A Quaternion Class
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 Q_bar = Q|r - Q|V
The magnitude of a quaternion Q is given by |Q|² = Q.Q_bar = Q_bar.Q = Q²|r + |Q|V|²
Therefore, the inverse of a quaternion is Q-1 = Q_bar /|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(f/2) + sin(f/2).
(where u is a unit vector // to the rotation axis,
cos(f/2) is the real part, sin(f/2).u is the vector part)
Note : The quaternion of identity is QI = cos(0) + sin(0)*(any vector) = 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 = Q1.Q2 represents the composition of the successive rotation R1 and R2 expressed in the current frame (the axis of rotation hold by Q2 is expressed in the frame as it is after R1 rotation).
Q = Q2.Q1 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(f/2) + sin(f/2).u = Q|r + Q|V, then :
Q-1 =cos(-f/2) + sin(-f/2).u = cos(f/2) - sin(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
= Q²|r + Q²|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 :
en.wikipedia.org/wiki/Quaternions_and_spatial_rotation .
en.wikipedia.org/wiki/Quaternion .
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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.
| virtual void | TObject::DoError(int level, const char* location, const char* fmt, va_list va) const |
| void | TObject::MakeZombie() |
| Double_t | fRealPart | Real part |
| TVector3 | fVectorPart | vector part |
| static TObject::(anonymous) | TObject::kBitMask | |
| static TObject::EStatusBits | TObject::kCanDelete | |
| static TObject::EStatusBits | TObject::kCannotPick | |
| static TObject::EStatusBits | TObject::kHasUUID | |
| static TObject::EStatusBits | TObject::kInvalidObject | |
| static TObject::(anonymous) | TObject::kIsOnHeap | |
| static TObject::EStatusBits | TObject::kIsReferenced | |
| static TObject::EStatusBits | TObject::kMustCleanup | |
| static TObject::EStatusBits | TObject::kNoContextMenu | |
| static TObject::(anonymous) | TObject::kNotDeleted | |
| static TObject::EStatusBits | TObject::kObjInCanvas | |
| static TObject::(anonymous) | TObject::kOverwrite | |
| static TObject::(anonymous) | TObject::kSingleKey | |
| static TObject::(anonymous) | TObject::kWriteDelete | |
| static TObject::(anonymous) | TObject::kZombie |
| Inheritance Chart: | |||||||||
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Get angle of quaternion (rad) N.B : this angle is half of the corresponding rotation angle
Set angle of quaternion (rad) - keep quaternion norm N.B : this angle is half of the corresponding rotation angle
set quaternion from vector and angle (rad) quaternion is set as unitary N.B : this angle is half of the corresponding rotation angle