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 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 :
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.
A rotation of angle \( f \) around a given axis, is represented by a unit quaternion Q :
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.
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.
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 (const Double_t *) | |
TQuaternion (const Float_t *) | |
TQuaternion (const TQuaternion &) | |
TQuaternion (const TVector3 &vector, Double_t real=0) | |
TQuaternion (Double_t real=0, Double_t X=0, Double_t Y=0, Double_t Z=0) | |
virtual | ~TQuaternion () |
TQuaternion | Conjugate () const |
TQuaternion & | DivideLeft (const TQuaternion &quaternion) |
left division | |
TQuaternion & | DivideLeft (const TVector3 &vector) |
left division | |
Double_t | GetQAngle () const |
Get angle of quaternion (rad) N.B : this angle is half of the corresponding rotation angle. | |
void | GetRXYZ (Double_t *carray) const |
void | GetRXYZ (Float_t *carray) const |
TQuaternion | Invert () const |
invert | |
TQuaternion | LeftProduct (const TQuaternion &quaternion) const |
left product | |
TQuaternion | LeftProduct (const TVector3 &vector) const |
left product | |
TQuaternion | LeftQuotient (const TQuaternion &quaternion) const |
left quotient | |
TQuaternion | LeftQuotient (const TVector3 &vector) const |
left quotient | |
TQuaternion & | MultiplyLeft (const TQuaternion &quaternion) |
left multiplication | |
TQuaternion & | MultiplyLeft (const TVector3 &vector) |
left multiplication | |
Double_t | Norm () const |
Double_t | Norm2 () const |
TQuaternion & | Normalize () |
Bool_t | operator!= (const TQuaternion &) const |
Bool_t | operator!= (const TVector3 &) const |
Bool_t | operator!= (Double_t r) const |
Double_t & | operator() (int) |
dereferencing operator | |
Double_t | operator() (int) const |
dereferencing operator const | |
TQuaternion | operator* (const TQuaternion &quaternion) const |
right product | |
TQuaternion | operator* (const TVector3 &vector) const |
right product | |
TQuaternion | operator* (Double_t real) const |
product of quaternion with a real | |
TQuaternion & | operator*= (const TQuaternion &quaternion) |
right multiplication | |
TQuaternion & | operator*= (const TVector3 &vector) |
right multiplication | |
TQuaternion & | operator*= (Double_t real) |
TQuaternion | operator+ (const TQuaternion &quaternion) const |
TQuaternion | operator+ (const TVector3 &vector) const |
sum of quaternion with a real | |
TQuaternion | operator+ (Double_t real) const |
sum of quaternion with a real | |
TQuaternion & | operator+= (const TQuaternion &quaternion) |
TQuaternion & | operator+= (const TVector3 &vector) |
TQuaternion & | operator+= (Double_t real) |
TQuaternion | operator- () const |
TQuaternion | operator- (const TQuaternion &quaternion) const |
TQuaternion | operator- (const TVector3 &vector) const |
substraction of real to quaternion | |
TQuaternion | operator- (Double_t real) const |
substraction of real to quaternion | |
TQuaternion & | operator-= (const TQuaternion &quaternion) |
TQuaternion & | operator-= (const TVector3 &vector) |
TQuaternion & | operator-= (Double_t real) |
TQuaternion | operator/ (const TQuaternion &quaternion) const |
right quotient | |
TQuaternion | operator/ (const TVector3 &vector) const |
right quotient | |
TQuaternion | operator/ (Double_t real) const |
division by a real | |
TQuaternion & | operator/= (const TQuaternion &quaternion) |
right division | |
TQuaternion & | operator/= (const TVector3 &vector) |
right division | |
TQuaternion & | operator/= (Double_t real) |
TQuaternion & | operator= (const TQuaternion &) |
TQuaternion & | operator= (const TVector3 &) |
TQuaternion & | operator= (Double_t r) |
Bool_t | operator== (const TQuaternion &) const |
Bool_t | operator== (const TVector3 &) const |
Bool_t | operator== (Double_t r) const |
Double_t & | operator[] (int) |
Double_t | operator[] (int) const |
void | Print (Option_t *option="") const |
Print Quaternion parameters. | |
Double_t | QMag () const |
Double_t | QMag2 () const |
void | Rotate (TVector3 &vect) const |
rotate vect by current quaternion | |
TVector3 | Rotation (const TVector3 &vect) const |
rotation of vect by current quaternion | |
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 | |
TQuaternion & | SetQAngle (Double_t angle) |
Set angle of quaternion (rad) - keep quaternion norm N.B : this angle is half of the corresponding rotation angle. | |
TQuaternion & | SetRV (Double_t r, TVector3 &vect) |
TQuaternion & | SetRXYZ (Double_t r, Double_t x, Double_t y, Double_t z) |
Public Member Functions inherited from TObject | |
TObject () | |
TObject constructor. | |
TObject (const TObject &object) | |
TObject copy ctor. | |
virtual | ~TObject () |
TObject destructor. | |
void | AbstractMethod (const char *method) const |
Use this method to implement an "abstract" method that you don't want to leave purely abstract. | |
virtual void | AppendPad (Option_t *option="") |
Append graphics object to current pad. | |
virtual void | Browse (TBrowser *b) |
Browse object. May be overridden for another default action. | |
ULong_t | CheckedHash () |
Check and record whether this class has a consistent Hash/RecursiveRemove setup (*) and then return the regular Hash value for this object. | |
virtual const char * | ClassName () const |
Returns name of class to which the object belongs. | |
virtual void | Clear (Option_t *="") |
virtual TObject * | Clone (const char *newname="") const |
Make a clone of an object using the Streamer facility. | |
virtual Int_t | Compare (const TObject *obj) const |
Compare abstract method. | |
virtual void | Copy (TObject &object) const |
Copy this to obj. | |
virtual void | Delete (Option_t *option="") |
Delete this object. | |
virtual Int_t | DistancetoPrimitive (Int_t px, Int_t py) |
Computes distance from point (px,py) to the object. | |
virtual void | Draw (Option_t *option="") |
Default Draw method for all objects. | |
virtual void | DrawClass () const |
Draw class inheritance tree of the class to which this object belongs. | |
virtual TObject * | DrawClone (Option_t *option="") const |
Draw a clone of this object in the current selected pad for instance with: gROOT->SetSelectedPad(gPad) . | |
virtual void | Dump () const |
Dump contents of object on stdout. | |
virtual void | Error (const char *method, const char *msgfmt,...) const |
Issue error message. | |
virtual void | Execute (const char *method, const char *params, Int_t *error=0) |
Execute method on this object with the given parameter string, e.g. | |
virtual void | Execute (TMethod *method, TObjArray *params, Int_t *error=0) |
Execute method on this object with parameters stored in the TObjArray. | |
virtual void | ExecuteEvent (Int_t event, Int_t px, Int_t py) |
Execute action corresponding to an event at (px,py). | |
virtual void | Fatal (const char *method, const char *msgfmt,...) const |
Issue fatal error message. | |
virtual TObject * | FindObject (const char *name) const |
Must be redefined in derived classes. | |
virtual TObject * | FindObject (const TObject *obj) const |
Must be redefined in derived classes. | |
virtual Option_t * | GetDrawOption () const |
Get option used by the graphics system to draw this object. | |
virtual const char * | GetIconName () const |
Returns mime type name of object. | |
virtual const char * | GetName () const |
Returns name of object. | |
virtual char * | GetObjectInfo (Int_t px, Int_t py) const |
Returns string containing info about the object at position (px,py). | |
virtual Option_t * | GetOption () const |
virtual const char * | GetTitle () const |
Returns title of object. | |
virtual UInt_t | GetUniqueID () const |
Return the unique object id. | |
virtual Bool_t | HandleTimer (TTimer *timer) |
Execute action in response of a timer timing out. | |
virtual ULong_t | Hash () const |
Return hash value for this object. | |
Bool_t | HasInconsistentHash () const |
Return true is the type of this object is known to have an inconsistent setup for Hash and RecursiveRemove (i.e. | |
virtual void | Info (const char *method, const char *msgfmt,...) const |
Issue info message. | |
virtual Bool_t | InheritsFrom (const char *classname) const |
Returns kTRUE if object inherits from class "classname". | |
virtual Bool_t | InheritsFrom (const TClass *cl) const |
Returns kTRUE if object inherits from TClass cl. | |
virtual void | Inspect () const |
Dump contents of this object in a graphics canvas. | |
void | InvertBit (UInt_t f) |
Bool_t | IsDestructed () const |
IsDestructed. | |
virtual Bool_t | IsEqual (const TObject *obj) const |
Default equal comparison (objects are equal if they have the same address in memory). | |
virtual Bool_t | IsFolder () const |
Returns kTRUE in case object contains browsable objects (like containers or lists of other objects). | |
R__ALWAYS_INLINE Bool_t | IsOnHeap () const |
virtual Bool_t | IsSortable () const |
R__ALWAYS_INLINE Bool_t | IsZombie () const |
virtual void | ls (Option_t *option="") const |
The ls function lists the contents of a class on stdout. | |
void | MayNotUse (const char *method) const |
Use this method to signal that a method (defined in a base class) may not be called in a derived class (in principle against good design since a child class should not provide less functionality than its parent, however, sometimes it is necessary). | |
virtual Bool_t | Notify () |
This method must be overridden to handle object notification. | |
void | Obsolete (const char *method, const char *asOfVers, const char *removedFromVers) const |
Use this method to declare a method obsolete. | |
void | operator delete (void *ptr) |
Operator delete. | |
void | operator delete[] (void *ptr) |
Operator delete []. | |
void * | operator new (size_t sz) |
void * | operator new (size_t sz, void *vp) |
void * | operator new[] (size_t sz) |
void * | operator new[] (size_t sz, void *vp) |
TObject & | operator= (const TObject &rhs) |
TObject assignment operator. | |
virtual void | Paint (Option_t *option="") |
This method must be overridden if a class wants to paint itself. | |
virtual void | Pop () |
Pop on object drawn in a pad to the top of the display list. | |
virtual Int_t | Read (const char *name) |
Read contents of object with specified name from the current directory. | |
virtual void | RecursiveRemove (TObject *obj) |
Recursively remove this object from a list. | |
void | ResetBit (UInt_t f) |
virtual void | SaveAs (const char *filename="", Option_t *option="") const |
Save this object in the file specified by filename. | |
virtual void | SavePrimitive (std::ostream &out, Option_t *option="") |
Save a primitive as a C++ statement(s) on output stream "out". | |
void | SetBit (UInt_t f) |
void | SetBit (UInt_t f, Bool_t set) |
Set or unset the user status bits as specified in f. | |
virtual void | SetDrawOption (Option_t *option="") |
Set drawing option for object. | |
virtual void | SetUniqueID (UInt_t uid) |
Set the unique object id. | |
virtual void | SysError (const char *method, const char *msgfmt,...) const |
Issue system error message. | |
R__ALWAYS_INLINE Bool_t | TestBit (UInt_t f) const |
Int_t | TestBits (UInt_t f) const |
virtual void | UseCurrentStyle () |
Set current style settings in this object This function is called when either TCanvas::UseCurrentStyle or TROOT::ForceStyle have been invoked. | |
virtual void | Warning (const char *method, const char *msgfmt,...) const |
Issue warning message. | |
virtual Int_t | Write (const char *name=0, Int_t option=0, Int_t bufsize=0) |
Write this object to the current directory. | |
virtual Int_t | Write (const char *name=0, Int_t option=0, Int_t bufsize=0) const |
Write this object to the current directory. | |
Public Attributes | |
Double_t | fRealPart |
TVector3 | fVectorPart |
Additional Inherited Members | |
Public Types inherited from TObject | |
enum | { kIsOnHeap = 0x01000000 , kNotDeleted = 0x02000000 , kZombie = 0x04000000 , kInconsistent = 0x08000000 , kBitMask = 0x00ffffff } |
enum | { kSingleKey = BIT(0) , kOverwrite = BIT(1) , kWriteDelete = BIT(2) } |
enum | EDeprecatedStatusBits { kObjInCanvas = BIT(3) } |
enum | EStatusBits { kCanDelete = BIT(0) , kMustCleanup = BIT(3) , kIsReferenced = BIT(4) , kHasUUID = BIT(5) , kCannotPick = BIT(6) , kNoContextMenu = BIT(8) , kInvalidObject = BIT(13) } |
Static Public Member Functions inherited from TObject | |
static Longptr_t | GetDtorOnly () |
Return destructor only flag. | |
static Bool_t | GetObjectStat () |
Get status of object stat flag. | |
static void | SetDtorOnly (void *obj) |
Set destructor only flag. | |
static void | SetObjectStat (Bool_t stat) |
Turn on/off tracking of objects in the TObjectTable. | |
Protected Types inherited from TObject | |
enum | { kOnlyPrepStep = BIT(3) } |
Protected Member Functions inherited from TObject | |
virtual void | DoError (int level, const char *location, const char *fmt, va_list va) const |
Interface to ErrorHandler (protected). | |
void | MakeZombie () |
#include <TQuaternion.h>
Definition at line 113 of file TQuaternion.cxx.
Definition at line 104 of file TQuaternion.cxx.
TQuaternion::TQuaternion | ( | const Double_t * | x0 | ) |
Definition at line 107 of file TQuaternion.cxx.
TQuaternion::TQuaternion | ( | const Float_t * | x0 | ) |
Definition at line 110 of file TQuaternion.cxx.
TQuaternion::TQuaternion | ( | const TQuaternion & | q | ) |
Definition at line 101 of file TQuaternion.cxx.
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Definition at line 116 of file TQuaternion.cxx.
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Definition at line 283 of file TQuaternion.h.
TQuaternion & TQuaternion::DivideLeft | ( | const TQuaternion & | quaternion | ) |
left division
Definition at line 412 of file TQuaternion.cxx.
TQuaternion & TQuaternion::DivideLeft | ( | const TVector3 & | vector | ) |
left division
Definition at line 288 of file TQuaternion.cxx.
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.
Definition at line 131 of file TQuaternion.h.
Definition at line 136 of file TQuaternion.h.
TQuaternion TQuaternion::Invert | ( | ) | const |
invert
Definition at line 479 of file TQuaternion.cxx.
TQuaternion TQuaternion::LeftProduct | ( | const TQuaternion & | quaternion | ) | const |
left product
Definition at line 394 of file TQuaternion.cxx.
TQuaternion TQuaternion::LeftProduct | ( | const TVector3 & | vector | ) | const |
left product
Definition at line 274 of file TQuaternion.cxx.
TQuaternion TQuaternion::LeftQuotient | ( | const TQuaternion & | quaternion | ) | const |
left quotient
Definition at line 443 of file TQuaternion.cxx.
TQuaternion TQuaternion::LeftQuotient | ( | const TVector3 & | vector | ) | const |
left quotient
Definition at line 318 of file TQuaternion.cxx.
TQuaternion & TQuaternion::MultiplyLeft | ( | const TQuaternion & | quaternion | ) |
left multiplication
Definition at line 378 of file TQuaternion.cxx.
TQuaternion & TQuaternion::MultiplyLeft | ( | const TVector3 & | vector | ) |
left multiplication
Definition at line 250 of file TQuaternion.cxx.
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Definition at line 265 of file TQuaternion.h.
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inline |
Definition at line 269 of file TQuaternion.h.
|
inline |
Definition at line 273 of file TQuaternion.h.
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inline |
Definition at line 230 of file TQuaternion.h.
Definition at line 199 of file TQuaternion.h.
Definition at line 150 of file TQuaternion.h.
dereferencing operator
Definition at line 138 of file TQuaternion.cxx.
dereferencing operator const
Definition at line 121 of file TQuaternion.cxx.
TQuaternion TQuaternion::operator* | ( | const TQuaternion & | quaternion | ) | const |
right product
Definition at line 403 of file TQuaternion.cxx.
TQuaternion TQuaternion::operator* | ( | const TVector3 & | vector | ) | const |
right product
Definition at line 281 of file TQuaternion.cxx.
TQuaternion TQuaternion::operator* | ( | Double_t | real | ) | const |
product of quaternion with a real
Definition at line 208 of file TQuaternion.cxx.
TQuaternion & TQuaternion::operator*= | ( | const TQuaternion & | quaternion | ) |
right multiplication
Definition at line 363 of file TQuaternion.cxx.
TQuaternion & TQuaternion::operator*= | ( | const TVector3 & | vector | ) |
right multiplication
Definition at line 262 of file TQuaternion.cxx.
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inline |
Definition at line 170 of file TQuaternion.h.
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inline |
Definition at line 254 of file TQuaternion.h.
TQuaternion TQuaternion::operator+ | ( | const TVector3 & | vector | ) | const |
sum of quaternion with a real
Definition at line 236 of file TQuaternion.cxx.
TQuaternion TQuaternion::operator+ | ( | Double_t | real | ) | const |
sum of quaternion with a real
Definition at line 194 of file TQuaternion.cxx.
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inline |
Definition at line 242 of file TQuaternion.h.
|
inline |
Definition at line 209 of file TQuaternion.h.
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inline |
Definition at line 160 of file TQuaternion.h.
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inline |
Definition at line 279 of file TQuaternion.h.
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inline |
Definition at line 259 of file TQuaternion.h.
TQuaternion TQuaternion::operator- | ( | const TVector3 & | vector | ) | const |
substraction of real to quaternion
Definition at line 243 of file TQuaternion.cxx.
TQuaternion TQuaternion::operator- | ( | Double_t | real | ) | const |
substraction of real to quaternion
Definition at line 201 of file TQuaternion.cxx.
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inline |
Definition at line 248 of file TQuaternion.h.
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inline |
Definition at line 214 of file TQuaternion.h.
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inline |
Definition at line 165 of file TQuaternion.h.
TQuaternion TQuaternion::operator/ | ( | const TQuaternion & | quaternion | ) | const |
right quotient
Definition at line 461 of file TQuaternion.cxx.
TQuaternion TQuaternion::operator/ | ( | const TVector3 & | vector | ) | const |
right quotient
Definition at line 334 of file TQuaternion.cxx.
TQuaternion TQuaternion::operator/ | ( | Double_t | real | ) | const |
division by a real
Definition at line 216 of file TQuaternion.cxx.
TQuaternion & TQuaternion::operator/= | ( | const TQuaternion & | quaternion | ) |
right division
Definition at line 427 of file TQuaternion.cxx.
TQuaternion & TQuaternion::operator/= | ( | const TVector3 & | vector | ) |
right division
Definition at line 303 of file TQuaternion.cxx.
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Definition at line 176 of file TQuaternion.h.
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inline |
Definition at line 234 of file TQuaternion.h.
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inline |
Definition at line 203 of file TQuaternion.h.
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inline |
Definition at line 154 of file TQuaternion.h.
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inline |
Definition at line 226 of file TQuaternion.h.
Definition at line 195 of file TQuaternion.h.
Definition at line 146 of file TQuaternion.h.
Definition at line 141 of file TQuaternion.h.
Definition at line 142 of file TQuaternion.h.
Print Quaternion parameters.
Reimplemented from TObject.
Definition at line 527 of file TQuaternion.cxx.
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inline |
Definition at line 99 of file TQuaternion.h.
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inline |
Definition at line 100 of file TQuaternion.h.
rotate vect by current quaternion
Definition at line 493 of file TQuaternion.cxx.
rotation of vect by current quaternion
Definition at line 500 of file TQuaternion.cxx.
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.
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.
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inline |
Definition at line 125 of file TQuaternion.h.
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inline |
Definition at line 119 of file TQuaternion.h.
Double_t TQuaternion::fRealPart |
Definition at line 110 of file TQuaternion.h.
TVector3 TQuaternion::fVectorPart |
Definition at line 111 of file TQuaternion.h.