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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>
{
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>(Scalar(1), Scalar(0), Scalar(0), Scalar(0)); }
/** \sa QuaternionBase::Identity(), MatrixBase::setIdentity()
*/
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()
*/
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;
template<class OtherDerived> Quaternion<Scalar> slerp(const 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, 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_STRONG_INLINE Vector3 _transformVector(const 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
*
* \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{
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_OPERATORS(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(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 */
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, (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> >
{
typedef QuaternionBase<Map<const Quaternion<_Scalar>, _Options> > Base;
public:
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_STRONG_INLINE Map(const Scalar* coeffs) : m_coeffs(coeffs) {}
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> >
{
typedef QuaternionBase<Map<Quaternion<_Scalar>, _Options> > Base;
public:
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_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 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, 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(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_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)
{
using std::cos;
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>
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;
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 = (max)(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 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 > Scalar(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::atan2;
using std::abs;
Quaternion<Scalar> d = (*this) * other.conjugate();
return Scalar(2) * atan2( d.vec().norm(), 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>
Quaternion<typename internal::traits<Derived>::Scalar>
QuaternionBase<Derived>::slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const
{
using std::acos;
using std::sin;
using std::abs;
static const Scalar one = Scalar(1) - NumTraits<Scalar>::epsilon();
Scalar d = this->dot(other);
Scalar absD = 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;
typedef DenseIndex Index;
template<class Derived> static inline void run(QuaternionBase<Derived>& q, const Other& mat)
{
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
{
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
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