// Rotation and transformation doxygen page /** \page TransformPage Vector Transformations Transformations classes are grouped in Rotations (in 3 dimensions), Lorentz transformations and Poincarre transformations, which are Translation/Rotation combinations. Each group has several members which may model physically equivalent trasformations but with different internal representations.
All the classes are non-template and use double precision as the scalar type The following types of transformation classes are defined:

- 3D Rotations:
  - ROOT::Math::Rotation3D, rotation described by a 3x3 matrix of doubles
  - ROOT::Math::EulerAngles rotation described by the three Euler angles (phi, theta and psi) following the GoldStein definition.
  - ROOT::Math::RotationZYX rotation described by three angles defining a rotation first along the Z axis, then along the rotated Y' axis and then along the rotated X'' axis.
  - ROOT::Math::AxisAngle, rotation described by a vector (axis) and an angle
  - ROOT::Math::Quaternion, rotation described by a quaternion (4 numbers)
  - ROOT::Math::RotationX, specialized rotation along the X axis
  - ROOT::Math::RotationY, specialized rotation along the Y axis
  - ROOT::Math::RotationZ, specialized rotation along the Z axis
- 3D Transformations (Rotations + Translations)
  - ROOT::Math::Translation3D, (only translation) described by a 3D Vector
  - ROOT::Math::Transform3D, (rotations and then translation) described by a 3x4 matrix (12 numbers)
- Lorentz Rotations and Boost
  - ROOT::Math::LorentzRotation , 4D rotation (3D rotation plus a boost) described by a 4x4 matrix
  - ROOT::Math::Boost, a Lorentz boost in an arbitrary direction and described by a 4x4 symmetrix matrix (10 numbers)
  - ROOT::Math::BoostX, a boost in the X axis direction
  - ROOT::Math::BoostY, a boost in the Y axis direction
  - ROOT::Math::BoostZ, a boost in the Z axis direction

#### Constructors

All rotations and transformations are default constructible (giving the identity transformation). All rotations are constructible taking a number of scalar argumnts matching the number (and order of components)

    Rotation3D    rI;                                // create a summy rotation (Identity matrix)
    RotationX     rX(M_PI);                          // create a rotationX with an angle PI
    EulerAngles   rE(phi, theta, psi);               // create a Euler rotation with phi,theta,psi angles
    XYZVector     u(ux,uy,uz);
    AxisAngle     rA(u, delta);                      // create a rotation based on direction u with delta angle

In addition, all rotations and transformations (other than the axial rotations) and transformations are constructible from (begin,end) iterators or from pointers whicih behave like iterators.

    double          data[9];
    Rotation3D      r(data, data+9);                 // create a rotation from a rotation matrix
    std::vector<double> w(12);
    Transform3D     t(w.begin(),w.end());            // create a Transform3D from the content of a std::vector

All rotations, except the axial rotations, are constructible and assigned from any other type of rotation (including the axial):

    Rotation3D    r(ROOT::Math::RotationX(PI));      // create a rotation 3D from a rotation along X axis of angle PI
    EulerAngles   r2(r);                             // construct an Euler Rotation from A Rotation3D
    AxisAngle     r3; r3 = r2;                       // assign an Axis Rotation from an Euler Rotation;

Transform3D (rotation + translation) can be constructed from a rotation and a translation vector

    Rotation3D r; XYZVector v;
    Transform3D   t1(r,v);                           // construct from rotation and then translation
    Transform3D   t2(v,r);                           // construct inverse from first translation then rotation
    Transform3D   t3(r);                             // construct from only a rotation (zero translation)
    Transform3D   t4(v);                             // construct from only translation (identity rotation)

#### Operations

All transformations can be applied to vector and points using the operator * or using the operator()

    XYZVector  v1(...);
    Rotation3D r(...);
    XYZVector v2 = r*v1;                             // rotate vector v1 using r
    v2 = r(v1)                                       // equivalent

Transformations can be combined using the operator * . Note that the rotations are not commutative ans therefore the order is important

    Rotation3D     r1(...);
    Rotation3D     r2(...);
    Rotation3D  r3 = r2*r1;                          // obtain a combine rotation r3 by applying first r1 then r2

We can combine rotations of different types, like Rotation3D with any other type of rotations. The product of two different axial rotations return a Rotation3D:

    RotationX        rx(1.);
    RotationY        ry(2.);
    Rotation3D  r = ry * rx;                         // rotation along X and then Y axis

It is also possible to invert all the transformation or return the inverse of a transformation

    Rotation3D           r1(...);
                     r1.Invert();                    // invert the rotation modifying its content
    Rotation3D  r2 =r1.Inverse();                    // return the inverse in a new rotation class

We have used rotation as examples, but all these operations can be applied to all the transformation classes. Rotation3D, Transform3D and Translation3D classes can all be combined via the operator *.

    Rotation3D     r(AxisAngle(phi,ux,uy,uz));   // rotation of an angle phi around u.
    Translation3D  d(dx,dy,dz);                  // translation of a vector d
    Transform3D    t1 = d * r;                   // transformation obtained applying first the rotation
    Transform3D    t2 = r * d;                   // transformation obtained applying first the translation

#### Set/GetComponents methods

Common methods to all the transformations are the Get and SetComponents. They can be used to retrieve all the scalar values on which the trasformation is based. They can be used with a signature based iterators or by using any foreign matrix which implements the operator(i,j) or a different signatures depending on the transformation type.

    RotationX  rx;  rx.SetComponents(1.)           // set agle of the X rotation
    double d[9] = {........}
    Rotation3D r;   r.SetComponents(d,d+9);        // set 9 components of 3D rotation
    double d[16];
    LorentzRotation lr;
    lr.GetComponents( d, d+16);                    // get 16 components of a LorentzRotation
    TMatrixD(3,4) m;
    Transform3D t;  t.GetComponens(m);             // fill matrix of size 3x4 with components of the transform3D t

For more detailed documentation on all methods see the reference doc for the specific transformation class. */