TQuaternion Class Reference

Quaternion is a 4-component mathematic object quite convenient when dealing with space rotation (or reference frame transformation).

In short, think of quaternion Q as a 3-vector augmented by a real number. \( Q = Q|_r + Q|_V \)

Quaternion multiplication :

Quaternion multiplication is given by :

\[ Q.Q' = (Q|_r + Q|_V )*( Q'|_r + Q'|_V) = [ Q|_r*Q'|_r - Q|_V*Q'|_V ] + [ Q|_r*Q'|_V + Q'|_r*Q|_V + Q|_V X Q'|_V ] \]

where :

• \( Q|_r*Q'|_r \) is a real number product of real numbers
• \( Q|_V*Q'|_V \) is a real number, scalar product of two 3-vectors
• \( Q|_r*Q'|_V \) is a 3-vector, scaling of a 3-vector by a real number
• \( Q|_VXQ'|_V \) is a 3-vector, cross product of two 3-vectors

Thus, quaternion product is a generalization of real number product and product of a vector by a real number. Product of two pure vectors gives a quaternion whose real part is the opposite of scalar product and the vector part the cross product.

The conjugate of a quaternion \( Q = Q|r + Q|V \) is \( \bar{Q} = Q|r - Q|V \)

The magnitude of a quaternion \( Q \) is given by \( |Q|^2 = Q.\bar{Q} = \bar{Q}.Q = Q^2|r + |Q|V|^2 \)

Therefore, the inverse of a quaternion is \( Q-1 = \bar{Q} /|Q|^2 \)

"unit" quaternion is a quaternion of magnitude 1 : \( |Q|^2 = 1. \) Unit quaternions are a subset of the quaternions set.

Quaternion and rotations :

A rotation of angle \( f \) around a given axis, is represented by a unit quaternion Q :

• The axis of the rotation is given by the vector part of Q.
• The ratio between the magnitude of the vector part and the real part of Q equals tan({f}{2}).

In other words : \( Q = Q|_r + Q|_V = cos(\frac{f}{2}) + sin(\frac{f}{2}) \). (where u is a unit vector // to the rotation axis, \( cos(\frac{f}{2}) \) is the real part, \( sin(\frac{f}{2}) \) .u is the vector part) Note : The quaternion of identity is \( Q_I = cos(0) + sin(0)*(AnyVector) = 1\) .

The composition of two rotations is described by the product of the two corresponding quaternions. As for 3-space rotations, quaternion multiplication is not commutative !

\( Q = Q_1.Q_2 \) represents the composition of the successive rotation R1 and R2 expressed in the current frame (the axis of rotation hold by \( Q_2 \) is expressed in the frame as it is after R1 rotation). \( Q = Q_2.Q_1 \) represents the composition of the successive rotation R1 and R2 expressed in the initial reference frame.

The inverse of a rotation is a rotation about the same axis but of opposite angle, thus if Q is a unit quaternion, \( Q = cos(\frac{f}{2}) + sin(\frac{f}{2}).u = Q|_r + Q|_V\) , then : \( Q^{-1} =cos(-\frac{f}{2}) + sin(-\frac{f}{2}).u = cos(\frac{f}{2}) - sin(\frac{f}{2}).u = Q|_r -Q|_V \) is its inverse quaternion.

One verifies that : \( Q.Q^{-1} = Q^{-1}.Q = Q|_r*Q|_r + Q|_V*Q|_V + Q|_r*Q|_V -Q|_r*Q|_V + Q|_VXQ|_V = Q\leq|_r + Q\leq|_V = 1 \)

The rotation of a vector V by the rotation described by a unit quaternion Q is obtained by the following operation : \( V' = Q*V*Q^{-1} \), considering V as a quaternion whose real part is null.

Numeric computation considerations :

Numerically, the quaternion multiplication involves 12 additions and 16 multiplications. It is therefore faster than 3x3 matrixes multiplication involving 18 additions and 27 multiplications.

On the contrary, rotation of a vector by the above formula ( \( Q*V*Q^{-1} \) ) involves 18 additions and 24 multiplications, whereas multiplication of a 3-vector by a 3x3 matrix involves only 6 additions and 9 multiplications.

When dealing with numerous composition of space rotation, it is therefore faster to use quaternion product. On the other hand if a huge set of vectors must be rotated by a given quaternion, it is more optimized to convert the quaternion into a rotation matrix once, and then use that later to rotate the set of vectors.

More information :

http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

http://en.wikipedia.org/wiki/Quaternion

This Class represents all quaternions (unit or non-unit) It possesses a Normalize() method to make a given quaternion unit The Rotate(TVector3&) and Rotation(TVector3&) methods can be used even for a non-unit quaternion, in that case, the proper normalization is applied to perform the rotation.

A TRotation constructor exists than takes a quaternion for parameter (even non-unit), in that cas the proper normalisation is applied.

Definition at line 11 of file TQuaternion.h.

Public Member Functions
	TQuaternion (Double_t real=0, Double_t X=0, Double_t Y=0, Double_t Z=0)
	TQuaternion (const TVector3 &vector, Double_t real=0)
	TQuaternion (const Double_t *)
	TQuaternion (const Float_t *)
	TQuaternion (const TQuaternion &)
virtual	~TQuaternion ()
TQuaternion	Conjugate () const
TQuaternion &	DivideLeft (const TVector3 &vector)
	left division More...
TQuaternion &	DivideLeft (const TQuaternion &quaternion)
	left division More...
Double_t	GetQAngle () const
	Get angle of quaternion (rad) N.B : this angle is half of the corresponding rotation angle. More...
void	GetRXYZ (Double_t *carray) const
void	GetRXYZ (Float_t *carray) const
TQuaternion	Invert () const
	invert More...
TQuaternion	LeftProduct (const TVector3 &vector) const
	left product More...
TQuaternion	LeftProduct (const TQuaternion &quaternion) const
	left product More...
TQuaternion	LeftQuotient (const TVector3 &vector) const
	left quotient More...
TQuaternion	LeftQuotient (const TQuaternion &quaternion) const
	left quotient More...
TQuaternion &	MultiplyLeft (const TVector3 &vector)
	left multiplication More...
TQuaternion &	MultiplyLeft (const TQuaternion &quaternion)
	left multiplication More...
Double_t	Norm () const
Double_t	Norm2 () const
TQuaternion &	Normalize ()
Bool_t	operator!= (Double_t r) const
Bool_t	operator!= (const TVector3 &) const
Bool_t	operator!= (const TQuaternion &) const
Double_t	operator() (int) const
	dereferencing operator const More...
Double_t &	operator() (int)
	dereferencing operator More...
TQuaternion	operator* (Double_t real) const
	product of quaternion with a real More...
TQuaternion	operator* (const TVector3 &vector) const
	right product More...
TQuaternion	operator* (const TQuaternion &quaternion) const
	right product More...
TQuaternion &	operator*= (Double_t real)
TQuaternion &	operator*= (const TVector3 &vector)
	right multiplication More...
TQuaternion &	operator*= (const TQuaternion &quaternion)
	right multiplication More...
TQuaternion	operator+ (Double_t real) const
	sum of quaternion with a real More...
TQuaternion	operator+ (const TVector3 &vector) const
	sum of quaternion with a real More...
TQuaternion	operator+ (const TQuaternion &quaternion) const
TQuaternion &	operator+= (Double_t real)
TQuaternion &	operator+= (const TVector3 &vector)
TQuaternion &	operator+= (const TQuaternion &quaternion)
TQuaternion	operator- (Double_t real) const
	substraction of real to quaternion More...
TQuaternion	operator- (const TVector3 &vector) const
	substraction of real to quaternion More...
TQuaternion	operator- (const TQuaternion &quaternion) const
TQuaternion	operator- () const
TQuaternion &	operator-= (Double_t real)
TQuaternion &	operator-= (const TVector3 &vector)
TQuaternion &	operator-= (const TQuaternion &quaternion)
TQuaternion	operator/ (Double_t real) const
	division by a real More...
TQuaternion	operator/ (const TVector3 &vector) const
	right quotient More...
TQuaternion	operator/ (const TQuaternion &quaternion) const
	right quotient More...
TQuaternion &	operator/= (Double_t real)
TQuaternion &	operator/= (const TVector3 &vector)
	right division More...
TQuaternion &	operator/= (const TQuaternion &quaternion)
	right division More...
TQuaternion &	operator= (Double_t r)
TQuaternion &	operator= (const TVector3 &)
TQuaternion &	operator= (const TQuaternion &)
Bool_t	operator== (Double_t r) const
Bool_t	operator== (const TVector3 &) const
Bool_t	operator== (const TQuaternion &) const
Double_t	operator[] (int) const
Double_t &	operator[] (int)
void	Print (Option_t *option="") const
	Print Quaternion parameters. More...
Double_t	QMag () const
Double_t	QMag2 () const
void	Rotate (TVector3 &vect) const
	rotate vect by current quaternion More...
TVector3	Rotation (const TVector3 &vect) const
	rotation of vect by current quaternion More...
TQuaternion &	SetAxisQAngle (TVector3 &v, Double_t QAngle)
	set quaternion from vector and angle (rad) quaternion is set as unitary N.B : this angle is half of the corresponding rotation angle More...
TQuaternion &	SetQAngle (Double_t angle)
	Set angle of quaternion (rad) - keep quaternion norm N.B : this angle is half of the corresponding rotation angle. More...
TQuaternion &	SetRV (Double_t r, TVector3 &vect)
TQuaternion &	SetRXYZ (Double_t r, Double_t x, Double_t y, Double_t z)

Public Attributes
Double_t	fRealPart
TVector3	fVectorPart

#include <TQuaternion.h>

Inheritance diagram for TQuaternion:

Constructor & Destructor Documentation

TQuaternion() [1/5]

TQuaternion::TQuaternion	(	Double_t	real = 0,
		Double_t	X = 0,
		Double_t	Y = 0,
		Double_t	Z = 0
	)

Definition at line 113 of file TQuaternion.cxx.

TQuaternion() [2/5]

TQuaternion::TQuaternion	(	const TVector3 &	vector,
		Double_t	real = 0
	)

Definition at line 104 of file TQuaternion.cxx.

TQuaternion() [3/5]

TQuaternion::TQuaternion

(

const Double_t *

x0

)

Definition at line 107 of file TQuaternion.cxx.

TQuaternion() [4/5]

TQuaternion::TQuaternion

(

const Float_t *

x0

)

Definition at line 110 of file TQuaternion.cxx.

TQuaternion() [5/5]

TQuaternion::TQuaternion

(

const TQuaternion &

q

)

Definition at line 101 of file TQuaternion.cxx.

~TQuaternion()

TQuaternion::~TQuaternion ( )

virtual

Definition at line 116 of file TQuaternion.cxx.

Member Function Documentation

Conjugate()

TQuaternion TQuaternion::Conjugate ( ) const

inline

Definition at line 283 of file TQuaternion.h.

DivideLeft() [1/2]

TQuaternion & TQuaternion::DivideLeft

(

const TVector3 &

vector

)

left division

Definition at line 288 of file TQuaternion.cxx.

DivideLeft() [2/2]

TQuaternion & TQuaternion::DivideLeft

(

const TQuaternion &

quaternion

)

left division

Definition at line 412 of file TQuaternion.cxx.

GetQAngle()

Double_t TQuaternion::GetQAngle

(

)

const

Get angle of quaternion (rad) N.B : this angle is half of the corresponding rotation angle.

Definition at line 155 of file TQuaternion.cxx.

GetRXYZ() [1/2]

void TQuaternion::GetRXYZ ( Double_t *  carray ) const

inline

Definition at line 131 of file TQuaternion.h.

GetRXYZ() [2/2]

void TQuaternion::GetRXYZ ( Float_t *  carray ) const

inline

Definition at line 136 of file TQuaternion.h.

Invert()

TQuaternion TQuaternion::Invert

(

)

const

invert

Definition at line 479 of file TQuaternion.cxx.

LeftProduct() [1/2]

TQuaternion TQuaternion::LeftProduct

(

const TVector3 &

vector

)

const

left product

Definition at line 274 of file TQuaternion.cxx.

LeftProduct() [2/2]

TQuaternion TQuaternion::LeftProduct

(

const TQuaternion &

quaternion

)

const

left product

Definition at line 394 of file TQuaternion.cxx.

LeftQuotient() [1/2]

TQuaternion TQuaternion::LeftQuotient

(

const TVector3 &

vector

)

const

left quotient

Definition at line 318 of file TQuaternion.cxx.

LeftQuotient() [2/2]

TQuaternion TQuaternion::LeftQuotient

(

const TQuaternion &

quaternion

)

const

left quotient

Definition at line 443 of file TQuaternion.cxx.

MultiplyLeft() [1/2]

TQuaternion & TQuaternion::MultiplyLeft

(

const TVector3 &

vector

)

left multiplication

Definition at line 250 of file TQuaternion.cxx.

MultiplyLeft() [2/2]

TQuaternion & TQuaternion::MultiplyLeft

(

const TQuaternion &

quaternion

)

left multiplication

Definition at line 378 of file TQuaternion.cxx.

Norm()

Double_t TQuaternion::Norm ( ) const

inline

Definition at line 265 of file TQuaternion.h.

Norm2()

Double_t TQuaternion::Norm2 ( ) const

inline

Definition at line 269 of file TQuaternion.h.

Normalize()

TQuaternion & TQuaternion::Normalize ( )

inline

Definition at line 273 of file TQuaternion.h.

operator!=() [1/3]

Bool_t TQuaternion::operator!= ( Double_t  r ) const

inline

Definition at line 150 of file TQuaternion.h.

operator!=() [2/3]

Bool_t TQuaternion::operator!= ( const TVector3 &  V ) const

inline

Definition at line 199 of file TQuaternion.h.

operator!=() [3/3]

Bool_t TQuaternion::operator!= ( const TQuaternion &  Q ) const

inline

Definition at line 230 of file TQuaternion.h.

operator()() [1/2]

Double_t TQuaternion::operator()

(

int

i

)

const

dereferencing operator const

Definition at line 121 of file TQuaternion.cxx.

operator()() [2/2]

Double_t & TQuaternion::operator()

(

int

i

)

dereferencing operator

Definition at line 138 of file TQuaternion.cxx.

operator*() [1/3]

TQuaternion TQuaternion::operator*

(

Double_t

real

)

const

product of quaternion with a real

Definition at line 208 of file TQuaternion.cxx.

operator*() [2/3]

TQuaternion TQuaternion::operator*

(

const TVector3 &

vector

)

const

right product

Definition at line 281 of file TQuaternion.cxx.

operator*() [3/3]

TQuaternion TQuaternion::operator*

(

const TQuaternion &

quaternion

)

const

right product

Definition at line 403 of file TQuaternion.cxx.

operator*=() [1/3]

TQuaternion & TQuaternion::operator*= ( Double_t  real )

inline

Definition at line 170 of file TQuaternion.h.

operator*=() [2/3]

TQuaternion & TQuaternion::operator*=

(

const TVector3 &

vector

)

right multiplication

Definition at line 262 of file TQuaternion.cxx.

operator*=() [3/3]

TQuaternion & TQuaternion::operator*=

(

const TQuaternion &

quaternion

)

right multiplication

Definition at line 363 of file TQuaternion.cxx.

operator+() [1/3]

TQuaternion TQuaternion::operator+

(

Double_t

real

)

const

sum of quaternion with a real

Definition at line 194 of file TQuaternion.cxx.

operator+() [2/3]

TQuaternion TQuaternion::operator+

(

const TVector3 &

vector

)

const

sum of quaternion with a real

Definition at line 236 of file TQuaternion.cxx.

operator+() [3/3]

TQuaternion TQuaternion::operator+ ( const TQuaternion &  quaternion ) const

inline

Definition at line 254 of file TQuaternion.h.

operator+=() [1/3]

TQuaternion & TQuaternion::operator+= ( Double_t  real )

inline

Definition at line 160 of file TQuaternion.h.

operator+=() [2/3]

TQuaternion & TQuaternion::operator+= ( const TVector3 &  vector )

inline

Definition at line 209 of file TQuaternion.h.

operator+=() [3/3]

TQuaternion & TQuaternion::operator+= ( const TQuaternion &  quaternion )

inline

Definition at line 242 of file TQuaternion.h.

operator-() [1/4]

TQuaternion TQuaternion::operator-

(

Double_t

real

)

const

substraction of real to quaternion

Definition at line 201 of file TQuaternion.cxx.

operator-() [2/4]

TQuaternion TQuaternion::operator-

(

const TVector3 &

vector

)

const

substraction of real to quaternion

Definition at line 243 of file TQuaternion.cxx.

operator-() [3/4]

TQuaternion TQuaternion::operator- ( const TQuaternion &  quaternion ) const

inline

Definition at line 259 of file TQuaternion.h.

operator-() [4/4]

TQuaternion TQuaternion::operator- ( ) const

inline

Definition at line 279 of file TQuaternion.h.

operator-=() [1/3]

TQuaternion & TQuaternion::operator-= ( Double_t  real )

inline

Definition at line 165 of file TQuaternion.h.

operator-=() [2/3]

TQuaternion & TQuaternion::operator-= ( const TVector3 &  vector )

inline

Definition at line 214 of file TQuaternion.h.

operator-=() [3/3]

TQuaternion & TQuaternion::operator-= ( const TQuaternion &  quaternion )

inline

Definition at line 248 of file TQuaternion.h.

operator/() [1/3]

TQuaternion TQuaternion::operator/

(

Double_t

real

)

const

division by a real

Definition at line 216 of file TQuaternion.cxx.

operator/() [2/3]

TQuaternion TQuaternion::operator/

(

const TVector3 &

vector

)

const

right quotient

Definition at line 334 of file TQuaternion.cxx.

operator/() [3/3]

TQuaternion TQuaternion::operator/

(

const TQuaternion &

quaternion

)

const

right quotient

Definition at line 461 of file TQuaternion.cxx.

operator/=() [1/3]

TQuaternion & TQuaternion::operator/= ( Double_t  real )

inline

Definition at line 176 of file TQuaternion.h.

operator/=() [2/3]

TQuaternion & TQuaternion::operator/=

(

const TVector3 &

vector

)

right division

Definition at line 303 of file TQuaternion.cxx.

operator/=() [3/3]

TQuaternion & TQuaternion::operator/=

(

const TQuaternion &

quaternion

)

right division

Definition at line 427 of file TQuaternion.cxx.

operator=() [1/3]

TQuaternion & TQuaternion::operator= ( Double_t  r )

inline

Definition at line 154 of file TQuaternion.h.

operator=() [2/3]

TQuaternion & TQuaternion::operator= ( const TVector3 &  vect )

inline

Definition at line 203 of file TQuaternion.h.

operator=() [3/3]

TQuaternion & TQuaternion::operator= ( const TQuaternion &  quat )

inline

Definition at line 234 of file TQuaternion.h.

operator==() [1/3]

Bool_t TQuaternion::operator== ( Double_t  r ) const

inline

Definition at line 146 of file TQuaternion.h.

operator==() [2/3]

Bool_t TQuaternion::operator== ( const TVector3 &  V ) const

inline

Definition at line 195 of file TQuaternion.h.

operator==() [3/3]

Bool_t TQuaternion::operator== ( const TQuaternion &  Q ) const

inline

Definition at line 226 of file TQuaternion.h.

operator[]() [1/2]

Double_t TQuaternion::operator[] ( int  i ) const

inline

Definition at line 142 of file TQuaternion.h.

operator[]() [2/2]

Double_t & TQuaternion::operator[] ( int  i )

inline

Definition at line 141 of file TQuaternion.h.

Print()

void TQuaternion::Print

(

Option_t *

option = ""

)

const

Print Quaternion parameters.

Definition at line 527 of file TQuaternion.cxx.

QMag()

Double_t TQuaternion::QMag ( ) const

inline

Definition at line 99 of file TQuaternion.h.

QMag2()

Double_t TQuaternion::QMag2 ( ) const

inline

Definition at line 100 of file TQuaternion.h.

Rotate()

void TQuaternion::Rotate

(

TVector3 &

vect

)

const

rotate vect by current quaternion

Definition at line 493 of file TQuaternion.cxx.

Rotation()

TVector3 TQuaternion::Rotation

(

const TVector3 &

vect

)

const

rotation of vect by current quaternion

Definition at line 500 of file TQuaternion.cxx.

SetAxisQAngle()

TQuaternion & TQuaternion::SetAxisQAngle	(	TVector3 &	v,
		Double_t	QAngle
	)

set quaternion from vector and angle (rad) quaternion is set as unitary N.B : this angle is half of the corresponding rotation angle

Definition at line 179 of file TQuaternion.cxx.

SetQAngle()

TQuaternion & TQuaternion::SetQAngle

(

Double_t

angle

)

Set angle of quaternion (rad) - keep quaternion norm N.B : this angle is half of the corresponding rotation angle.

Definition at line 165 of file TQuaternion.cxx.

SetRV()

TQuaternion & TQuaternion::SetRV ( Double_t  r, TVector3 &  vect  )

inline

Definition at line 125 of file TQuaternion.h.

SetRXYZ()

TQuaternion & TQuaternion::SetRXYZ ( Double_t  r, Double_t  x, Double_t  y, Double_t  z  )

inline

Definition at line 119 of file TQuaternion.h.

Member Data Documentation

fRealPart

Double_t TQuaternion::fRealPart

Definition at line 110 of file TQuaternion.h.

fVectorPart

TVector3 TQuaternion::fVectorPart

Definition at line 111 of file TQuaternion.h.

Libraries for TQuaternion:

---

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

• math/physics/inc/TQuaternion.h
• math/physics/src/TQuaternion.cxx