// @(#)root/physics:$Name:  $:$Id: TLorentzRotation.cxx,v 1.3 2003/09/03 06:08:34 brun Exp $
// Author: Peter Malzacher   19/06/99
//______________________________________________________________________________
//*-*-*-*-*-*-*-*-*-*-*-*The Physics Vector package *-*-*-*-*-*-*-*-*-*-*-*
//*-*                    ==========================                       *
//*-* The Physics Vector package consists of five classes:                *
//*-*   - TVector2                                                        *
//*-*   - TVector3                                                        *
//*-*   - TRotation                                                       *
//*-*   - TLorentzVector                                                  *
//*-*   - TLorentzRotation                                                *
//*-* It is a combination of CLHEPs Vector package written by             *
//*-* Leif Lonnblad, Andreas Nilsson and Evgueni Tcherniaev               *
//*-* and a ROOT package written by Pasha Murat.                          *
//*-* for CLHEP see:  http://wwwinfo.cern.ch/asd/lhc++/clhep/             *
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
//

TLorentzRotation

The TLorentzRotation class describes Lorentz transformations including Lorentz boosts and rotations (see TRotation)

            | xx  xy  xz  xt |
            |                |
            | yx  yy  yz  yt |
   lambda = |                |
            | zx  zy  zz  zt |
            |                |
            | tx  ty  tz  tt |
 

Declaration

By default it is initialized to the identity matrix, but it may also be intialized by an other TLorentzRotation,
by a pure TRotation or by a boost:

  TLorentzRotation l;      // l is initialized as identity
  TLorentzRotation m(l);   // m = l
  TRotation r;
  TLorentzRotation lr(r);
  TLorentzRotation lb1(bx,by,bz);
  TVector3 b;
  TLorentzRotation lb2(b);

The Matrix for a Lorentz boosts is:

 | 1+gamma'*bx*bx  gamma'*bx*by   gamma'*bx*bz  gamma*bx |
 |  gamma'*bx*bz  1+gamma'*by*by  gamma'*by*by  gamma*by |
 |  gamma'*bz*bx   gamma'*bz*by  1+gamma'*bz*bz gamma*bz |
 |    gamma*bx       gamma*by       gamma*bz     gamma   |

with the boost vector b=(bx,by,bz) and gamma=1/Sqrt(1-beta*beta) and gamma'=(gamma-1)/beta*beta.

Access to the matrix components/Comparisons

Access to the matrix components is possible through the member functions XX(), XY() .. TT(),
through the operator (int,int):

  Double_t xx;
  TLorentzRotation l;
  xx = l.XX();    // gets the xx component
  xx = l(0,0);    // gets the xx component

  if (l==m) {...}  // test for equality
  if (l !=m) {...} // test for inequality
  if (l.IsIdentity()) {...} // test for identity
 

Transformations of a LorentzRotation

Compound transformations
There are four possibilities to find the product of two TLorentzRotation transformations:

  TLorentzRotation a,b,c;
  c = b*a;                       // product
  c = a.MatrixMultiplication(b);  // a is unchanged
  a *= b;                        // Attention: a=a*b
  c = a.Transform(b)             // a=b*a then c=a
 

Lorentz boosts
  Double_t bx, by, bz;
  TVector3 v(bx,by,bz);
  TLorentzRotation l;
  l.Boost(v);
  l.Boost(bx,by,bz);
 
Rotations
  TVector3 axis;
  l.RotateX(TMath::Pi());   //  rotation around x-axis
  l.Rotate(.5,axis);               //  rotation around specified vector
Inverse transformation
The matrix for the inverse transformation of a TLorentzRotation is as follows:
            | xx  yx  zx -tx |
            |                |
            | xy  yy  zy -ty |
            |                |
            | xz  yz  zz -tz |
            |                |
            |-xt -yt -zt  tt |
To return the inverse transformation keeping the current unchanged use the memberfunction Inverse().
Invert() inverts the current TLorentzRotation:

  l1 = l2.Inverse();  // l1 is inverse of l2, l2 unchanged
  l1 = l2.Invert();   // invert l2, then  l1=l2

Transformation of a TLorentzVector

To apply TLorentzRotation to TLorentzVector you can use either the VectorMultiplication() member function or the * operator. You can also use the Transform() function and the *= operator of the TLorentzVector class.:

  TLorentzVector v;
  ...
  v=l.VectorMultiplication(v);
  v = l * v;

  v.Transform(l);
  v *= l;  // Attention v = l*v 


//

#include "TError.h"
#include "TLorentzRotation.h"

ClassImp(TLorentzRotation)

 TLorentzRotation::TLorentzRotation()
  : fxx(1.0), fxy(0.0), fxz(0.0), fxt(0.0),
    fyx(0.0), fyy(1.0), fyz(0.0), fyt(0.0),
    fzx(0.0), fzy(0.0), fzz(1.0), fzt(0.0),
    ftx(0.0), fty(0.0), ftz(0.0), ftt(1.0) {}

 TLorentzRotation::TLorentzRotation(const TRotation & r)
  : fxx(r.XX()), fxy(r.XY()), fxz(r.XZ()), fxt(0.0),
    fyx(r.YX()), fyy(r.YY()), fyz(r.YZ()), fyt(0.0),
    fzx(r.ZX()), fzy(r.ZY()), fzz(r.ZZ()), fzt(0.0),
    ftx(0.0),    fty(0.0),    ftz(0.0),    ftt(1.0) {}

 TLorentzRotation::TLorentzRotation(const TLorentzRotation & r) : TObject(r),
    fxx(r.fxx), fxy(r.fxy), fxz(r.fxz), fxt(r.fxt),
    fyx(r.fyx), fyy(r.fyy), fyz(r.fyz), fyt(r.fyt),
    fzx(r.fzx), fzy(r.fzy), fzz(r.fzz), fzt(r.fzt),
    ftx(r.ftx), fty(r.fty), ftz(r.ftz), ftt(r.ftt) {}

 TLorentzRotation::TLorentzRotation(
  Double_t rxx, Double_t rxy, Double_t rxz, Double_t rxt,
  Double_t ryx, Double_t ryy, Double_t ryz, Double_t ryt,
  Double_t rzx, Double_t rzy, Double_t rzz, Double_t rzt,
  Double_t rtx, Double_t rty, Double_t rtz, Double_t rtt)
  : fxx(rxx), fxy(rxy), fxz(rxz), fxt(rxt),
    fyx(ryx), fyy(ryy), fyz(ryz), fyt(ryt),
    fzx(rzx), fzy(rzy), fzz(rzz), fzt(rzt),
    ftx(rtx), fty(rty), ftz(rtz), ftt(rtt) {}

 TLorentzRotation::TLorentzRotation(Double_t bx,
                                   Double_t by,
                                   Double_t bz)
{
  SetBoost(bx, by, bz);
}

 TLorentzRotation::TLorentzRotation(const TVector3 & p) {
  SetBoost(p.X(), p.Y(), p.Z());
}

Double_t TLorentzRotation::operator () (int i, int j) const {
  if (i == 0) {
    if (j == 0) { return fxx; }
    if (j == 1) { return fxy; }
    if (j == 2) { return fxz; }
    if (j == 3) { return fxt; }
  } else if (i == 1) {
    if (j == 0) { return fyx; }
    if (j == 1) { return fyy; }
    if (j == 2) { return fyz; }
    if (j == 3) { return fyt; }
  } else if (i == 2) {
    if (j == 0) { return fzx; }
    if (j == 1) { return fzy; }
    if (j == 2) { return fzz; }
    if (j == 3) { return fzt; }
  } else if (i == 3) {
    if (j == 0) { return ftx; }
    if (j == 1) { return fty; }
    if (j == 2) { return ftz; }
    if (j == 3) { return ftt; }
  }
  Warning("operator()(i,j)","subscripting: bad indeces(%d,%d)",i,j);
  return 0.0;
}

 void TLorentzRotation::SetBoost(Double_t bx, Double_t by, Double_t bz) {
  Double_t bp2 = bx*bx + by*by + bz*bz;
  Double_t gamma = 1.0 / TMath::Sqrt(1.0 - bp2);
  Double_t bgamma = gamma * gamma / (1.0 + gamma);
  fxx = 1.0 + bgamma * bx * bx;
  fyy = 1.0 + bgamma * by * by;
  fzz = 1.0 + bgamma * bz * bz;
  fxy = fyx = bgamma * bx * by;
  fxz = fzx = bgamma * bx * bz;
  fyz = fzy = bgamma * by * bz;
  fxt = ftx = gamma * bx;
  fyt = fty = gamma * by;
  fzt = ftz = gamma * bz;
  ftt = gamma;
}

TLorentzRotation
 TLorentzRotation::MatrixMultiplication(const TLorentzRotation & b) const {
  return TLorentzRotation(
    fxx*b.fxx + fxy*b.fyx + fxz*b.fzx + fxt*b.ftx,
    fxx*b.fxy + fxy*b.fyy + fxz*b.fzy + fxt*b.fty,
    fxx*b.fxz + fxy*b.fyz + fxz*b.fzz + fxt*b.ftz,
    fxx*b.fxt + fxy*b.fyt + fxz*b.fzt + fxt*b.ftt,

    fyx*b.fxx + fyy*b.fyx + fyz*b.fzx + fyt*b.ftx,
    fyx*b.fxy + fyy*b.fyy + fyz*b.fzy + fyt*b.fty,
    fyx*b.fxz + fyy*b.fyz + fyz*b.fzz + fyt*b.ftz,
    fyx*b.fxt + fyy*b.fyt + fyz*b.fzt + fyt*b.ftt,

    fzx*b.fxx + fzy*b.fyx + fzz*b.fzx + fzt*b.ftx,
    fzx*b.fxy + fzy*b.fyy + fzz*b.fzy + fzt*b.fty,
    fzx*b.fxz + fzy*b.fyz + fzz*b.fzz + fzt*b.ftz,
    fzx*b.fxt + fzy*b.fyt + fzz*b.fzt + fzt*b.ftt,

    ftx*b.fxx + fty*b.fyx + ftz*b.fzx + ftt*b.ftx,
    ftx*b.fxy + fty*b.fyy + ftz*b.fzy + ftt*b.fty,
    ftx*b.fxz + fty*b.fyz + ftz*b.fzz + ftt*b.ftz,
    ftx*b.fxt + fty*b.fyt + ftz*b.fzt + ftt*b.ftt);
}

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