SO2< T > Class Template Reference

Class representing a member of the \(SO(2)\) manifold, or a 2D rotation. More...

#include <SO2.h>

Public Member Functions
	SO2 ()
	Create a rotation (with garbage data).
	SO2 (const Ref< const Vec2T > &arr)
	Create a transform from an array representing all rotation fields in \(\boldsymbol{q}\in\mathbb{R}^2\).
	SO2 (const SO2 &q)
	Copy constructor from another rotation.
	SO2 (const T *data)
	Create a rotation from a pointer array representing all rotation fields in \(\boldsymbol{q}\in\mathbb{R}^2\).
T &	operator[] (int i)
	Access a field from \(\boldsymbol{q}\in\mathbb{R}^2\).
const T &	w () const
	Access the real element of the rotation.
const T &	x () const
	Access the imaginary element of the rotation.
T &	w ()
	Access the real element of the rotation.
T &	x ()
	Access the imaginary element of the rotation.
const Vec2T	elements () const
	Access all elements of \(\boldsymbol{q}\in\mathbb{R}^2\).
Vec2T	array () const
	Access all elements of \(\boldsymbol{q}\in\mathbb{R}^2\).
T *	data ()
	Access pointer to all elements of \(\boldsymbol{q}\in\mathbb{R}^2\).
const T *	data () const
	Access pointer to all elements of \(\boldsymbol{q}\in\mathbb{R}^2\).
SO2	copy () const
	Get a deep copy of the current rotation.
void	normalize ()
	Normalize the elements of \(\boldsymbol{q}\) to make it a valid rotation.
SO2	normalized ()
	Obtain a normalized copy of the current rotation.
Mat2T	R () const
	Convert the rotation to matrix representation \(\begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}\).
SO2	inverse () const
	Obtain the inverse rotation \(\boldsymbol{R}_A^B\rightarrow \boldsymbol{R}_B^A\).
SO2 &	invert ()
	Invert the current rotation \(\boldsymbol{R}_A^B\rightarrow \boldsymbol{R}_B^A\).
T	angle () const
	Obtain the equivalent scalar angle \(\theta\) of the rotation.
template<typename Tout = T, typename T2 >
SO2< Tout >	otimes (const SO2< 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 >
SO2< Tout >	oplus (const Matrix< T2, 1, 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, 1, 1 >	ominus (const SO2< T2 > &q) const
	Implementation of group subtraction: \(\boldsymbol{q}_A^B\ominus \boldsymbol{q}_A^{B'} \rightarrow \boldsymbol{\theta}_B^{B'}\).
SO2 &	operator= (const SO2 &q)
	Copy constructor.
template<typename T2 >
SO2	operator* (const SO2< T2 > &q) const
	Invocation of otimes via multiplication.
template<typename T2 >
SO2 &	operator*= (const SO2< T2 > &q)
	Invocation of otimes via multiplication.
SO2 &	operator*= (const double &s)
	Scale a rotation by a scalar.
SO2 &	operator/= (const double &s)
	Scale a rotation by a scalar.
SO2	operator/ (const double &s) const
	Scale a rotation by a scalar.
template<typename Tout = T, typename T2 >
Matrix< Tout, 2, 1 >	operator* (const Matrix< T2, 2, 1 > &v) const
	Transform a vector via multiplication: \(\boldsymbol{q}_A^B\boldsymbol{t}^A \rightarrow \boldsymbol{t}^B\).
Vec2T	operator* (const Vec2T &v) const
	Transform a vector via multiplication: \(\boldsymbol{q}_A^B\boldsymbol{t}^A \rightarrow \boldsymbol{t}^B\).
SO2	operator+ (const Vec1T &v) const
	Invocation of oplus via addition.
SO2 &	operator+= (const Vec1T &v)
	Invocation of oplus via addition.
template<typename T2 >
Vec1T	operator- (const SO2< T2 > &q) const
	Invocation of ominus via subtraction.
template<typename T2 >
SO2< T2 >	cast () const
	Cast the underlying numeric type.

Static Public Member Functions
static SO2	random ()
	Obtain a random rotation.
static SO2	identity ()
	Obtain an identity \(SO(2)\) rotation.
static SO2	nans ()
	Obtain a rotation full of NaNs.
static SO2	fromAngle (const T angle)
	Convert an angle (in radians) into a rotation.
static SO2	fromR (const Mat2T &m)
	Convert a rotation matrix \(\boldsymbol{R}\in\mathbb{R}^{2\times 2}\) to a \(SO(2)\) rotation.
static SO2	fromTwoUnitVectors (const Vec2T &u, const Vec2T &v)
	Given two unit vectors \(\boldsymbol{u},\boldsymbol{v}\in \mathbb{R}^2\), returns the rotation that rotates \(\boldsymbol{u}\rightarrow\boldsymbol{v}\).
static SO2	fromComplex (const T qw, const T qx)
	Construct a rotation from the individual fields.
static SO2	fromComplex (const Vec2T &qvec)
	Construct a rotation from a rotation fields vector.
static Mat2T	hat (const Vec1T &omega)
	Hat operator implementation, which coverts the tangent-space vector representation to the corresponding Lie algebra: \(\mathbb{R}\rightarrow \mathfrak{so}(2)\).
static Vec1T	vee (const Mat2T &Omega)
	Vee operator implementation, which coverts the Lie algebra representation to a tangent-space vector representation: \(\mathfrak{so}(2) \rightarrow \mathbb{R}\).
static Mat2T	log (const SO2 &q)
	Logarithmic chart map implementation: \(SO(2) \rightarrow \mathfrak{so}(2)\).
static Vec1T	Log (const SO2 &q)
	Logarithmic chart map implementation: \(SO(2) \rightarrow \mathbb{R}\).
static SO2	exp (const Mat2T &Omega)
	Exponential chart map implementation: \(\mathfrak{so}(2) \rightarrow SO(2)\).
static SO2	Exp (const Vec1T &omega)
	Exponential chart map implementation: \(\mathbb{R} \rightarrow SO(2)\).

Public Attributes
EIGEN_MAKE_ALIGNED_OPERATOR_NEW Map< Vec2T >	arr_
	Memory-mapped array representing all rotation fields in \(\boldsymbol{q}\).

Detailed Description

template<typename T>
class SO2< T >

Class representing a member of the \(SO(2)\) manifold, or a 2D rotation.

Constructor & Destructor Documentation

SO2() [1/4]

template<typename T >

SO2< T >::SO2 ( )

inline

Create a rotation (with garbage data).

SO2() [2/4]

template<typename T >

SO2< T >::SO2 ( const Ref< const Vec2T > & arr )

inline

Create a transform from an array representing all rotation fields in \(\boldsymbol{q}\in\mathbb{R}^2\).

SO2() [3/4]

template<typename T >

SO2< T >::SO2 ( const SO2< T > & q )

inline

Copy constructor from another rotation.

SO2() [4/4]

template<typename T >

SO2< T >::SO2 ( const T * data )

inline

Create a rotation from a pointer array representing all rotation fields in \(\boldsymbol{q}\in\mathbb{R}^2\).

Member Function Documentation

angle()

template<typename T >

T SO2< T >::angle ( ) const

inline

Obtain the equivalent scalar angle \(\theta\) of the rotation.

\(-\pi \leq \theta \leq \pi\).

array()

template<typename T >

Vec2T SO2< T >::array ( ) const

inline

Access all elements of \(\boldsymbol{q}\in\mathbb{R}^2\).

cast()

template<typename T >

template<typename T2 >

SO2< T2 > SO2< T >::cast ( ) const

inline

Cast the underlying numeric type.

copy()

template<typename T >

SO2 SO2< T >::copy ( ) const

inline

Get a deep copy of the current rotation.

data() [1/2]

template<typename T >

T * SO2< T >::data ( )

inline

Access pointer to all elements of \(\boldsymbol{q}\in\mathbb{R}^2\).

data() [2/2]

template<typename T >

const T * SO2< T >::data ( ) const

inline

Access pointer to all elements of \(\boldsymbol{q}\in\mathbb{R}^2\).

elements()

template<typename T >

const Vec2T SO2< T >::elements ( ) const

inline

Access all elements of \(\boldsymbol{q}\in\mathbb{R}^2\).

exp()

template<typename T >

static SO2 SO2< T >::exp ( const Mat2T & Omega )

inlinestatic

Exponential chart map implementation: \(\mathfrak{so}(2) \rightarrow SO(2)\).

Exp()

template<typename T >

static SO2 SO2< T >::Exp ( const Vec1T & omega )

inlinestatic

Exponential chart map implementation: \(\mathbb{R} \rightarrow SO(2)\).

fromAngle()

template<typename T >

static SO2 SO2< T >::fromAngle ( const T angle )

inlinestatic

Convert an angle (in radians) into a rotation.

fromComplex() [1/2]

template<typename T >

static SO2 SO2< T >::fromComplex ( const T qw, const T qx )

inlinestatic

Construct a rotation from the individual fields.

Parameters

qw	Real component of the rotation \(\boldsymbol{q}\in SO(2)\).
qx	Imaginary component of the rotation \(\boldsymbol{q}\in SO(2)\).

fromComplex() [2/2]

template<typename T >

static SO2 SO2< T >::fromComplex ( const Vec2T & qvec )

inlinestatic

Construct a rotation from a rotation fields vector.

Parameters

qvec	The rotation represented as an array \(\boldsymbol{q}\in\mathbb{R}^2\).

fromR()

template<typename T >

static SO2 SO2< T >::fromR ( const Mat2T & m )

inlinestatic

Convert a rotation matrix \(\boldsymbol{R}\in\mathbb{R}^{2\times 2}\) to a \(SO(2)\) rotation.

fromTwoUnitVectors()

template<typename T >

static SO2 SO2< T >::fromTwoUnitVectors ( const Vec2T & u, const Vec2T & v )

inlinestatic

Given two unit vectors \(\boldsymbol{u},\boldsymbol{v}\in \mathbb{R}^2\), returns the rotation that rotates \(\boldsymbol{u}\rightarrow\boldsymbol{v}\).

hat()

template<typename T >

static Mat2T SO2< T >::hat ( const Vec1T & omega )

inlinestatic

Hat operator implementation, which coverts the tangent-space vector representation to the corresponding Lie algebra: \(\mathbb{R}\rightarrow \mathfrak{so}(2)\).

identity()

template<typename T >

static SO2 SO2< T >::identity ( )

inlinestatic

Obtain an identity \(SO(2)\) rotation.

inverse()

template<typename T >

SO2 SO2< T >::inverse ( ) const

inline

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

invert()

template<typename T >

SO2 & SO2< T >::invert ( )

inline

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

log()

template<typename T >

static Mat2T SO2< T >::log ( const SO2< T > & q )

inlinestatic

Logarithmic chart map implementation: \(SO(2) \rightarrow \mathfrak{so}(2)\).

Log()

template<typename T >

static Vec1T SO2< T >::Log ( const SO2< T > & q )

inlinestatic

Logarithmic chart map implementation: \(SO(2) \rightarrow \mathbb{R}\).

nans()

template<typename T >

static SO2 SO2< T >::nans ( )

inlinestatic

Obtain a rotation full of NaNs.

normalize()

template<typename T >

void SO2< T >::normalize ( )

inline

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

normalized()

template<typename T >

SO2 SO2< T >::normalized ( )

inline

Obtain a normalized copy of the current rotation.

ominus()

template<typename T >

template<typename Tout = T, typename T2 >

Matrix< Tout, 1, 1 > SO2< T >::ominus ( const SO2< 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, 2, 1 > SO2< T >::operator* ( const Matrix< T2, 2, 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 >

SO2 SO2< T >::operator* ( const SO2< T2 > & q ) const

inline

Invocation of otimes via multiplication.

operator*() [3/3]

template<typename T >

Vec2T SO2< T >::operator* ( const Vec2T & 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 >

SO2 & SO2< 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 >

SO2 & SO2< T >::operator*= ( const SO2< T2 > & q )

inline

Invocation of otimes via multiplication.

operator+()

template<typename T >

SO2 SO2< T >::operator+ ( const Vec1T & v ) const

inline

Invocation of oplus via addition.

operator+=()

template<typename T >

SO2 & SO2< T >::operator+= ( const Vec1T & v )

inline

Invocation of oplus via addition.

operator-()

template<typename T >

template<typename T2 >

Vec1T SO2< T >::operator- ( const SO2< T2 > & q ) const

inline

Invocation of ominus via subtraction.

operator/()

template<typename T >

SO2 SO2< 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 >

SO2 & SO2< 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 >

SO2 & SO2< T >::operator= ( const SO2< T > & q )

inline

Copy constructor.

operator[]()

template<typename T >

T & SO2< T >::operator[] ( int i )

inline

Access a field from \(\boldsymbol{q}\in\mathbb{R}^2\).

oplus()

template<typename T >

template<typename Tout = T, typename T2 >

SO2< Tout > SO2< T >::oplus ( const Matrix< T2, 1, 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 >

SO2< Tout > SO2< T >::otimes ( const SO2< T2 > & q ) const

inline

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

R()

template<typename T >

Mat2T SO2< T >::R ( ) const

inline

Convert the rotation to matrix representation \(\begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}\).

random()

template<typename T >

static SO2 SO2< T >::random ( )

inlinestatic

Obtain a random rotation.

Returns

A random rotation \(\boldsymbol{q}_B^W\in SO(2)\).

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

vee()

template<typename T >

static Vec1T SO2< T >::vee ( const Mat2T & Omega )

inlinestatic

Vee operator implementation, which coverts the Lie algebra representation to a tangent-space vector representation: \(\mathfrak{so}(2) \rightarrow \mathbb{R}\).

w() [1/2]

template<typename T >

T & SO2< T >::w ( )

inline

Access the real element of the rotation.

w() [2/2]

template<typename T >

const T & SO2< T >::w ( ) const

inline

Access the real element of the rotation.

x() [1/2]

template<typename T >

T & SO2< T >::x ( )

inline

Access the imaginary element of the rotation.

x() [2/2]

template<typename T >

const T & SO2< T >::x ( ) const

inline

Access the imaginary element of the rotation.

Member Data Documentation

arr_

template<typename T >

EIGEN_MAKE_ALIGNED_OPERATOR_NEW Map<Vec2T> SO2< T >::arr_

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

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

• include/SO2.h