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Diffstat (limited to 'Eigen/src/Geometry/Quaternion.h')
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diff --git a/Eigen/src/Geometry/Quaternion.h b/Eigen/src/Geometry/Quaternion.h new file mode 100644 index 000000000..8792e2da2 --- /dev/null +++ b/Eigen/src/Geometry/Quaternion.h @@ -0,0 +1,778 @@ +// 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> +{ + typedef RotationBase<Derived, 3> Base; +public: + 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; + 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 */ + inline Scalar x() const { return this->derived().coeffs().coeff(0); } + /** \returns the \c y coefficient */ + inline Scalar y() const { return this->derived().coeffs().coeff(1); } + /** \returns the \c z coefficient */ + inline Scalar z() const { return this->derived().coeffs().coeff(2); } + /** \returns the \c w coefficient */ + inline Scalar w() const { return this->derived().coeffs().coeff(3); } + + /** \returns a reference to the \c x coefficient */ + inline Scalar& x() { return this->derived().coeffs().coeffRef(0); } + /** \returns a reference to the \c y coefficient */ + inline Scalar& y() { return this->derived().coeffs().coeffRef(1); } + /** \returns a reference to the \c z coefficient */ + inline Scalar& z() { return this->derived().coeffs().coeffRef(2); } + /** \returns a reference to the \c w coefficient */ + inline Scalar& w() { return this->derived().coeffs().coeffRef(3); } + + /** \returns a read-only vector expression of the imaginary part (x,y,z) */ + inline const VectorBlock<const Coefficients,3> vec() const { return coeffs().template head<3>(); } + + /** \returns a vector expression of the imaginary part (x,y,z) */ + inline VectorBlock<Coefficients,3> vec() { return coeffs().template head<3>(); } + + /** \returns a read-only vector expression of the coefficients (x,y,z,w) */ + inline const typename internal::traits<Derived>::Coefficients& coeffs() const { return derived().coeffs(); } + + /** \returns a vector expression of the coefficients (x,y,z,w) */ + inline typename internal::traits<Derived>::Coefficients& coeffs() { return derived().coeffs(); } + + EIGEN_STRONG_INLINE QuaternionBase<Derived>& operator=(const QuaternionBase<Derived>& other); + template<class OtherDerived> 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); } + + Derived& operator=(const AngleAxisType& aa); + template<class OtherDerived> Derived& operator=(const MatrixBase<OtherDerived>& m); + + /** \returns a quaternion representing an identity rotation + * \sa MatrixBase::Identity() + */ + static inline Quaternion<Scalar> Identity() { return Quaternion<Scalar>(1, 0, 0, 0); } + + /** \sa QuaternionBase::Identity(), MatrixBase::setIdentity() + */ + inline QuaternionBase& setIdentity() { coeffs() << 0, 0, 0, 1; return *this; } + + /** \returns the squared norm of the quaternion's coefficients + * \sa QuaternionBase::norm(), MatrixBase::squaredNorm() + */ + inline Scalar squaredNorm() const { return coeffs().squaredNorm(); } + + /** \returns the norm of the quaternion's coefficients + * \sa QuaternionBase::squaredNorm(), MatrixBase::norm() + */ + inline Scalar norm() const { return coeffs().norm(); } + + /** Normalizes the quaternion \c *this + * \sa normalized(), MatrixBase::normalize() */ + inline void normalize() { coeffs().normalize(); } + /** \returns a normalized copy of \c *this + * \sa normalize(), MatrixBase::normalized() */ + 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> inline Scalar dot(const QuaternionBase<OtherDerived>& other) const { return coeffs().dot(other.coeffs()); } + + template<class OtherDerived> Scalar angularDistance(const QuaternionBase<OtherDerived>& other) const; + + /** \returns an equivalent 3x3 rotation matrix */ + Matrix3 toRotationMatrix() const; + + /** \returns the quaternion which transform \a a into \a b through a rotation */ + template<typename Derived1, typename Derived2> + Derived& setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b); + + template<class OtherDerived> EIGEN_STRONG_INLINE Quaternion<Scalar> operator* (const QuaternionBase<OtherDerived>& q) const; + template<class OtherDerived> EIGEN_STRONG_INLINE Derived& operator*= (const QuaternionBase<OtherDerived>& q); + + /** \returns the quaternion describing the inverse rotation */ + Quaternion<Scalar> inverse() const; + + /** \returns the conjugated quaternion */ + Quaternion<Scalar> conjugate() const; + + /** \returns an interpolation for a constant motion between \a other and \c *this + * \a t in [0;1] + * see http://en.wikipedia.org/wiki/Slerp + */ + template<class OtherDerived> Quaternion<Scalar> slerp(Scalar t, const QuaternionBase<OtherDerived>& other) const; + + /** \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> + bool isApprox(const QuaternionBase<OtherDerived>& other, RealScalar prec = NumTraits<Scalar>::dummy_precision()) const + { return coeffs().isApprox(other.coeffs(), prec); } + + /** return the result vector of \a v through the rotation*/ + EIGEN_STRONG_INLINE Vector3 _transformVector(Vector3 v) 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> + inline typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type cast() const + { + return typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type(derived()); + } + +#ifdef EIGEN_QUATERNIONBASE_PLUGIN +# include EIGEN_QUATERNIONBASE_PLUGIN +#endif +}; + +/*************************************************************************** +* Definition/implementation of Quaternion<Scalar> +***************************************************************************/ + +/** \geometry_module \ingroup Geometry_Module + * + * \class Quaternion + * + * \brief The quaternion class used to represent 3D orientations and rotations + * + * \param _Scalar the scalar type, i.e., the type of the coefficients + * + * 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, quatertions 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 + * + * \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{ + IsAligned = internal::traits<Coefficients>::Flags & AlignedBit, + Flags = IsAligned ? (AlignedBit | LvalueBit) : LvalueBit + }; +}; +} + +template<typename _Scalar, int _Options> +class Quaternion : public QuaternionBase<Quaternion<_Scalar,_Options> > +{ + typedef QuaternionBase<Quaternion<_Scalar,_Options> > Base; + enum { IsAligned = internal::traits<Quaternion>::IsAligned }; + +public: + typedef _Scalar Scalar; + + EIGEN_INHERIT_ASSIGNMENT_EQUAL_OPERATOR(Quaternion) + using Base::operator*=; + + typedef typename internal::traits<Quaternion>::Coefficients Coefficients; + typedef typename Base::AngleAxisType AngleAxisType; + + /** Default constructor leaving the quaternion uninitialized. */ + 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] + */ + inline Quaternion(Scalar w, Scalar x, Scalar y, Scalar z) : m_coeffs(x, y, z, w){} + + /** Constructs and initialize a quaternion from the array data */ + inline Quaternion(const Scalar* data) : m_coeffs(data) {} + + /** Copy constructor */ + template<class Derived> EIGEN_STRONG_INLINE Quaternion(const QuaternionBase<Derived>& other) { this->Base::operator=(other); } + + /** Constructs and initializes a quaternion from the angle-axis \a aa */ + 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> + explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; } + + /** Explicit copy constructor with scalar conversion */ + template<typename OtherScalar, int OtherOptions> + explicit inline Quaternion(const Quaternion<OtherScalar, OtherOptions>& other) + { m_coeffs = other.coeffs().template cast<Scalar>(); } + + template<typename Derived1, typename Derived2> + static Quaternion FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b); + + inline Coefficients& coeffs() { return m_coeffs;} + inline const Coefficients& coeffs() const { return m_coeffs;} + + EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF(IsAligned) + +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, _Options> > + { + typedef _Scalar Scalar; + typedef Map<Matrix<_Scalar,4,1>, _Options> Coefficients; + + typedef traits<Quaternion<_Scalar, _Options> > TraitsBase; + enum { + IsAligned = TraitsBase::IsAligned, + + Flags = TraitsBase::Flags + }; + }; +} + +namespace internal { + template<typename _Scalar, int _Options> + struct traits<Map<const Quaternion<_Scalar>, _Options> >: + traits<Quaternion<_Scalar> > + { + typedef _Scalar Scalar; + typedef Map<const Matrix<_Scalar,4,1>, _Options> Coefficients; + + typedef traits<Quaternion<_Scalar, _Options> > TraitsBase; + enum { + IsAligned = TraitsBase::IsAligned, + Flags = TraitsBase::Flags & ~LvalueBit + }; + }; +} + +/** \brief Quaternion expression mapping a constant memory buffer + * + * \param _Scalar the type of the Quaternion coefficients + * \param _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> > +{ + typedef QuaternionBase<Map<const Quaternion<_Scalar>, _Options> > Base; + + public: + typedef _Scalar Scalar; + typedef typename internal::traits<Map>::Coefficients Coefficients; + EIGEN_INHERIT_ASSIGNMENT_EQUAL_OPERATOR(Map) + using Base::operator*=; + + /** Constructs a Mapped Quaternion object from the pointer \a coeffs + * + * The pointer \a coeffs must reference the four coeffecients 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_STRONG_INLINE Map(const Scalar* coeffs) : m_coeffs(coeffs) {} + + inline const Coefficients& coeffs() const { return m_coeffs;} + + protected: + const Coefficients m_coeffs; +}; + +/** \brief Expression of a quaternion from a memory buffer + * + * \param _Scalar the type of the Quaternion coefficients + * \param _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> > +{ + typedef QuaternionBase<Map<Quaternion<_Scalar>, _Options> > Base; + + public: + typedef _Scalar Scalar; + typedef typename internal::traits<Map>::Coefficients Coefficients; + EIGEN_INHERIT_ASSIGNMENT_EQUAL_OPERATOR(Map) + using Base::operator*=; + + /** Constructs a Mapped Quaternion object from the pointer \a coeffs + * + * The pointer \a coeffs must reference the four coeffecients 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_STRONG_INLINE Map(Scalar* coeffs) : m_coeffs(coeffs) {} + + inline Coefficients& coeffs() { return m_coeffs; } + inline const Coefficients& coeffs() const { return m_coeffs; } + + protected: + Coefficients m_coeffs; +}; + +/** \ingroup Geometry_Module + * Map an unaligned array of single precision scalar as a quaternion */ +typedef Map<Quaternion<float>, 0> QuaternionMapf; +/** \ingroup Geometry_Module + * Map an unaligned array of double precision scalar as a quaternion */ +typedef Map<Quaternion<double>, 0> QuaternionMapd; +/** \ingroup Geometry_Module + * Map a 16-bits aligned array of double precision scalars as a quaternion */ +typedef Map<Quaternion<float>, Aligned> QuaternionMapAlignedf; +/** \ingroup Geometry_Module + * Map a 16-bits 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, int _Options> struct quat_product +{ + 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_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, + internal::traits<Derived>::IsAligned && internal::traits<OtherDerived>::IsAligned>::run(*this, other); +} + +/** \sa operator*(Quaternion) */ +template <class Derived> +template <class OtherDerived> +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_STRONG_INLINE typename QuaternionBase<Derived>::Vector3 +QuaternionBase<Derived>::_transformVector(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 litterature (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_STRONG_INLINE QuaternionBase<Derived>& QuaternionBase<Derived>::operator=(const QuaternionBase<Derived>& other) +{ + coeffs() = other.coeffs(); + return derived(); +} + +template<class Derived> +template<class OtherDerived> +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_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const AngleAxisType& aa) +{ + Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings + this->w() = internal::cos(ha); + this->vec() = internal::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> +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> +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> +inline Derived& QuaternionBase<Derived>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b) +{ + using std::max; + Vector3 v0 = a.normalized(); + Vector3 v1 = b.normalized(); + Scalar c = v1.dot(v0); + + // if dot == -1, vectors are nearly opposites + // => accuraletly 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 = max<Scalar>(c,-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() = internal::sqrt(w2); + this->vec() = axis * internal::sqrt(Scalar(1) - w2); + return derived(); + } + Vector3 axis = v0.cross(v1); + Scalar s = internal::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 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> +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> +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 > 0) + return Quaternion<Scalar>(conjugate().coeffs() / n2); + else + { + // return an invalid result to flag the error + return Quaternion<Scalar>(Coefficients::Zero()); + } +} + +/** \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> +inline Quaternion<typename internal::traits<Derived>::Scalar> +QuaternionBase<Derived>::conjugate() const +{ + return Quaternion<Scalar>(this->w(),-this->x(),-this->y(),-this->z()); +} + +/** \returns the angle (in radian) between two rotations + * \sa dot() + */ +template <class Derived> +template <class OtherDerived> +inline typename internal::traits<Derived>::Scalar +QuaternionBase<Derived>::angularDistance(const QuaternionBase<OtherDerived>& other) const +{ + using std::acos; + double d = internal::abs(this->dot(other)); + if (d>=1.0) + return Scalar(0); + return static_cast<Scalar>(2 * acos(d)); +} + +/** \returns the spherical linear interpolation between the two quaternions + * \c *this and \a other at the parameter \a t + */ +template <class Derived> +template <class OtherDerived> +Quaternion<typename internal::traits<Derived>::Scalar> +QuaternionBase<Derived>::slerp(Scalar t, const QuaternionBase<OtherDerived>& other) const +{ + using std::acos; + static const Scalar one = Scalar(1) - NumTraits<Scalar>::epsilon(); + Scalar d = this->dot(other); + Scalar absD = internal::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 = internal::sin(theta); + + scale0 = internal::sin( ( Scalar(1) - t ) * theta) / sinTheta; + scale1 = internal::sin( ( t * theta) ) / sinTheta; + } + if(d<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; + typedef DenseIndex Index; + template<class Derived> static inline void run(QuaternionBase<Derived>& q, const Other& mat) + { + // 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 + { + DenseIndex i = 0; + if (mat.coeff(1,1) > mat.coeff(0,0)) + i = 1; + if (mat.coeff(2,2) > mat.coeff(i,i)) + i = 2; + DenseIndex j = (i+1)%3; + DenseIndex 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> static inline void run(QuaternionBase<Derived>& q, const Other& vec) + { + q.coeffs() = vec; + } +}; + +} // end namespace internal + +} // end namespace Eigen + +#endif // EIGEN_QUATERNION_H |