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Diffstat (limited to 'eigen/Eigen/src/Eigen2Support/Geometry/Quaternion.h')
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1 files changed, 495 insertions, 0 deletions
diff --git a/eigen/Eigen/src/Eigen2Support/Geometry/Quaternion.h b/eigen/Eigen/src/Eigen2Support/Geometry/Quaternion.h new file mode 100644 index 0000000..4b6390c --- /dev/null +++ b/eigen/Eigen/src/Eigen2Support/Geometry/Quaternion.h @@ -0,0 +1,495 @@ +// This file is part of Eigen, a lightweight C++ template library +// for linear algebra. +// +// Copyright (C) 2008 Gael Guennebaud <g.gael@free.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/. + +// no include guard, we'll include this twice from All.h from Eigen2Support, and it's internal anyway + +namespace Eigen { + +template<typename Other, + int OtherRows=Other::RowsAtCompileTime, + int OtherCols=Other::ColsAtCompileTime> +struct ei_quaternion_assign_impl; + +/** \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 + */ + +template<typename _Scalar> struct ei_traits<Quaternion<_Scalar> > +{ + typedef _Scalar Scalar; +}; + +template<typename _Scalar> +class Quaternion : public RotationBase<Quaternion<_Scalar>,3> +{ + typedef RotationBase<Quaternion<_Scalar>,3> Base; + +public: + EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,4) + + using Base::operator*; + + /** the scalar type of the coefficients */ + typedef _Scalar Scalar; + + /** the type of the Coefficients 4-vector */ + typedef 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 m_coeffs.coeff(0); } + /** \returns the \c y coefficient */ + inline Scalar y() const { return m_coeffs.coeff(1); } + /** \returns the \c z coefficient */ + inline Scalar z() const { return m_coeffs.coeff(2); } + /** \returns the \c w coefficient */ + inline Scalar w() const { return m_coeffs.coeff(3); } + + /** \returns a reference to the \c x coefficient */ + inline Scalar& x() { return m_coeffs.coeffRef(0); } + /** \returns a reference to the \c y coefficient */ + inline Scalar& y() { return m_coeffs.coeffRef(1); } + /** \returns a reference to the \c z coefficient */ + inline Scalar& z() { return m_coeffs.coeffRef(2); } + /** \returns a reference to the \c w coefficient */ + inline Scalar& w() { return m_coeffs.coeffRef(3); } + + /** \returns a read-only vector expression of the imaginary part (x,y,z) */ + inline const Block<const Coefficients,3,1> vec() const { return m_coeffs.template start<3>(); } + + /** \returns a vector expression of the imaginary part (x,y,z) */ + inline Block<Coefficients,3,1> vec() { return m_coeffs.template start<3>(); } + + /** \returns a read-only vector expression of the coefficients (x,y,z,w) */ + inline const Coefficients& coeffs() const { return m_coeffs; } + + /** \returns a vector expression of the coefficients (x,y,z,w) */ + inline Coefficients& coeffs() { return m_coeffs; } + + /** 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; } + + /** Copy constructor */ + inline Quaternion(const Quaternion& other) { m_coeffs = other.m_coeffs; } + + /** 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. + * \sa operator=(MatrixBase<Derived>) + */ + template<typename Derived> + explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; } + + Quaternion& operator=(const Quaternion& other); + Quaternion& operator=(const AngleAxisType& aa); + template<typename Derived> + Quaternion& operator=(const MatrixBase<Derived>& m); + + /** \returns a quaternion representing an identity rotation + * \sa MatrixBase::Identity() + */ + static inline Quaternion Identity() { return Quaternion(1, 0, 0, 0); } + + /** \sa Quaternion::Identity(), MatrixBase::setIdentity() + */ + inline Quaternion& setIdentity() { m_coeffs << 0, 0, 0, 1; return *this; } + + /** \returns the squared norm of the quaternion's coefficients + * \sa Quaternion::norm(), MatrixBase::squaredNorm() + */ + inline Scalar squaredNorm() const { return m_coeffs.squaredNorm(); } + + /** \returns the norm of the quaternion's coefficients + * \sa Quaternion::squaredNorm(), MatrixBase::norm() + */ + inline Scalar norm() const { return m_coeffs.norm(); } + + /** Normalizes the quaternion \c *this + * \sa normalized(), MatrixBase::normalize() */ + inline void normalize() { m_coeffs.normalize(); } + /** \returns a normalized version of \c *this + * \sa normalize(), MatrixBase::normalized() */ + inline Quaternion normalized() const { return Quaternion(m_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() + */ + inline Scalar eigen2_dot(const Quaternion& other) const { return m_coeffs.eigen2_dot(other.m_coeffs); } + + inline Scalar angularDistance(const Quaternion& other) const; + + Matrix3 toRotationMatrix(void) const; + + template<typename Derived1, typename Derived2> + Quaternion& setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b); + + inline Quaternion operator* (const Quaternion& q) const; + inline Quaternion& operator*= (const Quaternion& q); + + Quaternion inverse(void) const; + Quaternion conjugate(void) const; + + Quaternion slerp(Scalar t, const Quaternion& other) const; + + template<typename Derived> + Vector3 operator* (const MatrixBase<Derived>& vec) 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<Quaternion,Quaternion<NewScalarType> >::type cast() const + { return typename internal::cast_return_type<Quaternion,Quaternion<NewScalarType> >::type(*this); } + + /** Copy constructor with scalar type conversion */ + template<typename OtherScalarType> + inline explicit Quaternion(const Quaternion<OtherScalarType>& other) + { m_coeffs = other.coeffs().template cast<Scalar>(); } + + /** \returns \c true if \c *this is approximately equal to \a other, within the precision + * determined by \a prec. + * + * \sa MatrixBase::isApprox() */ + bool isApprox(const Quaternion& other, typename NumTraits<Scalar>::Real prec = precision<Scalar>()) const + { return m_coeffs.isApprox(other.m_coeffs, prec); } + +protected: + Coefficients m_coeffs; +}; + +/** \ingroup Geometry_Module + * single precision quaternion type */ +typedef Quaternion<float> Quaternionf; +/** \ingroup Geometry_Module + * double precision quaternion type */ +typedef Quaternion<double> Quaterniond; + +// Generic Quaternion * Quaternion product +template<typename Scalar> inline Quaternion<Scalar> +ei_quaternion_product(const Quaternion<Scalar>& a, const Quaternion<Scalar>& 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 <typename Scalar> +inline Quaternion<Scalar> Quaternion<Scalar>::operator* (const Quaternion& other) const +{ + return ei_quaternion_product(*this,other); +} + +/** \sa operator*(Quaternion) */ +template <typename Scalar> +inline Quaternion<Scalar>& Quaternion<Scalar>::operator*= (const Quaternion& other) +{ + return (*this = *this * other); +} + +/** 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: + * - Quaternion: 30n + * - Via a Matrix3: 24 + 15n + */ +template <typename Scalar> +template<typename Derived> +inline typename Quaternion<Scalar>::Vector3 +Quaternion<Scalar>::operator* (const MatrixBase<Derived>& 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; + uv = 2 * this->vec().cross(v); + return v + this->w() * uv + this->vec().cross(uv); +} + +template<typename Scalar> +inline Quaternion<Scalar>& Quaternion<Scalar>::operator=(const Quaternion& other) +{ + m_coeffs = other.m_coeffs; + return *this; +} + +/** Set \c *this from an angle-axis \a aa and returns a reference to \c *this + */ +template<typename Scalar> +inline Quaternion<Scalar>& Quaternion<Scalar>::operator=(const AngleAxisType& aa) +{ + Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings + this->w() = ei_cos(ha); + this->vec() = ei_sin(ha) * aa.axis(); + return *this; +} + +/** 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<typename Scalar> +template<typename Derived> +inline Quaternion<Scalar>& Quaternion<Scalar>::operator=(const MatrixBase<Derived>& xpr) +{ + ei_quaternion_assign_impl<Derived>::run(*this, xpr.derived()); + return *this; +} + +/** Convert the quaternion to a 3x3 rotation matrix */ +template<typename Scalar> +inline typename Quaternion<Scalar>::Matrix3 +Quaternion<Scalar>::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 *this to be a quaternion representing a rotation sending the vector \a a to the vector \a b. + * + * \returns a reference to *this. + * + * Note that the two input vectors do \b not have to be normalized. + */ +template<typename Scalar> +template<typename Derived1, typename Derived2> +inline Quaternion<Scalar>& Quaternion<Scalar>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b) +{ + Vector3 v0 = a.normalized(); + Vector3 v1 = b.normalized(); + Scalar c = v0.eigen2_dot(v1); + + // if dot == 1, vectors are the same + if (ei_isApprox(c,Scalar(1))) + { + // set to identity + this->w() = 1; this->vec().setZero(); + return *this; + } + // if dot == -1, vectors are opposites + if (ei_isApprox(c,Scalar(-1))) + { + this->vec() = v0.unitOrthogonal(); + this->w() = 0; + return *this; + } + + Vector3 axis = v0.cross(v1); + Scalar s = ei_sqrt((Scalar(1)+c)*Scalar(2)); + Scalar invs = Scalar(1)/s; + this->vec() = axis * invs; + this->w() = s * Scalar(0.5); + + return *this; +} + +/** \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 Quaternion::conjugate() + */ +template <typename Scalar> +inline Quaternion<Scalar> Quaternion<Scalar>::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(conjugate().coeffs() / n2); + else + { + // return an invalid result to flag the error + return Quaternion(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 Quaternion::inverse() + */ +template <typename Scalar> +inline Quaternion<Scalar> Quaternion<Scalar>::conjugate() const +{ + return Quaternion(this->w(),-this->x(),-this->y(),-this->z()); +} + +/** \returns the angle (in radian) between two rotations + * \sa eigen2_dot() + */ +template <typename Scalar> +inline Scalar Quaternion<Scalar>::angularDistance(const Quaternion& other) const +{ + double d = ei_abs(this->eigen2_dot(other)); + if (d>=1.0) + return 0; + return Scalar(2) * std::acos(d); +} + +/** \returns the spherical linear interpolation between the two quaternions + * \c *this and \a other at the parameter \a t + */ +template <typename Scalar> +Quaternion<Scalar> Quaternion<Scalar>::slerp(Scalar t, const Quaternion& other) const +{ + static const Scalar one = Scalar(1) - machine_epsilon<Scalar>(); + Scalar d = this->eigen2_dot(other); + Scalar absD = ei_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 = std::acos(absD); + Scalar sinTheta = ei_sin(theta); + + scale0 = ei_sin( ( Scalar(1) - t ) * theta) / sinTheta; + scale1 = ei_sin( ( t * theta) ) / sinTheta; + if (d<0) + scale1 = -scale1; + } + + return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs()); +} + +// set from a rotation matrix +template<typename Other> +struct ei_quaternion_assign_impl<Other,3,3> +{ + typedef typename Other::Scalar Scalar; + static inline void run(Quaternion<Scalar>& 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 > 0) + { + t = ei_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 + { + int i = 0; + if (mat.coeff(1,1) > mat.coeff(0,0)) + i = 1; + if (mat.coeff(2,2) > mat.coeff(i,i)) + i = 2; + int j = (i+1)%3; + int k = (j+1)%3; + + t = ei_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 ei_quaternion_assign_impl<Other,4,1> +{ + typedef typename Other::Scalar Scalar; + static inline void run(Quaternion<Scalar>& q, const Other& vec) + { + q.coeffs() = vec; + } +}; + +} // end namespace Eigen |