Squashed 'third_party/eigen/' content from commit 61d72f6
Change-Id: Iccc90fa0b55ab44037f018046d2fcffd90d9d025
git-subtree-dir: third_party/eigen
git-subtree-split: 61d72f6383cfa842868c53e30e087b0258177257
diff --git a/Eigen/src/Eigen2Support/Geometry/Quaternion.h b/Eigen/src/Eigen2Support/Geometry/Quaternion.h
new file mode 100644
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+++ b/Eigen/src/Eigen2Support/Geometry/Quaternion.h
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+// 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