 0> QuaternionMapd;
	/** \ingroup Geometry_Module
	* Map a 16-byte aligned array of single precision scalars as a quaternion */
	typedef Map<Quaternion<float>, Aligned> QuaternionMapAlignedf;
	/** \ingroup Geometry_Module
	* Map a 16-byte aligned array of double precision scalars as a quaternion */
	typedef Map<Quaternion<double>, Aligned> QuaternionMapAlignedd;

	/***************************************************************************
	* Implementation of QuaternionBase methods
	***************************************************************************/

	// Generic Quaternion * Quaternion product
	// This product can be specialized for a given architecture via the Arch template argument.
	namespace internal {
	template<int Arch, class Derived1, class Derived2, typename Scalar> struct quat_product
	{
	EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived1>& a, const QuaternionBase<Derived2>& b){
	return Quaternion<Scalar>
	(
	a.w() * b.w() - a.x() * b.x() - a.y() * b.y() - a.z() * b.z(),
	a.w() * b.x() + a.x() * b.w() + a.y() * b.z() - a.z() * b.y(),
	a.w() * b.y() + a.y() * b.w() + a.z() * b.x() - a.x() * b.z(),
	a.w() * b.z() + a.z() * b.w() + a.x() * b.y() - a.y() * b.x()
	);
	}
	};
	}

	/** \returns the concatenation of two rotations as a quaternion-quaternion product */
	template <class Derived>
	template <class OtherDerived>
	EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion<typename internal::traits<Derived>::Scalar>
	QuaternionBase<Derived>::operator* (const QuaternionBase<OtherDerived>& other) const
	{
	EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename OtherDerived::Scalar>::value),
	YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
	return internal::quat_product<Architecture::Target, Derived, OtherDerived,
	typename internal::traits<Derived>::Scalar>::run(*this, other);
	}

	/** \sa operator(Quaternion) /
	template <class Derived>
	template <class OtherDerived>
	EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator*= (const QuaternionBase<OtherDerived>& other)
	{
	derived() = derived() * other.derived();
	return derived();
	}

	/** Rotation of a vector by a quaternion.
	* \remarks If the quaternion is used to rotate several points (>1)
	* then it is much more efficient to first convert it to a 3x3 Matrix.
	* Comparison of the operation cost for n transformations:
	* - Quaternion2: 30n
	* - Via a Matrix3: 24 + 15n
	*/
	template <class Derived>
	EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename QuaternionBase<Derived>::Vector3
	QuaternionBase<Derived>::_transformVector(const Vector3& v) const
	{
	// Note that this algorithm comes from the optimization by hand
	// of the conversion to a Matrix followed by a Matrix/Vector product.
	// It appears to be much faster than the common algorithm found
	// in the literature (30 versus 39 flops). It also requires two
	// Vector3 as temporaries.
	Vector3 uv = this->vec().cross(v);
	uv += uv;
	return v + this->w() * uv + this->vec().cross(uv);
	}

	template<class Derived>
	EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE QuaternionBase<Derived>& QuaternionBase<Derived>::operator=(const QuaternionBase<Derived>& other)
	{
	coeffs() = other.coeffs();
	return derived();
	}

	template<class Derived>
	template<class OtherDerived>
	EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const QuaternionBase<OtherDerived>& other)
	{
	coeffs() = other.coeffs();
	return derived();
	}

	/** Set \c this from an angle-axis \a aa and returns a reference to \c this
	*/
	template<class Derived>
	EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const AngleAxisType& aa)
	{
	EIGEN_USING_STD(cos)
	EIGEN_USING_STD(sin)
	Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings
	this->w() = cos(ha);
	this->vec() = sin(ha) * aa.axis();
	return derived();
	}

	/** Set \c *this from the expression \a xpr:
	* - if \a xpr is a 4x1 vector, then \a xpr is assumed to be a quaternion
	* - if \a xpr is a 3x3 matrix, then \a xpr is assumed to be rotation matrix
	* and \a xpr is converted to a quaternion
	*/

	template<class Derived>
	template<class MatrixDerived>
	EIGEN_DEVICE_FUNC inline Derived& QuaternionBase<Derived>::operator=(const MatrixBase<MatrixDerived>& xpr)
	{
	EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename MatrixDerived::Scalar>::value),
	YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
	internal::quaternionbase_assign_impl<MatrixDerived>::run(*this, xpr.derived());
	return derived();
	}

	/** Convert the quaternion to a 3x3 rotation matrix. The quaternion is required to
	* be normalized, otherwise the result is undefined.
	*/
	template<class Derived>
	EIGEN_DEVICE_FUNC inline typename QuaternionBase<Derived>::Matrix3
	QuaternionBase<Derived>::toRotationMatrix(void) const
	{
	// NOTE if inlined, then gcc 4.2 and 4.4 get rid of the temporary (not gcc 4.3 !!)
	// if not inlined then the cost of the return by value is huge ~ +35%,
	// however, not inlining this function is an order of magnitude slower, so
	// it has to be inlined, and so the return by value is not an issue
	Matrix3 res;

	const Scalar tx = Scalar(2)*this->x();
	const Scalar ty = Scalar(2)*this->y();
	const Scalar tz = Scalar(2)*this->z();
	const Scalar twx = tx*this->w();
	const Scalar twy = ty*this->w();
	const Scalar twz = tz*this->w();
	const Scalar txx = tx*this->x();
	const Scalar txy = ty*this->x();
	const Scalar txz = tz*this->x();
	const Scalar tyy = ty*this->y();
	const Scalar tyz = tz*this->y();
	const Scalar tzz = tz*this->z();

	res.coeffRef(0,0) = Scalar(1)-(tyy+tzz);
	res.coeffRef(0,1) = txy-twz;
	res.coeffRef(0,2) = txz+twy;
	res.coeffRef(1,0) = txy+twz;
	res.coeffRef(1,1) = Scalar(1)-(txx+tzz);
	res.coeffRef(1,2) = tyz-twx;
	res.coeffRef(2,0) = txz-twy;
	res.coeffRef(2,1) = tyz+twx;
	res.coeffRef(2,2) = Scalar(1)-(txx+tyy);

	return res;
	}

	/** Sets \c *this to be a quaternion representing a rotation between
	* the two arbitrary vectors \a a and \a b. In other words, the built
	* rotation represent a rotation sending the line of direction \a a
	* to the line of direction \a b, both lines passing through the origin.
	*
	* \returns a reference to \c *this.
	*
	* Note that the two input vectors do \b not have to be normalized, and
	* do not need to have the same norm.
	*/
	template<class Derived>
	template<typename Derived1, typename Derived2>
	EIGEN_DEVICE_FUNC inline Derived& QuaternionBase<Derived>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
	{
	EIGEN_USING_STD(sqrt)
	Vector3 v0 = a.normalized();
	Vector3 v1 = b.normalized();
	Scalar c = v1.dot(v0);

	// if dot == -1, vectors are nearly opposites
	// => accurately compute the rotation axis by computing the
	// intersection of the two planes. This is done by solving:
	// x^T v0 = 0
	// x^T v1 = 0
	// under the constraint:
	// \|\|x\|\| = 1
	// which yields a singular value problem
	if (c < Scalar(-1)+NumTraits<Scalar>::dummy_precision())
	{
	c = numext::maxi(c,Scalar(-1));
	Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
	JacobiSVD<Matrix<Scalar,2,3> > svd(m, ComputeFullV);
	Vector3 axis = svd.matrixV().col(2);

	Scalar w2 = (Scalar(1)+c)*Scalar(0.5);
	this->w() = sqrt(w2);
	this->vec() = axis * sqrt(Scalar(1) - w2);
	return derived();
	}
	Vector3 axis = v0.cross(v1);
	Scalar s = sqrt((Scalar(1)+c)*Scalar(2));
	Scalar invs = Scalar(1)/s;
	this->vec() = axis * invs;
	this->w() = s * Scalar(0.5);

	return derived();
	}

	/** \returns a random unit quaternion following a uniform distribution law on SO(3)
	*
	* \note The implementation is based on http://planning.cs.uiuc.edu/node198.html
	*/
	template<typename Scalar, int Options>
	EIGEN_DEVICE_FUNC Quaternion<Scalar,Options> Quaternion<Scalar,Options>::UnitRandom()
	{
	EIGEN_USING_STD(sqrt)
	EIGEN_USING_STD(sin)
	EIGEN_USING_STD(cos)
	const Scalar u1 = internal::random<Scalar>(0, 1),
	u2 = internal::random<Scalar>(0, 2*EIGEN_PI),
	u3 = internal::random<Scalar>(0, 2*EIGEN_PI);
	const Scalar a = sqrt(Scalar(1) - u1),
	b = sqrt(u1);
	return Quaternion (a * sin(u2), a * cos(u2), b * sin(u3), b * cos(u3));
	}


	/** Returns a quaternion representing a rotation between
	* the two arbitrary vectors \a a and \a b. In other words, the built
	* rotation represent a rotation sending the line of direction \a a
	* to the line of direction \a b, both lines passing through the origin.
	*
	* \returns resulting quaternion
	*
	* Note that the two input vectors do \b not have to be normalized, and
	* do not need to have the same norm.
	*/
	template<typename Scalar, int Options>
	template<typename Derived1, typename Derived2>
	EIGEN_DEVICE_FUNC Quaternion<Scalar,Options> Quaternion<Scalar,Options>::FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
	{
	Quaternion quat;
	quat.setFromTwoVectors(a, b);
	return quat;
	}


	/** \returns the multiplicative inverse of \c *this
	* Note that in most cases, i.e., if you simply want the opposite rotation,
	* and/or the quaternion is normalized, then it is enough to use the conjugate.
	*
	* \sa QuaternionBase::conjugate()
	*/
	template <class Derived>
	EIGEN_DEVICE_FUNC inline Quaternion<typename internal::traits<Derived>::Scalar> QuaternionBase<Derived>::inverse() const
	{
	// FIXME should this function be called multiplicativeInverse and conjugate() be called inverse() or opposite() ??
	Scalar n2 = this->squaredNorm();
	if (n2 > Scalar(0))
	return Quaternion<Scalar>(conjugate().coeffs() / n2);
	else
	{
	// return an invalid result to flag the error
	return Quaternion<Scalar>(Coefficients::Zero());
	}
	}

	// Generic conjugate of a Quaternion
	namespace internal {
	template<int Arch, class Derived, typename Scalar> struct quat_conj
	{
	EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived>& q){
	return Quaternion<Scalar>(q.w(),-q.x(),-q.y(),-q.z());
	}
	};
	}

	/** \returns the conjugate of the \c *this which is equal to the multiplicative inverse
	* if the quaternion is normalized.
	* The conjugate of a quaternion represents the opposite rotation.
	*
	* \sa Quaternion2::inverse()
	*/
	template <class Derived>
	EIGEN_DEVICE_FUNC inline Quaternion<typename internal::traits<Derived>::Scalar>
	QuaternionBase<Derived>::conjugate() const
	{
	return internal::quat_conj<Architecture::Target, Derived,
	typename internal::traits<Derived>::Scalar>::run(*this);

	}

	/** \returns the angle (in radian) between two rotations
	* \sa dot()
	*/
	template <class Derived>
	template <class OtherDerived>
	EIGEN_DEVICE_FUNC inline typename internal::traits<Derived>::Scalar
	QuaternionBase<Derived>::angularDistance(const QuaternionBase<OtherDerived>& other) const
	{
	EIGEN_USING_STD(atan2)
	Quaternion<Scalar> d = (this) other.conjugate();
	return Scalar(2) * atan2( d.vec().norm(), numext::abs(d.w()) );
	}



	/** \returns the spherical linear interpolation between the two quaternions
	* \c *this and \a other at the parameter \a t in [0;1].
	*
	* This represents an interpolation for a constant motion between \c *this and \a other,
	* see also http://en.wikipedia.org/wiki/Slerp.
	*/
	template <class Derived>
	template <class OtherDerived>
	EIGEN_DEVICE_FUNC Quaternion<typename internal::traits<Derived>::Scalar>
	QuaternionBase<Derived>::slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const
	{
	EIGEN_USING_STD(acos)
	EIGEN_USING_STD(sin)
	const Scalar one = Scalar(1) - NumTraits<Scalar>::epsilon();
	Scalar d = this->dot(other);
	Scalar absD = numext::abs(d);

	Scalar scale0;
	Scalar scale1;

	if(absD>=one)
	{
	scale0 = Scalar(1) - t;
	scale1 = t;
	}
	else
	{
	// theta is the angle between the 2 quaternions
	Scalar theta = acos(absD);
	Scalar sinTheta = sin(theta);

	scale0 = sin( ( Scalar(1) - t ) * theta) / sinTheta;
	scale1 = sin( ( t * theta) ) / sinTheta;
	}
	if(d<Scalar(0)) scale1 = -scale1;

	return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs());
	}

	namespace internal {

	// set from a rotation matrix
	template<typename Other>
	struct quaternionbase_assign_impl<Other,3,3>
	{
	typedef typename Other::Scalar Scalar;
	template<class Derived> EIGEN_DEVICE_FUNC static inline void run(QuaternionBase<Derived>& q, const Other& a_mat)
	{
	const typename internal::nested_eval<Other,2>::type mat(a_mat);
	EIGEN_USING_STD(sqrt)
	// This algorithm comes from "Quaternion Calculus and Fast Animation",
	// Ken Shoemake, 1987 SIGGRAPH course notes
	Scalar t = mat.trace();
	if (t > Scalar(0))
	{
	t = sqrt(t + Scalar(1.0));
	q.w() = Scalar(0.5)*t;
	t = Scalar(0.5)/t;
	q.x() = (mat.coeff(2,1) - mat.coeff(1,2)) * t;
	q.y() = (mat.coeff(0,2) - mat.coeff(2,0)) * t;
	q.z() = (mat.coeff(1,0) - mat.coeff(0,1)) * t;
	}
	else
	{
	Index i = 0;
	if (mat.coeff(1,1) > mat.coeff(0,0))
	i = 1;
	if (mat.coeff(2,2) > mat.coeff(i,i))
	i = 2;
	Index j = (i+1)%3;
	Index k = (j+1)%3;

	t = sqrt(mat.coeff(i,i)-mat.coeff(j,j)-mat.coeff(k,k) + Scalar(1.0));
	q.coeffs().coeffRef(i) = Scalar(0.5) * t;
	t = Scalar(0.5)/t;
	q.w() = (mat.coeff(k,j)-mat.coeff(j,k))*t;
	q.coeffs().coeffRef(j) = (mat.coeff(j,i)+mat.coeff(i,j))*t;
	q.coeffs().coeffRef(k) = (mat.coeff(k,i)+mat.coeff(i,k))*t;
	}
	}
	};

	// set from a vector of coefficients assumed to be a quaternion
	template<typename Other>
	struct quaternionbase_assign_impl<Other,4,1>
	{
	typedef typename Other::Scalar Scalar;
	template<class Derived> EIGEN_DEVICE_FUNC static inline void run(QuaternionBase<Derived>& q, const Other& vec)
	{
	q.coeffs() = vec;
	}
	};

	} // end namespace internal

	} // end namespace Eigen

	#endif // EIGEN_QUATERNION_H

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