Quaternion storage and rotation algebra used by OPALX geometry code. More...

#include <Quaternion.hpp>

Inherits ippl::Vector< double, 4 >.

Collaboration diagram for Quaternion:

Public Member Functions
	Quaternion ()
	Construct the identity rotation quaternion \((1, 0, 0, 0)\).

	Quaternion (const Quaternion &)

	Quaternion (const double &, const double &, const double &, const double &)
	Construct a quaternion from its four components \((w, x, y, z)\).

	Quaternion (const ippl::Vector< double, 3 > &)
	Construct a pure quaternion \((0, \mathbf{v})\) from a 3-vector.

	Quaternion (const double &, const ippl::Vector< double, 3 > &)
	Construct a quaternion from scalar part \(w\) and vector part \(\mathbf{v}\).

	Quaternion (const matrix3x3_t &)
	Construct a quaternion from a 3x3 rotation matrix.

Quaternion	inverse () const
	Return the multiplicative inverse.

Quaternion	conjugate () const
	Return the quaternion conjugate \(q^\ast = (w, -\mathbf{v})\).

Quaternion	operator* (const double &) const

Quaternion	operator* (const Quaternion &) const
	Hamilton product of two quaternions.

Quaternion &	operator= (const Quaternion &)=default

Quaternion &	operator*= (const Quaternion &)

Quaternion	operator/ (const double &) const

double	Norm () const
	Return \(\\| q \\|^2 = w^2 + x^2 + y^2 + z^2\).

double	length () const
	Return the Euclidean norm \(\\| q \\|\).

Quaternion &	normalize ()
	Normalize the quaternion in place.

bool	isUnit () const
	Return true if \(\\| q \\|^2 \approx 1\) within a fixed tolerance.

bool	isPure () const
	Return true if the scalar part is approximately zero.

bool	isPureUnit () const
	Return true if the quaternion is both pure and unit length.

double	real () const
	Return the scalar part \(w\).

ippl::Vector< double, 3 >	imag () const
	Return the vector part \(\mathbf{v} = (x, y, z)\).

ippl::Vector< double, 3 >	rotate (const ippl::Vector< double, 3 > &) const
	Rotate a spatial vector by a unit quaternion.

matrix3x3_t	getRotationMatrix () const
	Convert the quaternion to a 3x3 rotation matrix.

Detailed Description

Quaternion storage and rotation algebra used by OPALX geometry code.

The quaternion is stored as

\[ q = (w, x, y, z) = \left(w, \mathbf{v}\right) \]

with scalar part \(w\) and vector part \(\mathbf{v} = (x, y, z)\).

For a unit quaternion, the active rotation of a spatial vector \(\mathbf{r}\) is computed as

\[ \mathbf{r}' = \left(q \, (0, \mathbf{r}) \, q^\ast\right)_{\mathrm{imag}}, \]

where \(q^\ast\) is the quaternion conjugate and the imaginary part is mapped back to \(\mathbb{R}^3\).

OPALX uses quaternions as the rotational kernel of CoordinateSystemTrafo. The conversion to a rotation matrix is defined for unit quaternions; non-unit inputs are normalized before the matrix is built.

Definition at line 29 of file Quaternion.hpp.

Constructor & Destructor Documentation

◆ Quaternion() [1/6]

Quaternion::Quaternion ( )

inline

Construct the identity rotation quaternion \((1, 0, 0, 0)\).

Definition at line 140 of file Quaternion.hpp.

Referenced by operator*=().

◆ Quaternion() [2/6]

Quaternion::Quaternion ( const Quaternion & quat )

inline

Definition at line 142 of file Quaternion.hpp.

◆ Quaternion() [3/6]

Quaternion::Quaternion	(	const double &	x0,
		const double &	x1,
		const double &	x2,
		const double &	x3
	)

inline

Construct a quaternion from its four components \((w, x, y, z)\).

Definition at line 144 of file Quaternion.hpp.

◆ Quaternion() [4/6]

Quaternion::Quaternion ( const ippl::Vector< double, 3 > & vec )

inline

Construct a pure quaternion \((0, \mathbf{v})\) from a 3-vector.

Definition at line 150 of file Quaternion.hpp.

◆ Quaternion() [5/6]

Quaternion::Quaternion	(	const double &	realPart,
		const ippl::Vector< double, 3 > &	vec
	)

inline

Construct a quaternion from scalar part \(w\) and vector part \(\mathbf{v}\).

Definition at line 155 of file Quaternion.hpp.

◆ Quaternion() [6/6]

Quaternion::Quaternion ( const matrix3x3_t & M )

Construct a quaternion from a 3x3 rotation matrix.

Definition at line 22 of file Quaternion.cpp.

Member Function Documentation

◆ conjugate()

Quaternion Quaternion::conjugate ( ) const

inline

Return the quaternion conjugate \(q^\ast = (w, -\mathbf{v})\).

Definition at line 170 of file Quaternion.hpp.

References imag(), and real().

Referenced by OpalBeamline::compileCompatibilityPlacement(), ParallelTracker::computeSpaceChargeFields(), Util::getTaitBryantAngles(), inverse(), CoordinateSystemTrafo::invert(), CoordinateSystemTrafo::operator*=(), CoordinateSystemTrafo::print(), rotate(), OpalBeamline::save3DInput(), TEST_F(), TEST_F(), TEST_F(), and OpalElement::update().

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◆ getRotationMatrix()

matrix3x3_t Quaternion::getRotationMatrix ( ) const

Convert the quaternion to a 3x3 rotation matrix.

The quaternion is normalized first, so the returned matrix represents the same rotation for any non-zero scalar multiple of \(q\).

Definition at line 138 of file Quaternion.cpp.

References normalize().

Referenced by CoordinateSystemTrafo::operator*=(), TEST_F(), TEST_F(), and TEST_F().

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◆ imag()

ippl::Vector< double, 3 > Quaternion::imag ( ) const

inline

Return the vector part \(\mathbf{v} = (x, y, z)\).

Definition at line 178 of file Quaternion.hpp.

Referenced by conjugate(), operator*(), operator*=(), operator/(), and TEST_F().

◆ inverse()

Quaternion Quaternion::inverse ( ) const

Return the multiplicative inverse.

The inverse is

\[ q^{-1} = \frac{q^\ast}{\| q \|^2}. \]

Definition at line 112 of file Quaternion.cpp.

References conjugate(), and Norm().

Referenced by TEST_F(), and TEST_F().

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◆ isPure()

bool Quaternion::isPure ( ) const

inline

Return true if the scalar part is approximately zero.

Definition at line 166 of file Quaternion.hpp.

Referenced by isPureUnit(), TEST_F(), and TEST_F().

◆ isPureUnit()

bool Quaternion::isPureUnit ( ) const

inline

Return true if the quaternion is both pure and unit length.

Definition at line 168 of file Quaternion.hpp.

References isPure(), and isUnit().

Referenced by TEST_F().

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◆ isUnit()

bool Quaternion::isUnit ( ) const

inline

Return true if \(\| q \|^2 \approx 1\) within a fixed tolerance.

Definition at line 164 of file Quaternion.hpp.

References Norm().

Referenced by isPureUnit(), rotate(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), and TEST_F().

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◆ length()

double Quaternion::length ( ) const

inline

Return the Euclidean norm \(\| q \|\).

Definition at line 162 of file Quaternion.hpp.

References Norm().

Referenced by normalize(), and TEST_F().

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◆ Norm()

double Quaternion::Norm ( ) const

inline

Return \(\| q \|^2 = w^2 + x^2 + y^2 + z^2\).

Definition at line 160 of file Quaternion.hpp.

References dot().

Referenced by inverse(), isUnit(), length(), normalize(), rotate(), TEST_F(), TEST_F(), TEST_F(), and TEST_F().

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◆ normalize()

Quaternion & Quaternion::normalize ( )

Normalize the quaternion in place.

After normalization,

\[ \| q \| = 1. \]

Definition at line 100 of file Quaternion.cpp.

References length(), and Norm().

Referenced by getRotationMatrix(), CoordinateSystemTrafo::operator*=(), TEST_F(), and TEST_F().

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◆ operator*() [1/2]

Quaternion Quaternion::operator* ( const double & d ) const

Definition at line 71 of file Quaternion.cpp.

References imag(), and real().

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◆ operator*() [2/2]

Quaternion Quaternion::operator* ( const Quaternion & other ) const

Hamilton product of two quaternions.

For \(q_1 = (w_1, \mathbf{v}_1)\) and \(q_2 = (w_2, \mathbf{v}_2)\),

\[ q_1 q_2 = \left(w_1 w_2 - \mathbf{v}_1 \cdot \mathbf{v}_2, w_1 \mathbf{v}_2 + w_2 \mathbf{v}_1 + \mathbf{v}_1 \times \mathbf{v}_2\right). \]

Definition at line 77 of file Quaternion.cpp.

◆ operator*=()

Quaternion & Quaternion::operator*= ( const Quaternion & other )

Definition at line 82 of file Quaternion.cpp.

References cross(), imag(), and Quaternion().

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◆ operator/()

Quaternion Quaternion::operator/ ( const double & d ) const

Definition at line 94 of file Quaternion.cpp.

References imag(), and real().

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◆ operator=()

Quaternion & Quaternion::operator= ( const Quaternion & )

default

◆ real()

double Quaternion::real ( ) const

inline

Return the scalar part \(w\).

Definition at line 176 of file Quaternion.hpp.

Referenced by conjugate(), operator*(), operator/(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), and TEST_F().

◆ rotate()

ippl::Vector< double, 3 > Quaternion::rotate ( const ippl::Vector< double, 3 > & vec ) const

Rotate a spatial vector by a unit quaternion.

This method requires a unit quaternion and evaluates

\[ \mathbf{r}' = \left(q \, (0, \mathbf{r}) \, q^\ast\right)_{\mathrm{imag}}. \]

Definition at line 125 of file Quaternion.cpp.

References conjugate(), isUnit(), and Norm().

Referenced by OpalBeamline::compileCompatibilityPlacement(), Util::getTaitBryantAngles(), CoordinateSystemTrafo::invert(), CoordinateSystemTrafo::operator*=(), CoordinateSystemTrafo::print(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), TEST_F(), and TEST_F().

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The documentation for this class was generated from the following files:

Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ Quaternion() [1/6]

◆ Quaternion() [2/6]

◆ Quaternion() [3/6]

◆ Quaternion() [4/6]

◆ Quaternion() [5/6]

◆ Quaternion() [6/6]

Member Function Documentation

◆ conjugate()

◆ getRotationMatrix()

◆ imag()

◆ inverse()

◆ isPure()

◆ isPureUnit()

◆ isUnit()

◆ length()

◆ Norm()

◆ normalize()

◆ operator*() [1/2]

◆ operator*() [2/2]

◆ operator*=()

◆ operator/()

◆ operator=()

◆ real()

◆ rotate()