#include <so3.h>

Classes
struct	Invert

Public Member Functions
	SO3 ()
	Default constructor. Initialises the matrix to the identity (no rotation) More...

template<int S, typename P , typename A >
	SO3 (const Vector< S, P, A > &v)
	Construct from the axis of rotation (and angle given by the magnitude). More...

template<int R, int C, typename P , typename A >
	SO3 (const Matrix< R, C, P, A > &rhs)
	Construct from a rotation matrix. More...

template<int S1, int S2, typename P1 , typename P2 , typename A1 , typename A2 >
	SO3 (const Vector< S1, P1, A1 > &a, const Vector< S2, P2, A2 > &b)

template<int R, int C, typename P , typename A >
SO3 &	operator= (const Matrix< R, C, P, A > &rhs)

void	coerce ()
	Modifies the matrix to make sure it is a valid rotation matrix. More...

Vector< 3, Precision >	ln () const

SO3	inverse () const
	Returns the inverse of this matrix (=the transpose, so this is a fast operation) More...

SO3 &	operator*= (const SO3 &rhs)
	Right-multiply by another rotation matrix. More...

template<typename P >
SO3< typename Internal::MultiplyType< Precision, P >::type >	operator* (const SO3< P > &rhs) const
	Right-multiply by another rotation matrix. More...

const Matrix< 3, 3, Precision > &	get_matrix () const
	Returns the SO3 as a Matrix<3> More...

template<int S, typename A >
Vector< 3, Precision >	adjoint (const Vector< S, Precision, A > &vect) const

template<typename PA , typename PB >
	SO3 (const SO3< PA > &a, const SO3< PB > &b)

template<int S, typename VP , typename VA >
SO3< Precision >	exp (const Vector< S, VP, VA > &w)

Static Public Member Functions
template<int S, typename VP , typename A >
static SO3	exp (const Vector< S, VP, A > &vect)

static Matrix< 3, 3, Precision >	generator (int i)

template<typename Base >
static Vector< 3, Precision >	generator_field (int i, const Vector< 3, Precision, Base > &pos)
	Returns the i-th generator times pos. More...

Private Member Functions
	SO3 (const SO3 &so3, const Invert &)

Private Attributes
Matrix< 3, 3, Precision >	my_matrix

Friends
class	SE3< Precision >

std::istream &	operator>> (std::istream &is, SO3< Precision > &rhs)

std::istream &	operator>> (std::istream &is, SE3< Precision > &rhs)

Related Functions
(Note that these are not member functions.)
template<typename Precision >
std::ostream &	operator<< (std::ostream &os, const SO3< Precision > &rhs)

template<typename Precision >
std::istream &	operator>> (std::istream &is, SO3< Precision > &rhs)

template<typename Precision , int S, typename VP , typename VA , typename MA >
void	rodrigues_so3_exp (const Vector< S, VP, VA > &w, const Precision A, const Precision B, Matrix< 3, 3, Precision, MA > &R)

template<int S, typename P , typename PV , typename A >
Vector< 3, typename Internal::MultiplyType< P, PV >::type >	operator* (const SO3< P > &lhs, const Vector< S, PV, A > &rhs)

template<int S, typename P , typename PV , typename A >
Vector< 3, typename Internal::MultiplyType< PV, P >::type >	operator* (const Vector< S, PV, A > &lhs, const SO3< P > &rhs)

template<int R, int C, typename P , typename PM , typename A >
Matrix< 3, C, typename Internal::MultiplyType< P, PM >::type >	operator* (const SO3< P > &lhs, const Matrix< R, C, PM, A > &rhs)

template<int R, int C, typename P , typename PM , typename A >
Matrix< R, 3, typename Internal::MultiplyType< PM, P >::type >	operator* (const Matrix< R, C, PM, A > &lhs, const SO3< P > &rhs)

Detailed Description

template<typename Precision = double>
class TooN::SO3< Precision >

Class to represent a three-dimensional rotation matrix. Three-dimensional rotation matrices are members of the Special Orthogonal Lie group SO3. This group can be parameterised three numbers (a vector in the space of the Lie Algebra). In this class, the three parameters are the finite rotation vector, i.e. a three-dimensional vector whose direction is the axis of rotation and whose length is the angle of rotation in radians. Exponentiating this vector gives the matrix, and the logarithm of the matrix gives this vector.

Definition at line 39 of file so3.h.

Constructor & Destructor Documentation

template<typename Precision = double>

TooN::SO3< Precision >::SO3 ( )

inline

Default constructor. Initialises the matrix to the identity (no rotation)

Definition at line 60 of file so3.h.

template<typename Precision = double>

template<int S, typename P , typename A >

TooN::SO3< Precision >::SO3 ( const Vector< S, P, A > & v )

inline

Construct from the axis of rotation (and angle given by the magnitude).

Definition at line 64 of file so3.h.

template<typename Precision = double>

template<int R, int C, typename P , typename A >

TooN::SO3< Precision >::SO3 ( const Matrix< R, C, P, A > & rhs )

inline

Construct from a rotation matrix.

Definition at line 68 of file so3.h.

template<typename Precision = double>

template<int S1, int S2, typename P1 , typename P2 , typename A1 , typename A2 >

TooN::SO3< Precision >::SO3	(	const Vector< S1, P1, A1 > &	a,
		const Vector< S2, P2, A2 > &	b
	)

inline

creates an SO3 as a rotation that takes Vector a into the direction of Vector b with the rotation axis along a ^ b. If |a ^ b| == 0, it creates the identity rotation. An assertion will fail if Vector a and Vector b are in exactly opposite directions.

Parameters

a	source Vector
b	target Vector

Definition at line 76 of file so3.h.

template<typename Precision = double>

template<typename PA , typename PB >

TooN::SO3< Precision >::SO3	(	const SO3< PA > &	a,
		const SO3< PB > &	b
	)

inline

Definition at line 179 of file so3.h.

template<typename Precision = double>

TooN::SO3< Precision >::SO3	(	const SO3< Precision > &	so3,
		const Invert &
	)

inlineprivate

Definition at line 183 of file so3.h.

Member Function Documentation

template<typename Precision = double>

template<int S, typename A >

Vector<3, Precision> TooN::SO3< Precision >::adjoint ( const Vector< S, Precision, A > & vect ) const

inline

Transfer a vector in the Lie Algebra from one co-ordinate frame to another such that for a matrix $ M $ , the adjoint $Adj()$ obeys $e^{\text{Adj}(v)} = Me^{v}M^{-1}$

Definition at line 172 of file so3.h.

template<typename Precision = double>

void TooN::SO3< Precision >::coerce ( )

inline

Modifies the matrix to make sure it is a valid rotation matrix.

Definition at line 108 of file so3.h.

template<typename Precision = double>

template<int S, typename VP , typename A >

static SO3 TooN::SO3< Precision >::exp ( const Vector< S, VP, A > & vect )

inlinestatic

Exponentiate a vector in the Lie algebra to generate a new SO3. See the Detailed Description for details of this vector.

template<typename Precision = double>

template<int S, typename VP , typename VA >

SO3<Precision> TooN::SO3< Precision >::exp ( const Vector< S, VP, VA > & w )

inline

Perform the exponential of the matrix $\sum_i w_iG_i$

Parameters

w	Weightings of the generator matrices.

Definition at line 250 of file so3.h.

template<typename Precision = double>

static Matrix<3,3, Precision> TooN::SO3< Precision >::generator ( int i )

inlinestatic

Returns the i-th generator. The generators of a Lie group are the basis for the space of the Lie algebra. For SO3, the generators are three $3\times3$ matrices representing the three possible (linearised) rotations.

Definition at line 149 of file so3.h.

template<typename Precision = double>

template<typename Base >

static Vector<3,Precision> TooN::SO3< Precision >::generator_field	(	int	i,
		const Vector< 3, Precision, Base > &	pos
	)

inlinestatic

Returns the i-th generator times pos.

Definition at line 158 of file so3.h.

template<typename Precision = double>

const Matrix<3,3, Precision>& TooN::SO3< Precision >::get_matrix ( ) const

inline

Returns the SO3 as a Matrix<3>

Definition at line 143 of file so3.h.

template<typename Precision = double>

SO3 TooN::SO3< Precision >::inverse ( ) const

inline

Returns the inverse of this matrix (=the transpose, so this is a fast operation)

Definition at line 128 of file so3.h.

template<typename Precision >

Vector< 3, Precision > TooN::SO3< Precision >::ln ( ) const

inline

Take the logarithm of the matrix, generating the corresponding vector in the Lie Algebra. See the Detailed Description for details of this vector.

Definition at line 284 of file so3.h.

template<typename Precision = double>

template<typename P >

SO3<typename Internal::MultiplyType<Precision, P>::type> TooN::SO3< Precision >::operator* ( const SO3< P > & rhs ) const

inline

Right-multiply by another rotation matrix.

Definition at line 138 of file so3.h.

template<typename Precision = double>

SO3& TooN::SO3< Precision >::operator*= ( const SO3< Precision > & rhs )

inline

Right-multiply by another rotation matrix.

Definition at line 131 of file so3.h.

template<typename Precision = double>

template<int R, int C, typename P , typename A >

SO3& TooN::SO3< Precision >::operator= ( const Matrix< R, C, P, A > & rhs )

inline

Assignment operator from a general matrix. This also calls coerce() to make sure that the matrix is a valid rotation matrix.

Definition at line 101 of file so3.h.

Friends And Related Function Documentation

template<int S, typename P , typename PV , typename A >

Vector< 3, typename Internal::MultiplyType< P, PV >::type > operator*	(	const SO3< P > &	lhs,
		const Vector< S, PV, A > &	rhs
	)

Definition at line 332 of file so3.h.

template<int S, typename P , typename PV , typename A >

Vector< 3, typename Internal::MultiplyType< PV, P >::type > operator*	(	const Vector< S, PV, A > &	lhs,
		const SO3< P > &	rhs
	)

Definition at line 339 of file so3.h.

template<int R, int C, typename P , typename PM , typename A >

Matrix< 3, C, typename Internal::MultiplyType< P, PM >::type > operator*	(	const SO3< P > &	lhs,
		const Matrix< R, C, PM, A > &	rhs
	)

Definition at line 346 of file so3.h.

template<int R, int C, typename P , typename PM , typename A >

Matrix< R, 3, typename Internal::MultiplyType< PM, P >::type > operator*	(	const Matrix< R, C, PM, A > &	lhs,
		const SO3< P > &	rhs
	)

Definition at line 353 of file so3.h.

template<typename Precision >

std::ostream & operator<<	(	std::ostream &	os,
		const SO3< Precision > &	rhs
	)

Definition at line 191 of file so3.h.

template<typename Precision = double>

std::istream& operator>>	(	std::istream &	is,
		SO3< Precision > &	rhs
	)

friend

template<typename Precision = double>

std::istream& operator>>	(	std::istream &	is,
		SE3< Precision > &	rhs
	)

friend

template<typename Precision >

std::istream & operator>>	(	std::istream &	is,
		SO3< Precision > &	rhs
	)

Definition at line 198 of file so3.h.

template<typename Precision , int S, typename VP , typename VA , typename MA >

void rodrigues_so3_exp	(	const Vector< S, VP, VA > &	w,
		const Precision	A,
		const Precision	B,
		Matrix< 3, 3, Precision, MA > &	R
	)

Compute a rotation exponential using the Rodrigues Formula. The rotation axis is given by $\vec{w}$ , and the rotation angle must be computed using $\theta = |\vec{w}|$ . This is provided as a separate function primarily to allow fast and rough matrix exponentials using fast and rough approximations to A and B.