quaternions.h File Reference

Quaternions and Wigner matrices. More...

#include "qpms_types.h"
#include "vectors.h"
#include "tiny_inlines.h"

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Macros
#define	QPMS_QUAT_ATOL   (1e-10)
	Just some arbitrarily chosen "default" value for quaternion comparison tolerance.

Typedefs
using	q = qpms_quat_normalise(q)

Functions
static qpms_quat_t	qpms_quat_2c_from_4d (qpms_quat4d_t q)
	Conversion from the 4*double to the 2*complex quaternion.
static qpms_quat4d_t	qpms_quat_4d_from_2c (qpms_quat_t q)
	Conversion from the 2*complex to the 4*double quaternion.
static qpms_quat_t	qpms_quat_mult (qpms_quat_t p, qpms_quat_t q)
	Quaternion multiplication. More...
static qpms_quat_t	qpms_quat_add (qpms_quat_t p, qpms_quat_t q)
	Quaternion addition.
static qpms_quat_t	qpms_quat_sub (qpms_quat_t p, qpms_quat_t q)
	Quaternion substraction.
static qpms_quat_t	qpms_quat_exp (const qpms_quat_t q)
	Exponential function of a quaternion.
Quaternion scaling with a real static number qpms_quat_t	qpms_quat_rscale (double s, qpms_quat_t q)
Quaternion static conjugation qpms_quat_t	qpms_quat_conj (const qpms_quat_t q)
Quaternion static norm double	qpms_quat_norm (const qpms_quat_t q)
Test approximate equality of static quaternions bool	qpms_quat_isclose (const qpms_quat_t p, const qpms_quat_t q, double atol)
Standardises a quaternion to have the largest component static positive qpms_quat_t	qpms_quat_standardise (qpms_quat_t p, double atol)
Test approximate equality of standardised so that f q f is considered equal to f $q static f bool	qpms_quat_isclose2 (const qpms_quat_t p, const qpms_quat_t q, double atol)
Norm of the quaternion	imaginary (vector) part. static inline double qpms_quat_imnorm(const qpms_quat_t q)
Quaternion normalisation to unit static norm qpms_quat_t	qpms_quat_normalise (qpms_quat_t q)
Logarithm of a static quaternion qpms_quat_t	qpms_quat_log (const qpms_quat_t q)
Quaternion power to a real static exponent qpms_quat_t	qpms_quat_pow (const qpms_quat_t q, const double exponent)
Quaternion static inversion qpms_quat_t	qpms_quat_inv (const qpms_quat_t q)
Make a pure imaginary quaternion from a cartesian static vector qpms_quat_t	qpms_quat_from_cart3 (const cart3_t c)
Make a cartesian vector from the imaginary part of a static quaternion cart3_t	qpms_quat_to_cart3 (const qpms_quat_t q)
return	qpms_quat_to_cart3 (qpms_quat_mult(q, qpms_quat_mult(vv, qc)))
Versor quaternion from rotation	vector (norm of the vector is the rotation angle). static inline qpms_quat_t qpms_quat_from_rotvector(cart3_t v)
Wigner D matrix element from a rotator quaternion for integer a l complex double	qpms_wignerD_elem (qpms_quat_t q, qpms_l_t l, qpms_m_t mp, qpms_m_t m)
A VSWF representation element of the	O (3) group. complex double qpms_vswf_irot_elem_from_irot3(const qpms_irot3_t q
static int	qpms_irot3_checkdet (const qpms_irot3_t p)
Improper rotation static multiplication qpms_irot3_t	qpms_irot3_mult (const qpms_irot3_t p, const qpms_irot3_t q)
Improper rotation inverse static operation qpms_irot3_t	qpms_irot3_inv (qpms_irot3_t p)
Improper rotation power f p n static f qpms_irot3_t	qpms_irot3_pow (const qpms_irot3_t p, int n)
Test approximate equality of static irot3 bool	qpms_irot3_isclose (const qpms_irot3_t p, const qpms_irot3_t q, double atol)
Apply an improper rotation onto a cartesian static vector cart3_t	qpms_irot3_apply_cart3 (const qpms_irot3_t p, const cart3_t v)
versor representing rotation around z static axis qpms_quat_t	qpms_quat_zrot_angle (double angle)
versor representing rotation f C_N k i e of angle f pi k N f around z static axis qpms_quat_t	qpms_quat_zrot_Nk (double N, double k)
Rotation around z static axis qpms_irot3_t	qpms_irot3_zrot_angle (double angle)
Rotation f C_N k i e of angle f pi k N f around z static axis qpms_irot3_t	qpms_irot3_zrot_Nk (double N, double k)

Variables
Quaternion real static unit const qpms_quat_t	QPMS_QUAT_1 = {1, 0}
Quaternion imaginary unit static i const qpms_quat_t	QPMS_QUAT_I = {0, I}
Quaternion imaginury unik static j const qpms_quat_t	QPMS_QUAT_J = {0, 1}
Quaternion imaginary unit static k const qpms_quat_t	QPMS_QUAT_K = {I, 0}
Test approximate equality of standardised	quaternions
const qpms_quat_t	qc = qpms_quat_conj(q)
const qpms_quat_t	vv = qpms_quat_from_cart3(v)
A VSWF representation element of the qpms_l_t	l
	< The O(3) element in the quaternion representation.
A VSWF representation element of the qpms_l_t qpms_m_t	mp
A VSWF representation element of the qpms_l_t qpms_m_t qpms_m_t	m
A VSWF representation element of the qpms_l_t qpms_m_t qpms_m_t bool	pseudo
	< Determines the sign of improper rotations. True for magnetic waves, false otherwise.
static Identity const qpms_irot3_t	QPMS_IROT3_IDENTITY = {{1, 0}, 1}
f pi f rotation around x static axis const qpms_irot3_t	QPMS_IROT3_XROT_PI = {{0, I}, 1}
f pi f rotation around y static axis const qpms_irot3_t	QPMS_IROT3_YROT_PI = {{0, 1}, 1}
f pi f rotation around z static axis const qpms_irot3_t	QPMS_IROT3_ZROT_PI = {{I, 0}, 1}
Spatial static inversion const qpms_irot3_t	QPMS_IROT3_INVERSION = {{1, 0}, -1}
yz plane mirror static symmetry const qpms_irot3_t	QPMS_IROT3_XFLIP = {{0, I}, -1}
xz plane mirror static symmetry const qpms_irot3_t	QPMS_IROT3_YFLIP = {{0, 1}, -1}
xy plane mirror static symmetry const qpms_irot3_t	QPMS_IROT3_ZFLIP = {{I, 0}, -1}
versor representing rotation f C_N k	f

Detailed Description

Quaternions and Wigner matrices.

Function Documentation

O()

A VSWF representation element of the O

(

3

)

const

TODO more doc.

qpms_quat_inv()

Quaternion static inversion qpms_quat_t qpms_quat_inv ( const qpms_quat_t  q )

inlinestatic

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

qpms_quat_mult()

static qpms_quat_t qpms_quat_mult ( qpms_quat_t  p, qpms_quat_t  q  )

inlinestatic

Quaternion multiplication.

\[ (P Q)_a = P_a Q_a - \bar P_b Q_b, \]

\[ (P Q)_b = P_b Q_a + \bar P_a Q_b. \]

qpms_quat_standardise()

Standardises a quaternion to have the largest component static positive qpms_quat_t qpms_quat_standardise ( qpms_quat_t  p, double  atol  )

inlinestatic

This is to remove the ambiguity stemming from the double cover of SO(3).

qpms_wignerD_elem()

Wigner D matrix element from a rotator quaternion for integer a l complex double qpms_wignerD_elem	(	qpms_quat_t	q,
		qpms_l_t	l,
		qpms_m_t	mp,
		qpms_m_t	m
	)

The D matrix are calculated using formulae (3), (4), (6), (7) from http://moble.github.io/spherical_functions/WignerDMatrices.html