SO3< T > Class Template Reference

Class representing a member of the \(SO(3)\) manifold, or a 3D rotation. More...

#include <SO3.h>

Public Member Functions
	SO3 ()
	Create a rotation (with garbage data).
	SO3 (const Ref< const Vec4T > &arr)
	Create a transform from an array representing all rotation fields in \(\boldsymbol{q}\in\mathbb{R}^4\).
	SO3 (const SO3 &q)
	Copy constructor from another rotation.
	SO3 (const T *data)
	Create a rotation from a pointer array representing all rotation fields in \(\boldsymbol{q}\in\mathbb{R}^4\).
T &	operator[] (int i)
	Access a field from \(\boldsymbol{q}\in\mathbb{R}^4\).
const T &	w () const
	Access the real element of the rotation.
const T &	x () const
	Access the first imaginary element of the rotation.
const T &	y () const
	Access the second imaginary element of the rotation.
const T &	z () const
	Access the third imaginary element of the rotation.
T &	w ()
	Access the real element of the rotation.
T &	x ()
	Access the first imaginary element of the rotation.
T &	y ()
	Access the second imaginary element of the rotation.
T &	z ()
	Access the third imaginary element of the rotation.
const Vec4T	elements () const
	Access all elements of \(\boldsymbol{q}\in\mathbb{R}^4\).
Vec4T	array () const
	Access all elements of \(\boldsymbol{q}\in\mathbb{R}^4\).
T *	data ()
	Access pointer to all elements of \(\boldsymbol{q}\in\mathbb{R}^4\).
const T *	data () const
	Access pointer to all elements of \(\boldsymbol{q}\in\mathbb{R}^4\).
SO3	copy () const
	Get a deep copy of the current rotation.
void	normalize ()
	Normalize the elements of \(\boldsymbol{q}\) to make it a valid rotation.
SO3	normalized ()
	Obtain a normalized copy of the current rotation.
Mat3T	R () const
	Convert the rotation to matrix representation \(\boldsymbol{R}\in \mathbb{R}^{3\times 3}\).
SO3	inverse () const
	Obtain the inverse rotation \(\boldsymbol{R}_A^B\rightarrow \boldsymbol{R}_B^A\).
SO3 &	invert ()
	Invert the current rotation \(\boldsymbol{R}_A^B\rightarrow \boldsymbol{R}_B^A\).
T	roll () const
	Obtain the roll component of the yaw-pitch-roll successive-axes Euler angle representation.
T	pitch () const
	Obtain the pitch component of the yaw-pitch-roll successive-axes Euler angle representation.
T	yaw () const
	Obtain the yaw component of the yaw-pitch-roll successive-axes Euler angle representation.
Vec3T	toEuler () const
	Obtain the Euler angle representation as a 3-vector.
Mat4T	qMatLeft () const
	Obtain an equivalent 4-by-4 matrix representation of \(\boldsymbol{q}\rightarrow [\boldsymbol{q}]_L\) such that \(\boldsymbol{q}\otimes \boldsymbol{q}_{\text{other}}=[\boldsymbol{q}]_L \boldsymbol{q}_{\text{other}}\).
template<typename Tout = T, typename T2 >
SO3< Tout >	otimes (const SO3< T2 > &q) const
	Implementation of group composition: \(\boldsymbol{q}_B^C \otimes \boldsymbol{q}_A^B\rightarrow \boldsymbol{q}_A^C\).
template<typename Tout = T, typename T2 >
SO3< Tout >	oplus (const Matrix< T2, 3, 1 > &delta) const
	Implementation of tangent space group perturbations: \(\boldsymbol{q}_A^B\oplus \boldsymbol{\theta}_B^{B'} \rightarrow \boldsymbol{q}_A^{B'}\).
template<typename Tout = T, typename T2 >
Matrix< Tout, 3, 1 >	ominus (const SO3< T2 > &q) const
	Implementation of group subtraction: \(\boldsymbol{q}_A^B\ominus \boldsymbol{q}_A^{B'} \rightarrow \boldsymbol{\theta}_B^{B'}\).
SO3 &	operator= (const SO3 &q)
	Copy constructor.
template<typename T2 >
SO3	operator* (const SO3< T2 > &q) const
	Invocation of otimes via multiplication.
template<typename T2 >
SO3 &	operator*= (const SO3< T2 > &q)
	Invocation of otimes via multiplication.
SO3 &	operator*= (const double &s)
	Scale a rotation by a scalar.
SO3 &	operator/= (const double &s)
	Scale a rotation by a scalar.
SO3	operator/ (const double &s) const
	Scale a rotation by a scalar.
template<typename Tout = T, typename T2 >
Matrix< Tout, 3, 1 >	operator* (const Matrix< T2, 3, 1 > &v) const
	Transform a vector via multiplication: \(\boldsymbol{q}_A^B\boldsymbol{t}^A \rightarrow \boldsymbol{t}^B\).
Vec3T	operator* (const Vec3T &v) const
	Transform a vector via multiplication: \(\boldsymbol{q}_A^B\boldsymbol{t}^A \rightarrow \boldsymbol{t}^B\).
SO3	operator+ (const Vec3T &v) const
	Invocation of oplus via addition.
SO3 &	operator+= (const Vec3T &v)
	Invocation of oplus via addition.
template<typename T2 >
Vec3T	operator- (const SO3< T2 > &q) const
	Invocation of ominus via subtraction.
template<typename T2 >
SO3< T2 >	cast () const
	Cast the underlying numeric type.

Static Public Member Functions
static SO3	random ()
	Obtain a random rotation.
static SO3	identity ()
	Obtain an identity \(SO(3)\) rotation.
static SO3	nans ()
	Obtain a rotation full of NaNs.
static SO3	fromAxisAngle (const Vec3T &axis, const T angle)
	Convert an angle (in radians) with accompanying axis (defined by a vector) into a rotation.
static SO3	fromEuler (const T roll, const T pitch, const T yaw)
	Construct a rotation from yaw-pitch-roll successive-axes Euler angles.
static SO3	fromR (const Mat3T &m)
	Convert a rotation matrix \(\boldsymbol{R}\in\mathbb{R}^{3\times 3}\) to a \(SO(3)\) rotation.
static SO3	fromTwoUnitVectors (const Vec3T &u, const Vec3T &v)
	Given two unit vectors \(\boldsymbol{u},\boldsymbol{v}\in \mathbb{R}^3\), returns the rotation that rotates \(\boldsymbol{u}\rightarrow\boldsymbol{v}\).
static SO3	fromQuat (const T qw, const T qx, const T qy, const T qz)
	Construct a rotation from the individual fields.
static SO3	fromQuat (const Vec4T &qvec)
	Construct a rotation from a rotation fields vector.
static SO3	fromQuat (const Quaternion< T > quat)
	Construct a rotation from an Eigen quaternion.
static Mat3T	hat (const Vec3T &omega)
	Hat operator implementation, which coverts the tangent-space vector representation to the corresponding Lie algebra: \(\mathbb{R}^3\rightarrow \mathfrak{so}(3)\).
static Vec3T	vee (const Mat3T &Omega)
	Vee operator implementation, which coverts the Lie algebra representation to a tangent-space vector representation: \(\mathfrak{so}(3) \rightarrow \mathbb{R}^3\).
static Mat3T	log (const SO3 &q)
	Logarithmic chart map implementation: \(SO(3) \rightarrow \mathfrak{so}(3)\).
static Vec3T	Log (const SO3 &q)
	Logarithmic chart map implementation: \(SO(3) \rightarrow \mathbb{R}^3\).
static SO3	exp (const Mat3T &Omega)
	Exponential chart map implementation: \(\mathfrak{so}(3) \rightarrow SO(3)\).
static SO3	Exp (const Vec3T &omega)
	Exponential chart map implementation: \(\mathbb{R}^3 \rightarrow SO(3)\).

Public Attributes
EIGEN_MAKE_ALIGNED_OPERATOR_NEW Map< Vec4T >	arr_
	Memory-mapped array representing all rotation fields in \(\boldsymbol{q}\).

Detailed Description

template<typename T>
class SO3< T >

Class representing a member of the \(SO(3)\) manifold, or a 3D rotation.

Constructor & Destructor Documentation

SO3() [1/4]

template<typename T >

SO3< T >::SO3 ( )

inline

Create a rotation (with garbage data).

SO3() [2/4]

template<typename T >

SO3< T >::SO3 ( const Ref< const Vec4T > & arr )

inline

Create a transform from an array representing all rotation fields in \(\boldsymbol{q}\in\mathbb{R}^4\).

SO3() [3/4]

template<typename T >

SO3< T >::SO3 ( const SO3< T > & q )

inline

Copy constructor from another rotation.

SO3() [4/4]

template<typename T >

SO3< T >::SO3 ( const T * data )

inline

Create a rotation from a pointer array representing all rotation fields in \(\boldsymbol{q}\in\mathbb{R}^4\).

Member Function Documentation

array()

template<typename T >

Vec4T SO3< T >::array ( ) const

inline

Access all elements of \(\boldsymbol{q}\in\mathbb{R}^4\).

cast()

template<typename T >

template<typename T2 >

SO3< T2 > SO3< T >::cast ( ) const

inline

Cast the underlying numeric type.

copy()

template<typename T >

SO3 SO3< T >::copy ( ) const

inline

Get a deep copy of the current rotation.

data() [1/2]

template<typename T >

T * SO3< T >::data ( )

inline

Access pointer to all elements of \(\boldsymbol{q}\in\mathbb{R}^4\).

data() [2/2]

template<typename T >

const T * SO3< T >::data ( ) const

inline

Access pointer to all elements of \(\boldsymbol{q}\in\mathbb{R}^4\).

elements()

template<typename T >

const Vec4T SO3< T >::elements ( ) const

inline

Access all elements of \(\boldsymbol{q}\in\mathbb{R}^4\).

exp()

template<typename T >

static SO3 SO3< T >::exp ( const Mat3T & Omega )

inlinestatic

Exponential chart map implementation: \(\mathfrak{so}(3) \rightarrow SO(3)\).

Exp()

template<typename T >

static SO3 SO3< T >::Exp ( const Vec3T & omega )

inlinestatic

Exponential chart map implementation: \(\mathbb{R}^3 \rightarrow SO(3)\).

fromAxisAngle()

template<typename T >

static SO3 SO3< T >::fromAxisAngle ( const Vec3T & axis, const T angle )

inlinestatic

Convert an angle (in radians) with accompanying axis (defined by a vector) into a rotation.

fromEuler()

template<typename T >

static SO3 SO3< T >::fromEuler ( const T roll, const T pitch, const T yaw )

inlinestatic

Construct a rotation from yaw-pitch-roll successive-axes Euler angles.

fromQuat() [1/3]

template<typename T >

static SO3 SO3< T >::fromQuat ( const Quaternion< T > quat )

inlinestatic

Construct a rotation from an Eigen quaternion.

Parameters

quat	The Eigen quaternion.

fromQuat() [2/3]

template<typename T >

static SO3 SO3< T >::fromQuat ( const T qw, const T qx, const T qy, const T qz )

inlinestatic

Construct a rotation from the individual fields.

Parameters

qw	Real component of the rotation \(\boldsymbol{q}\in SO(3)\).
qx	First imaginary component of the rotation \(\boldsymbol{q}\in SO(3)\).
qy	Second imaginary component of the rotation \(\boldsymbol{q}\in SO(3)\).
qz	Third imaginary component of the rotation \(\boldsymbol{q}\in SO(3)\).

fromQuat() [3/3]

template<typename T >

static SO3 SO3< T >::fromQuat ( const Vec4T & qvec )

inlinestatic

Construct a rotation from a rotation fields vector.

Parameters

qvec	The rotation represented as an array \(\boldsymbol{q}\in\mathbb{R}^4\).

fromR()

template<typename T >

static SO3 SO3< T >::fromR ( const Mat3T & m )

inlinestatic

Convert a rotation matrix \(\boldsymbol{R}\in\mathbb{R}^{3\times 3}\) to a \(SO(3)\) rotation.

fromTwoUnitVectors()

template<typename T >

static SO3 SO3< T >::fromTwoUnitVectors ( const Vec3T & u, const Vec3T & v )

inlinestatic

Given two unit vectors \(\boldsymbol{u},\boldsymbol{v}\in \mathbb{R}^3\), returns the rotation that rotates \(\boldsymbol{u}\rightarrow\boldsymbol{v}\).

Naturally, there is roll ambiguity in such a transform, so watch out.

hat()

template<typename T >

static Mat3T SO3< T >::hat ( const Vec3T & omega )

inlinestatic

Hat operator implementation, which coverts the tangent-space vector representation to the corresponding Lie algebra: \(\mathbb{R}^3\rightarrow \mathfrak{so}(3)\).

identity()

template<typename T >

static SO3 SO3< T >::identity ( )

inlinestatic

Obtain an identity \(SO(3)\) rotation.

inverse()

template<typename T >

SO3 SO3< T >::inverse ( ) const

inline

Obtain the inverse rotation \(\boldsymbol{R}_A^B\rightarrow \boldsymbol{R}_B^A\).

invert()

template<typename T >

SO3 & SO3< T >::invert ( )

inline

Invert the current rotation \(\boldsymbol{R}_A^B\rightarrow \boldsymbol{R}_B^A\).

log()

template<typename T >

static Mat3T SO3< T >::log ( const SO3< T > & q )

inlinestatic

Logarithmic chart map implementation: \(SO(3) \rightarrow \mathfrak{so}(3)\).

Log()

template<typename T >

static Vec3T SO3< T >::Log ( const SO3< T > & q )

inlinestatic

Logarithmic chart map implementation: \(SO(3) \rightarrow \mathbb{R}^3\).

nans()

template<typename T >

static SO3 SO3< T >::nans ( )

inlinestatic

Obtain a rotation full of NaNs.

normalize()

template<typename T >

void SO3< T >::normalize ( )

inline

Normalize the elements of \(\boldsymbol{q}\) to make it a valid rotation.

normalized()

template<typename T >

SO3 SO3< T >::normalized ( )

inline

Obtain a normalized copy of the current rotation.

ominus()

template<typename T >

template<typename Tout = T, typename T2 >

Matrix< Tout, 3, 1 > SO3< T >::ominus ( const SO3< T2 > & q ) const

inline

Implementation of group subtraction: \(\boldsymbol{q}_A^B\ominus \boldsymbol{q}_A^{B'} \rightarrow \boldsymbol{\theta}_B^{B'}\).

operator*() [1/3]

template<typename T >

template<typename Tout = T, typename T2 >

Matrix< Tout, 3, 1 > SO3< T >::operator* ( const Matrix< T2, 3, 1 > & v ) const

inline

Transform a vector via multiplication: \(\boldsymbol{q}_A^B\boldsymbol{t}^A \rightarrow \boldsymbol{t}^B\).

operator*() [2/3]

template<typename T >

template<typename T2 >

SO3 SO3< T >::operator* ( const SO3< T2 > & q ) const

inline

Invocation of otimes via multiplication.

operator*() [3/3]

template<typename T >

Vec3T SO3< T >::operator* ( const Vec3T & v ) const

inline

Transform a vector via multiplication: \(\boldsymbol{q}_A^B\boldsymbol{t}^A \rightarrow \boldsymbol{t}^B\).

operator*=() [1/2]

template<typename T >

SO3 & SO3< T >::operator*= ( const double & s )

inline

Scale a rotation by a scalar.

Under the hood, this converts the rotation into a tangent-space vector, scales the vector, then converts the scaled vector back to a rotation.

operator*=() [2/2]

template<typename T >

template<typename T2 >

SO3 & SO3< T >::operator*= ( const SO3< T2 > & q )

inline

Invocation of otimes via multiplication.

operator+()

template<typename T >

SO3 SO3< T >::operator+ ( const Vec3T & v ) const

inline

Invocation of oplus via addition.

operator+=()

template<typename T >

SO3 & SO3< T >::operator+= ( const Vec3T & v )

inline

Invocation of oplus via addition.

operator-()

template<typename T >

template<typename T2 >

Vec3T SO3< T >::operator- ( const SO3< T2 > & q ) const

inline

Invocation of ominus via subtraction.

operator/()

template<typename T >

SO3 SO3< T >::operator/ ( const double & s ) const

inline

Scale a rotation by a scalar.

Under the hood, this converts the rotation into a tangent-space vector, scales the vector, then converts the scaled vector back to a rotation.

operator/=()

template<typename T >

SO3 & SO3< T >::operator/= ( const double & s )

inline

Scale a rotation by a scalar.

Under the hood, this converts the rotation into a tangent-space vector, scales the vector, then converts the scaled vector back to a rotation.

operator=()

template<typename T >

SO3 & SO3< T >::operator= ( const SO3< T > & q )

inline

Copy constructor.

operator[]()

template<typename T >

T & SO3< T >::operator[] ( int i )

inline

Access a field from \(\boldsymbol{q}\in\mathbb{R}^4\).

oplus()

template<typename T >

template<typename Tout = T, typename T2 >

SO3< Tout > SO3< T >::oplus ( const Matrix< T2, 3, 1 > & delta ) const

inline

Implementation of tangent space group perturbations: \(\boldsymbol{q}_A^B\oplus \boldsymbol{\theta}_B^{B'} \rightarrow \boldsymbol{q}_A^{B'}\).

otimes()

template<typename T >

template<typename Tout = T, typename T2 >

SO3< Tout > SO3< T >::otimes ( const SO3< T2 > & q ) const

inline

Implementation of group composition: \(\boldsymbol{q}_B^C \otimes \boldsymbol{q}_A^B\rightarrow \boldsymbol{q}_A^C\).

pitch()

template<typename T >

T SO3< T >::pitch ( ) const

inline

Obtain the pitch component of the yaw-pitch-roll successive-axes Euler angle representation.

qMatLeft()

template<typename T >

Mat4T SO3< T >::qMatLeft ( ) const

inline

Obtain an equivalent 4-by-4 matrix representation of \(\boldsymbol{q}\rightarrow [\boldsymbol{q}]_L\) such that \(\boldsymbol{q}\otimes \boldsymbol{q}_{\text{other}}=[\boldsymbol{q}]_L \boldsymbol{q}_{\text{other}}\).

R()

template<typename T >

Mat3T SO3< T >::R ( ) const

inline

Convert the rotation to matrix representation \(\boldsymbol{R}\in \mathbb{R}^{3\times 3}\).

random()

template<typename T >

static SO3 SO3< T >::random ( )

inlinestatic

Obtain a random rotation.

Returns

A random rotation \(\boldsymbol{q}_B^W\in SO(3)\).

The rotation \(\mathbf{q}_B^W\) will be normalized.

roll()

template<typename T >

T SO3< T >::roll ( ) const

inline

Obtain the roll component of the yaw-pitch-roll successive-axes Euler angle representation.

toEuler()

template<typename T >

Vec3T SO3< T >::toEuler ( ) const

inline

Obtain the Euler angle representation as a 3-vector.

vee()

template<typename T >

static Vec3T SO3< T >::vee ( const Mat3T & Omega )

inlinestatic

Vee operator implementation, which coverts the Lie algebra representation to a tangent-space vector representation: \(\mathfrak{so}(3) \rightarrow \mathbb{R}^3\).

w() [1/2]

template<typename T >

T & SO3< T >::w ( )

inline

Access the real element of the rotation.

w() [2/2]

template<typename T >

const T & SO3< T >::w ( ) const

inline

Access the real element of the rotation.

x() [1/2]

template<typename T >

T & SO3< T >::x ( )

inline

Access the first imaginary element of the rotation.

x() [2/2]

template<typename T >

const T & SO3< T >::x ( ) const

inline

Access the first imaginary element of the rotation.

y() [1/2]

template<typename T >

T & SO3< T >::y ( )

inline

Access the second imaginary element of the rotation.

y() [2/2]

template<typename T >

const T & SO3< T >::y ( ) const

inline

Access the second imaginary element of the rotation.

yaw()

template<typename T >

T SO3< T >::yaw ( ) const

inline

Obtain the yaw component of the yaw-pitch-roll successive-axes Euler angle representation.

z() [1/2]

template<typename T >

T & SO3< T >::z ( )

inline

Access the third imaginary element of the rotation.

z() [2/2]

template<typename T >

const T & SO3< T >::z ( ) const

inline

Access the third imaginary element of the rotation.

Member Data Documentation

arr_

template<typename T >

EIGEN_MAKE_ALIGNED_OPERATOR_NEW Map<Vec4T> SO3< T >::arr_

Memory-mapped array representing all rotation fields in \(\boldsymbol{q}\).

---

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

• include/SO3.h