numeric::EulerAngles< typename > Class Template Reference

Euler angles 3-D orientation representation. More...

#include <EulerAngles.hh>

Public Member Functions
	EulerAngles ()
	Default constructor. More...
	EulerAngles (xyzVector< T > const &v)
	Copy constructor. More...
	EulerAngles (Value const &phi, Value const &psi, Value const &theta)
	Triple value constructor. More...
	EulerAngles (xyzMatrix< T > rotation_matrix)
	Construct from rotation matrix. More...
	~EulerAngles ()
	Destructor. More...
void	from_rotation_matrix (xyzMatrix< T > matrix)
xyzMatrix< T >	to_rotation_matrix ()
	Construct rotation matrix from three euler angles that describe the frame. See the description for from_rotation_matrix to understand the Z-X-Z transformation convention. More...
Value &	phi ()
	Value phi in radians. More...
void	phi (Value const &value)
	Set value phi in radians. More...
Value &	phi_radians ()
	Value phi in radians. More...
void	phi_radians (Value const &value)
	Set value phi in radians. More...
Value	phi_degrees ()
	Value phi in degrees. More...
void	phi_degrees (Value const &value)
	Set value phi in degrees. More...
Value &	psi ()
	Value psi in radians. More...
void	psi (Value const &value)
	Set value psi in radians. More...
Value &	psi_radians ()
	Value psi in radians. More...
void	psi_radians (Value const &value)
	Set value psi in radians. More...
Value	psi_degrees ()
	Value psi in degrees. More...
void	psi_degrees (Value const &value)
	Set value psi in degrees. More...
Value &	theta ()
	Value theta in radians. More...
void	theta (Value const &value)
	Set value theta in radians. More...
Value &	theta_radians ()
	Value theta in radians. More...
void	theta_radians (Value const &value)
	Set value theta in radians. More...
Value	theta_degrees ()
	Value theta in degrees. More...
void	theta_degrees (Value const &value)
	Set value theta in degrees. More...
Public Member Functions inherited from numeric::xyzVector< T >
	xyzVector ()
	Default constructor. More...
	xyzVector (xyzVector const &v)
	Copy constructor. More...
	xyzVector (xyzVector< U > const &v)
	Copy constructor. More...
	xyzVector (Value const &t)
	Uniform value constructor. More...
	xyzVector (Value const &x_a, Value const &y_a, Value const &z_a)
	Triple value constructor. More...
	xyzVector (U const *p)
	Pointer to contiguous values constructor. More...
	~xyzVector ()
	Destructor. More...
xyzVector &	operator= (xyzVector const &v)
	Copy assignment. More...
xyzVector &	operator= (xyzVector< U > const &v)
	Copy assignment. More...
xyzVector &	operator= (U const *p)
	Assignment from pointer to contiguous values. More...
xyzVector &	operator= (Value const &t)
	= Value More...
xyzVector &	operator+= (xyzVector< U > const &v)
	+= xyzVector More...
xyzVector &	operator+= (Value const &t)
	+= Value More...
xyzVector &	operator-= (xyzVector< U > const &v)
	-= xyzVector More...
xyzVector &	operator-= (Value const &t)
	-= Value More...
xyzVector &	scaled_assign (Value const &t, xyzVector< U > const &v)
	Assign Value * xyzVector. More...
xyzVector &	scaled_add (Value const &t, xyzVector< U > const &v)
	Add Value * xyzVector. More...
xyzVector &	scaled_sub (Value const &t, xyzVector< U > const &v)
	Subtract Value * xyzVector. More...
xyzVector &	operator*= (Value const &t)
	*= Value More...
xyzVector &	operator/= (Value const &t)
	/= Value More...
xyzVector &	assign (Value const &x_a, Value const &y_a, Value const &z_a)
	Triple value assignment. More...
std::string	to_string () const
	to_string, useful for utility exits More...
void	show (std::ostream &output=std::cout) const
	Show. More...
xyzVector &	clear ()
	Clear. More...
xyzVector &	zero ()
	Zero. More...
xyzVector &	negate ()
	Negate. More...
xyzVector	operator- () const
	-xyzVector (negated copy) More...
xyzVector	negated () const
	Negated copy. More...
void	negated (xyzVector &a) const
	Negated: Return via argument (slightly faster) More...
xyzVector &	min (xyzVector const &v)
	Set minimum coordinates wrt another xyzVector. More...
xyzVector &	max (xyzVector const &v)
	Set maximum coordinates wrt another xyzVector. More...
xyzVector &	normalize ()
	Normalize. More...
xyzVector &	normalize (Value const &length_a)
	Normalize to a length. More...
T	minimum_value () const
	get the lowest of the three values in the vector More...
T	maximum_value () const
	get the highest of the three values in the vector More...
void	normalized (xyzVector &a) const
	Normalized. More...
void	normalized (Value const &length_a, xyzVector &a) const
	Normalized to a length. More...
xyzVector	normalized () const
	Normalized copy. More...
xyzVector	normalized (Value const &length_a) const
	Normalized to a length copy. More...
xyzVector &	normalize_or_zero ()
	Normalize: zero xyzVector if length is zero. More...
xyzVector &	normalize_or_zero (Value const &length_a)
	Normalize to a length: zero xyzVector if length is zero. More...
void	normalized_or_zero (xyzVector &a) const
	Normalized: zero xyzVector if length is zero. More...
void	normalized_or_zero (Value const &length_a, xyzVector &a) const
	Normalized to a length: zero xyzVector if length is zero. More...
xyzVector	normalized_or_zero () const
	Normalized copy: Zero xyzVector if length is zero. More...
xyzVector	normalized_or_zero (Value const &length_a) const
	Normalized to a length copy: Zero xyzVector if length is zero. More...
xyzVector &	normalize_any ()
	Normalize: arbitrary normalized xyzVector if length is zero. More...
xyzVector &	normalize_any (Value const &length_a)
	Normalize to a length: arbitrary normalized xyzVector if length is zero. More...
void	normalized_any (xyzVector &a) const
	Normalized: arbitrary normalized xyzVector if length is zero. More...
void	normalized_any (Value const &length_a, xyzVector &a) const
	Normalized to a length: arbitrary normalized xyzVector if length is zero. More...
xyzVector	normalized_any () const
	Normalized copy: Arbitrary normalized xyzVector if length is zero. More...
xyzVector	normalized_any (Value const &length_a) const
	Normalized to a length copy: Arbitrary normalized xyzVector if length is zero. More...
xyzVector &	project_normal (xyzVector const &v)
	Project normal. More...
xyzVector	projected_normal (xyzVector const &v) const
	Projected normal copy. More...
void	projected_normal (xyzVector const &v, xyzVector &a) const
	Projected normal. More...
xyzVector &	project_parallel (xyzVector const &v)
	Project parallel. More...
xyzVector	projected_parallel (xyzVector const &v) const
	Projected parallel copy. More...
void	projected_parallel (xyzVector const &v, xyzVector &a)
	Projected parallel. More...
Value	distance (xyzVector const &v) const
	Distance. More...
Value	distance_squared (xyzVector const &v) const
	Distance. More...
Value	dot (xyzVector const &v) const
	Distance squared. More...
Value	dot_product (xyzVector const &v) const
	Dot product. More...
Value	inner_product (xyzVector const &v) const
	Inner product ( == dot product ) More...
xyzVector	cross (xyzVector const &v) const
	Cross product. More...
xyzVector	cross_product (xyzVector const &v) const
	Cross product. More...
bool	is_zero () const
	Is zero? More...
bool	is_normalized () const
	Is exactly normalized? More...
bool	is_normalized (Value const &tol) const
	Is normalized to within a tolerance? More...
bool	is_unit () const
	Is exactly a unit vector? More...
bool	is_unit (Value const &tol) const
	Is a unit vector to within a tolerance? More...
Value const &	x () const
	Value x const. More...
Value &	x ()
	Value x. More...
void	x (Value const &x_a)
	x assignment More...
Value const &	y () const
	Value y const. More...
Value &	y ()
	Value y. More...
void	y (Value const &y_a)
	y assignment More...
Value const &	z () const
	Value z const. More...
Value &	z ()
	Value z. More...
void	z (Value const &z_a)
	z assignment More...
Value	length () const
	Length. More...
Value	length_squared () const
	Length squared. More...
Value	norm () const
	Norm. More...
Value	norm_squared () const
	Norm squared. More...
Value	magnitude () const
	Magnitude. More...
Value	magnitude_squared () const
	Magnitude squared. More...
Value const &	at (int const i) const
	xyzVector.at: 0-based index with bounds checking More...
Value &	at (int const i)
	xyzVector.at: 0-based index with bounds checking More...
Value const &	operator[] (int const i) const
	xyzVector[ i ] const: 0-based index More...
Value &	operator[] (int const i)
	xyzVector[ i ]: 0-based index More...
Value const &	operator() (int const i) const
	xyzVector( i ) const: 1-based index More...
Value &	operator() (int const i)
	xyzVector( i ): 1-based index More...
bool	equal_length (xyzVector const &v)
	Equal length? More...
bool	not_equal_length (xyzVector const &v)
	Not equal length? More...
bool	longer (xyzVector const &v)
	Longer? More...
bool	longer_or_equal (xyzVector const &v)
	Longer or equal length? More...
bool	shorter (xyzVector const &v)
	Shorter? More...
bool	shorter_or_equal (xyzVector const &v)
	Shorter or equal length? More...

Static Public Member Functions
static T	angular_distance_between (EulerAngles< T > a1, EulerAngles< T > a2)
	Get angular distance between two sets of Euler Angles. More...
static EulerAngles< T >	from_degrees (T phi, T psi, T theta)
static EulerAngles< T >	from_degrees (xyzVector< T > vector)
static EulerAngles< T >	from_radians (T phi, T psi, T theta)
static EulerAngles< T >	from_radians (xyzVector< T > vector)

Private Types
typedef xyzVector< T >	Vector
typedef xyzMatrix< T >	Matrix
typedef T	Value

Additional Inherited Members
Public Types inherited from numeric::xyzVector< T >
typedef T	Value
typedef T &	Reference
typedef T const &	ConstReference
typedef T *	Pointer
typedef T const *	ConstPointer
typedef T	value_type
typedef T &	reference
typedef T const &	const_reference
typedef T *	pointer
typedef T const *	const_pointer
typedef void	iterator_category
typedef void	difference_type

Detailed Description

template<typename>
class numeric::EulerAngles< typename >

Euler angles 3-D orientation representation.

Remarks

The three euler angles (in radians) that describing a rotation operation of a Z axis rotation by the angle phi (position 1), followed by an X axis rotation by the angle theta (position 3), followed by another Z axis rotation by the angle psi (position 2). this->code is a modified version of Alex Z's code from r++.

The range of phi is ( -pi, pi ]; The range of psi is ( -pi, pi ]; The range of theta is [ 0, pi ];

Member Typedef Documentation

template<typename >

typedef xyzMatrix<T> numeric::EulerAngles< typename >::Matrix

private

template<typename >

typedef T numeric::EulerAngles< typename >::Value

private

template<typename >

typedef xyzVector<T> numeric::EulerAngles< typename >::Vector

private

Constructor & Destructor Documentation

template<typename >

numeric::EulerAngles< typename >::EulerAngles ( )

inline

Default constructor.

Note

Values are initialized to zero

template<typename >

numeric::EulerAngles< typename >::EulerAngles ( xyzVector< T > const &  v )

inline

Copy constructor.

template<typename >

numeric::EulerAngles< typename >::EulerAngles ( Value const &  phi, Value const &  psi, Value const &  theta  )

inline

Triple value constructor.

template<typename >

numeric::EulerAngles< typename >::EulerAngles ( xyzMatrix< T >  rotation_matrix )

inline

Construct from rotation matrix.

References numeric::EulerAngles< typename >::from_rotation_matrix().

template<typename >

numeric::EulerAngles< typename >::~EulerAngles ( )

inline

Destructor.

Member Function Documentation

template<typename >

static T numeric::EulerAngles< typename >::angular_distance_between ( EulerAngles< T >  a1, EulerAngles< T >  a2  )

inlinestatic

Get angular distance between two sets of Euler Angles.

References numeric::rotation_angle().

template<typename >

static EulerAngles<T> numeric::EulerAngles< typename >::from_degrees ( T  phi, T  psi, T  theta  )

inlinestatic

References numeric::conversions::radians().

template<typename >

static EulerAngles<T> numeric::EulerAngles< typename >::from_degrees ( xyzVector< T >  vector )

inlinestatic

References numeric::conversions::radians(), numeric::xyzVector< typename >::x(), numeric::xyzVector< typename >::y(), and numeric::xyzVector< typename >::z().

template<typename >

static EulerAngles<T> numeric::EulerAngles< typename >::from_radians ( T  phi, T  psi, T  theta  )

inlinestatic

References numeric::EulerAngles< typename >::phi(), numeric::EulerAngles< typename >::psi(), and numeric::EulerAngles< typename >::theta().

template<typename >

static EulerAngles<T> numeric::EulerAngles< typename >::from_radians ( xyzVector< T >  vector )

inlinestatic

template<typename >

void numeric::EulerAngles< typename >::from_rotation_matrix ( xyzMatrix< T >  matrix )

inline

The equivalent rotation matrix representation of the euler angles would be:

FIGURE 1: R = [ cos(psi)cos(phi)-cos(theta)sin(phi)sin(psi) cos(psi)sin(phi)+cos(theta)cos(phi)sin(psi) sin(psi)sin(theta) -sin(psi)cos(phi)-cos(theta)sin(phi)cos(psi) -sin(psi)sin(phi)+cos(theta)cos(phi)cos(psi) cos(psi)sin(theta) sin(theta)sin(phi) -sin(theta)cos(phi) cos(theta) ]

The zz_ coordinate gives away theta. Theta may be computed as acos( zz_ ), or, as Alex does it, asin( sqrt( 1 - zz^2)) Since there is redundancy in theta, this->function chooses a theta with a positive sin(theta): i.e. quadrants I and II. Assuming we have a positive sin theta pushes phi and psi into conforming angles.

NOTE on theta: asin returns a value in the range [ -pi/2, pi/2 ], and we have artificially created a positive sin(theta), so we will get a asin( pos_sin_theta ), we have a value in the range [ 0, pi/2 ]. To convert this->into the actual angle theta, we examine the zz sign. If zz is negative, we chose the quadrant II theta. That is, asin( pos_sin_theta) returned an angle, call it theta'. Now, if cos( theta ) is negative, then we want to choose the positive x-axis rotation that's equivalent to -theta'. To do so, we reflect q through the y axis (See figure 2 below) to get p and then measure theta as pi - theta'.

FIGURE 2:

II | I | p. | .q (cos(-theta'), std::abs(sin(theta'))) . | .

theta'( . | . ) theta' = asin( std::abs(sin(theta))

| IV | The angle between the positive x axis and p is pi - theta'.

Since zx and zy contain only phi terms and a constant sin( theta ) term, phi is given by atan2( sin_phi, cos_phi ) = atan2( c*sin_phi, c*cos_phi ) = atan2( zx, -zy ) for c positive and non-zero. If sin_theta is zero, or very close to zero, we're at gimbal lock.

Moreover, since xz and yz contain only psi terms, psi may also be deduced using atan2.

There are 2 degenerate cases (gimbal lock)

1. theta close to 0 (North Pole singularity), or
2. theta close to pi (South Pole singularity) For these, we take: phi=acos(xx), theta = 0 (resp. Pi/2), psi = 0

References numeric::EulerAngles< typename >::phi(), numeric::constants::d::pi, numeric::EulerAngles< typename >::psi(), numeric::sin_cos_range(), basic::T(), numeric::EulerAngles< typename >::theta(), numeric::xyzMatrix< typename >::xx(), numeric::xyzMatrix< typename >::xz(), numeric::xyzMatrix< typename >::yz(), numeric::xyzMatrix< typename >::zx(), numeric::xyzMatrix< typename >::zy(), and numeric::xyzMatrix< typename >::zz().

Referenced by numeric::EulerAngles< typename >::EulerAngles().

template<typename >

Value& numeric::EulerAngles< typename >::phi ( )

inline

Value phi in radians.

References numeric::xyzVector< T >::x().

Referenced by numeric::EulerAngles< typename >::from_radians(), numeric::EulerAngles< typename >::from_rotation_matrix(), and numeric::EulerAngles< typename >::to_rotation_matrix().

template<typename >

void numeric::EulerAngles< typename >::phi ( Value const &  value )

inline

Set value phi in radians.

References numeric::xyzVector< T >::x().

template<typename >

Value numeric::EulerAngles< typename >::phi_degrees ( )

inline

Value phi in degrees.

References numeric::constants::d::radians_to_degrees, and numeric::xyzVector< T >::x().

template<typename >

void numeric::EulerAngles< typename >::phi_degrees ( Value const &  value )

inline

Set value phi in degrees.

References numeric::constants::d::degrees_to_radians, and numeric::xyzVector< T >::x().

template<typename >

Value& numeric::EulerAngles< typename >::phi_radians ( )

inline

Value phi in radians.

References numeric::xyzVector< T >::x().

template<typename >

void numeric::EulerAngles< typename >::phi_radians ( Value const &  value )

inline

Set value phi in radians.

References numeric::xyzVector< T >::x().

template<typename >

Value& numeric::EulerAngles< typename >::psi ( )

inline

Value psi in radians.

References numeric::xyzVector< T >::y().

Referenced by numeric::EulerAngles< typename >::from_radians(), numeric::EulerAngles< typename >::from_rotation_matrix(), and numeric::EulerAngles< typename >::to_rotation_matrix().

template<typename >

void numeric::EulerAngles< typename >::psi ( Value const &  value )

inline

Set value psi in radians.

References numeric::xyzVector< T >::y().

template<typename >

Value numeric::EulerAngles< typename >::psi_degrees ( )

inline

Value psi in degrees.

References numeric::constants::d::radians_to_degrees, and numeric::xyzVector< T >::y().

template<typename >

void numeric::EulerAngles< typename >::psi_degrees ( Value const &  value )

inline

Set value psi in degrees.

References numeric::constants::d::degrees_to_radians, and numeric::xyzVector< T >::y().

template<typename >

Value& numeric::EulerAngles< typename >::psi_radians ( )

inline

Value psi in radians.

References numeric::xyzVector< T >::y().

template<typename >

void numeric::EulerAngles< typename >::psi_radians ( Value const &  value )

inline

Set value psi in radians.

References numeric::xyzVector< T >::y().

template<typename >

Value& numeric::EulerAngles< typename >::theta ( )

inline

Value theta in radians.

References numeric::xyzVector< T >::z().

Referenced by numeric::EulerAngles< typename >::from_radians(), numeric::EulerAngles< typename >::from_rotation_matrix(), and numeric::EulerAngles< typename >::to_rotation_matrix().

template<typename >

void numeric::EulerAngles< typename >::theta ( Value const &  value )

inline

Set value theta in radians.

References numeric::xyzVector< T >::z().

template<typename >

Value numeric::EulerAngles< typename >::theta_degrees ( )

inline

Value theta in degrees.

References numeric::constants::d::radians_to_degrees, and numeric::xyzVector< T >::z().

template<typename >

void numeric::EulerAngles< typename >::theta_degrees ( Value const &  value )

inline

Set value theta in degrees.

References numeric::constants::d::degrees_to_radians, and numeric::xyzVector< T >::z().

template<typename >

Value& numeric::EulerAngles< typename >::theta_radians ( )

inline

Value theta in radians.

References numeric::xyzVector< T >::z().

template<typename >

void numeric::EulerAngles< typename >::theta_radians ( Value const &  value )

inline

Set value theta in radians.

References numeric::xyzVector< T >::z().

template<typename >

xyzMatrix<T> numeric::EulerAngles< typename >::to_rotation_matrix ( )

inline

Construct rotation matrix from three euler angles that describe the frame. See the description for from_rotation_matrix to understand the Z-X-Z transformation convention.

References numeric::EulerAngles< typename >::phi(), numeric::EulerAngles< typename >::psi(), basic::T(), numeric::EulerAngles< typename >::theta(), numeric::xyzMatrix< typename >::xx(), numeric::xyzMatrix< typename >::xy(), numeric::xyzMatrix< typename >::xz(), numeric::xyzMatrix< typename >::yx(), numeric::xyzMatrix< typename >::yy(), numeric::xyzMatrix< typename >::yz(), numeric::xyzMatrix< typename >::zx(), numeric::xyzMatrix< typename >::zy(), and numeric::xyzMatrix< typename >::zz().

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

• src/numeric/EulerAngles.fwd.hh
• src/numeric/EulerAngles.hh