 Rosetta  2021.16
numeric::EulerAngles< typename > Class Template Reference

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

`#include <EulerAngles.hh>`

Inheritance diagram for numeric::EulerAngles< typename >: [legend]

## 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< Tto_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...

Valuephi ()

void phi (Value const &value)
Set value phi in radians. More...

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...

Valuepsi ()

void psi (Value const &value)
Set value psi in radians. More...

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...

Valuetheta ()

void theta (Value const &value)
Set value theta in radians. More...

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...

xyzVectoroperator= (xyzVector const &v)
Copy assignment. More...

xyzVectoroperator= (xyzVector< U > const &v)
Copy assignment. More...

xyzVectoroperator= (U const *p)
Assignment from pointer to contiguous values. More...

xyzVectoroperator= (Value const &t)
= Value More...

xyzVectoroperator+= (xyzVector< U > const &v)
+= xyzVector More...

xyzVectoroperator+= (Value const &t)
+= Value More...

xyzVectoroperator-= (xyzVector< U > const &v)
-= xyzVector More...

xyzVectoroperator-= (Value const &t)
-= Value More...

xyzVectorscaled_assign (Value const &t, xyzVector< U > const &v)
Assign Value * xyzVector. More...

xyzVectorscaled_add (Value const &t, xyzVector< U > const &v)

xyzVectorscaled_sub (Value const &t, xyzVector< U > const &v)
Subtract Value * xyzVector. More...

xyzVectoroperator*= (Value const &t)
*= Value More...

xyzVectoroperator/= (Value const &t)
/= Value More...

xyzVectorassign (Value const &x_a, Value const &y_a, Value const &z_a)
Triple value assignment. More...

bool is_finite () const
Test if there's any non-finite (inf, NAN) values in the vector. More...

std::string to_string () const
to_string, useful for utility exits More...

void show (std::ostream &output=std::cout) const
Show. More...

xyzVectorclear ()
Clear. More...

xyzVectorzero ()
Zero. More...

xyzVectornegate ()
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...

xyzVectormin (xyzVector const &v)
Set minimum coordinates wrt another xyzVector. More...

xyzVectormax (xyzVector const &v)
Set maximum coordinates wrt another xyzVector. More...

xyzVectornormalize ()
Normalize. More...

xyzVectornormalize (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...

xyzVectornormalize_or_zero ()
Normalize: zero xyzVector if length is zero. More...

xyzVectornormalize_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...

xyzVectornormalize_any ()
Normalize: arbitrary normalized xyzVector if length is zero. More...

xyzVectornormalize_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...

xyzVectorproject_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...

xyzVectorproject_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...

Valuex ()
Value x. More...

void x (Value const &x_a)
x assignment More...

Value const & y () const
Value y const. More...

Valuey ()
Value y. More...

void y (Value const &y_a)
y assignment More...

Value const & z () const
Value z const. More...

Valuez ()
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...

Valueat (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...

Valueoperator[] (int const i)
xyzVector[ i ]: 0-based index More...

Value const & operator() (int const i) const
xyzVector( i ) const: 1-based index More...

Valueoperator() (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 not_equal_length (xyzVector< U > const &a, xyzVector< U > const &b)
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< Tfrom_degrees (T phi, T psi, T theta)

static EulerAngles< Tfrom_degrees (xyzVector< T > vector)

static EulerAngles< Tfrom_radians (T phi, T psi, T theta)

static EulerAngles< Tfrom_radians (xyzVector< T > vector)

## Private Types

typedef xyzVector< TVector

typedef xyzMatrix< TMatrix

typedef T Value Public Types inherited from numeric::xyzVector< T >
typedef T Value

typedef TReference

typedef T const & ConstReference

typedef TPointer

typedef T const * ConstPointer

typedef T value_type

typedef Treference

typedef T const & const_reference

typedef Tpointer

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 numeric::EulerAngles< typename >::Matrix
private
template<typename >
 typedef T numeric::EulerAngles< typename >::Value
private
template<typename >
 typedef xyzVector 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.

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.

template<typename >
 static EulerAngles numeric::EulerAngles< typename >::from_degrees ( T phi, T psi, T theta )
inlinestatic
template<typename >
 static EulerAngles numeric::EulerAngles< typename >::from_degrees ( xyzVector< T > vector )
inlinestatic
template<typename >
 static EulerAngles numeric::EulerAngles< typename >::from_radians ( T phi, T psi, T theta )
inlinestatic
template<typename >
 static EulerAngles 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

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

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

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

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

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

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

Value phi in degrees.

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

Set value phi in degrees.

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

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

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

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

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

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

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

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

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

Value psi in degrees.

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

Set value psi in degrees.

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

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

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

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

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

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

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

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

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

Value theta in degrees.

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

Set value theta in degrees.

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

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

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