Logarithmic and exponential maps on SO(N), quaternions, and conversions among rotation representations: rotation matrix, unit quaternion and principal rotation vector. More...

Classes
class	minitensor::Quaternion< T >

Functions
template<typename T >
KOKKOS_INLINE_FUNCTION Quaternion< T >	minitensor::operator* (Quaternion< T > const &p, Quaternion< T > const &q)

template<typename T >
KOKKOS_INLINE_FUNCTION bool	minitensor::operator== (Quaternion< T > const &p, Quaternion< T > const &q)

template<typename T >
KOKKOS_INLINE_FUNCTION bool	minitensor::operator!= (Quaternion< T > const &p, Quaternion< T > const &q)

template<typename T >
KOKKOS_INLINE_FUNCTION Quaternion< T >	minitensor::conjugate (Quaternion< T > const &q)

template<typename T >
KOKKOS_INLINE_FUNCTION T	minitensor::norm (Quaternion< T > const &q)

template<typename T >
KOKKOS_INLINE_FUNCTION Quaternion< T >	minitensor::inverse (Quaternion< T > const &q)

template<typename T >
KOKKOS_INLINE_FUNCTION Quaternion< T >	minitensor::unit (Quaternion< T > const &q)

template<typename T >
std::ostream &	minitensor::operator<< (std::ostream &os, Quaternion< T > const &q)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Quaternion< T >	minitensor::q_of_rt (Tensor< T, N > const &R)

template<typename T >
KOKKOS_INLINE_FUNCTION Vector< T, 3 >	minitensor::rv_of_q (Quaternion< T > const &q)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Quaternion< T >	minitensor::q_of_rv (Vector< T, N > const &rv)

template<typename T >
KOKKOS_INLINE_FUNCTION Tensor< T, 3 >	minitensor::rt_of_q (Quaternion< T > const &q)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Vector< T, 3 >	minitensor::rv_of_rt (Tensor< T, N > const &R)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, 3 >	minitensor::rt_of_rv (Vector< T, N > const &rv)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Vector< T, N >	minitensor::rv_continue (Vector< T, N > const &old, Vector< T, N > const &prev)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::log_rotation (Tensor< T, N > const &R)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::log_rotation_pi (Tensor< T, N > const &R)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::exp_skew_symmetric (Tensor< T, N > const &r)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Vector< T, N >	minitensor::vee (Tensor< T, N > const &W)

Detailed Description

Logarithmic and exponential maps on SO(N), quaternions, and conversions among rotation representations: rotation matrix, unit quaternion and principal rotation vector.

Function Documentation

◆ operator*()

template<typename T >

KOKKOS_INLINE_FUNCTION Quaternion< T > minitensor::operator*	(	Quaternion< T > const &	p,
		Quaternion< T > const &	q
	)

Hamilton product of two quaternions

Parameters

p	quaternion
q	quaternion

Returns: \( p q = (p_s q_s - p_v \cdot q_v; \; p_s q_v + q_s p_v + p_v \times q_v) \)

Definition at line 393 of file MiniTensor_Quaternion.h.

◆ operator==()

template<typename T >

KOKKOS_INLINE_FUNCTION bool minitensor::operator==	(	Quaternion< T > const &	p,
		Quaternion< T > const &	q
	)

Quaternion equality, tested component-wise.

Parameters

p	quaternion
q	quaternion

Definition at line 411 of file MiniTensor_Quaternion.h.

◆ operator!=()

template<typename T >

KOKKOS_INLINE_FUNCTION bool minitensor::operator!=	(	Quaternion< T > const &	p,
		Quaternion< T > const &	q
	)

Quaternion inequality, tested component-wise.

Parameters

p	quaternion
q	quaternion

Definition at line 422 of file MiniTensor_Quaternion.h.

◆ conjugate()

template<typename T >

KOKKOS_INLINE_FUNCTION Quaternion< T > minitensor::conjugate ( Quaternion< T > const & q )

Quaternion conjugate

Parameters

q	quaternion

Returns: \( \bar{q} = (q_s; -q_v) \)

Definition at line 433 of file MiniTensor_Quaternion.h.

◆ norm()

template<typename T >

KOKKOS_INLINE_FUNCTION T minitensor::norm ( Quaternion< T > const & q )

Quaternion norm

Parameters

q	quaternion

Returns: \( \sqrt{q_s^2 + q_v \cdot q_v} \)

Definition at line 444 of file MiniTensor_Quaternion.h.

◆ inverse()

template<typename T >

KOKKOS_INLINE_FUNCTION Quaternion< T > minitensor::inverse ( Quaternion< T > const & q )

Quaternion inverse

Parameters

q	quaternion

Returns: \( q^{-1} = \bar{q} / |q|^2 \)

Definition at line 456 of file MiniTensor_Quaternion.h.

◆ unit()

template<typename T >

KOKKOS_INLINE_FUNCTION Quaternion< T > minitensor::unit ( Quaternion< T > const & q )

Quaternion normalized to unit norm

Parameters

q	quaternion

Returns: \( q / |q| \)

Definition at line 470 of file MiniTensor_Quaternion.h.

◆ operator<<()

template<typename T >

std::ostream & minitensor::operator<<	(	std::ostream &	os,
		Quaternion< T > const &	q
	)

Quaternion output, scalar-first components separated by commas.

Parameters

os	output stream
q	quaternion

Returns: os

Definition at line 482 of file MiniTensor_Quaternion.h.

◆ q_of_rt()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Quaternion< T > minitensor::q_of_rt ( Tensor< T, N > const & R )

Quaternion of a rotation matrix, using Spurrier's singularity-free algorithm. Selects the largest of the trace and the diagonal entries of the matrix and computes the quaternion components accordingly, which ensures a stable evaluation for all rotation angles including \( \theta = \pi \).

Parameters

R	rotation matrix with \( R \in SO(3) \)

Returns: unit quaternion \( q \) such that rt_of_q(q) recovers R

Definition at line 502 of file MiniTensor_Quaternion.h.

◆ rv_of_q()

template<typename T >

KOKKOS_INLINE_FUNCTION Vector< T, 3 > minitensor::rv_of_q ( Quaternion< T > const & q )

Principal rotation vector of a quaternion. The sign of the quaternion is first normalized so that \( q_s \geq 0 \), which selects the principal rotation \( |rv| \leq \pi \). Small rotations use an alternate evaluation to avoid numerical instability.

Parameters

q	unit quaternion

Returns: rotation vector \( rv \) with \( |rv| \leq \pi \)

Definition at line 616 of file MiniTensor_Quaternion.h.

◆ q_of_rv()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Quaternion< T > minitensor::q_of_rv ( Vector< T, N > const & rv )

Quaternion of a rotation vector, computed as \( q_s = \cos(|rv|/2) \), \( q_v = \frac{1}{2}\,\psi(|rv|/2)\, rv \) with \( \psi(x) = \sin(x)/x \), using asymptotic expansions of \( \psi \) for small angles.

Parameters

rv	rotation vector

Returns: unit quaternion

Definition at line 663 of file MiniTensor_Quaternion.h.

◆ rt_of_q()

template<typename T >

KOKKOS_INLINE_FUNCTION Tensor< T, 3 > minitensor::rt_of_q ( Quaternion< T > const & q )

Rotation matrix of a quaternion, based on the identity \( R = 2 q_v \otimes q_v + 2 q_s \mathrm{skew}(q_v) + (2 q_s^2 - 1) I \).

Parameters

q	unit quaternion

Returns: rotation matrix \( R \in SO(3) \)

Definition at line 686 of file MiniTensor_Quaternion.h.

◆ rv_of_rt()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Vector< T, 3 > minitensor::rv_of_rt ( Tensor< T, N > const & R )

Principal rotation vector of a rotation matrix, computed through the quaternion representation as rv_of_q(q_of_rt(R)). Unlike log_rotation, this path is singularity-free for all rotation angles including \( \theta = \pi \).

Parameters

R	rotation matrix with \( R \in SO(3) \)

Returns: rotation vector \( rv \) with \( |rv| \leq \pi \)

Definition at line 707 of file MiniTensor_Quaternion.h.

◆ rt_of_rv()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, 3 > minitensor::rt_of_rv ( Vector< T, N > const & rv )

Rotation matrix of a rotation vector, computed through the quaternion representation as rt_of_q(q_of_rv(rv)).

Parameters

rv	rotation vector

Returns: rotation matrix \( R \in SO(3) \)

Definition at line 718 of file MiniTensor_Quaternion.h.

◆ rv_continue()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Vector< T, N > minitensor::rv_continue	(	Vector< T, N > const &	old,
		Vector< T, N > const &	prev
	)

Rotation vector equivalent to old (representing the same rotation) but as close as possible to prev. Resolves the \( 2 \pi k \) ambiguity of the rotation-vector representation, which enforces continuity in incremental rotation updates such as time stepping.

Parameters

old	rotation vector to continue
prev	previous rotation vector

Returns: rotation vector equivalent to old and closest to prev

Definition at line 729 of file MiniTensor_Quaternion.h.

◆ log_rotation()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::log_rotation ( Tensor< T, N > const & R )

Logarithmic map of a rotation

Parameters

R	with \( R \in SO(3) \)

Returns: \( r = \log R \) with \( r \in so(3) \)

Definition at line 71 of file MiniTensor_Rotations.h.

◆ log_rotation_pi()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::log_rotation_pi ( Tensor< T, N > const & R )

Logarithmic map of a 180 degree rotation

Parameters

R	with \( R \in SO(3) \)

Returns: \( r = \log R \) with \( r \in so(3) \)

Definition at line 141 of file MiniTensor_Rotations.h.

◆ exp_skew_symmetric()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::exp_skew_symmetric ( Tensor< T, N > const & r )

Exponential map of a skew-symmetric tensor

Parameters

r	\( r \in so(3) \)

Returns: \( R = \exp R \) with \( R \in SO(3) \)

Definition at line 229 of file MiniTensor_Rotations.h.

◆ vee()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Vector< T, N > minitensor::vee ( Tensor< T, N > const & W )

Axial vector of a skew-symmetric tensor, the inverse of skew(Vector). Defined for 3D only.

Parameters

W	skew-symmetric tensor \( W \in so(3) \)

Returns: \( w \) such that \( W u = w \times u \) for all \( u \)

Definition at line 292 of file MiniTensor_Rotations.h.

Classes

Functions

Detailed Description

Function Documentation

◆ operator*()

◆ operator==()

◆ operator!=()

◆ conjugate()

◆ norm()

◆ inverse()

◆ unit()

◆ operator<<()

◆ q_of_rt()

◆ rv_of_q()

◆ q_of_rv()

◆ rt_of_q()

◆ rv_of_rt()

◆ rt_of_rv()

◆ rv_continue()

◆ log_rotation()

◆ log_rotation_pi()

◆ exp_skew_symmetric()

◆ vee()