    // @(#)root/physics:$Name:  $:$Id: TRotation.cxx,v 1.8 2005/06/28 10:16:12 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/             *
    //*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
    //

## TRotation

The TRotation class describes a rotation of objects of the TVector3 class. It is a 3*3 matrix of Double_t:

`| xx xy xz |`
`| yx yy yz |`
`| zx zy zz |`

It describes a so called active rotation, i.e. rotation of objects inside a static system of coordinates. In case you want to rotate the frame and want to know the coordinates of objects in the rotated system, you should apply the inverse rotation to the objects. If you want to transform coordinates from the rotated frame to the original frame you have to apply the direct transformation.

A rotation around a specified axis means counterclockwise rotation around the positive direction of the axis.

### Declaration, Access, Comparisons

`TRotation r; // r initialized as identity`
`TRotation m(r); // m = r`

There is no direct way to to set the matrix elements - to ensure that a `TRotation` object always describes a real rotation. But you can get the values by the member functions `XX()..ZZ()` or the`(,)` operator:

`Double_t xx = r.XX(); // the same as xx=r(0,0)`
`xx = r(0,0);`

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

### Rotation around axes

The following matrices desrcibe counterclockwise rotations around coordinate axes

`| 1 0 0 |`
`Rx(a) = | 0 cos(a) -sin(a) |`
`| 0 sin(a) cos(a) |`

`| cos(a) 0 sin(a) |`
`Ry(a) = | 0 1 0 |`
`| -sin(a) 0 cos(a) |`

`| cos(a) -sin(a) 0 |`
`Rz(a) = | sin(a) cos(a) 0 |`
`| 0 0 1 |`
and are implemented as member functions `RotateX()`, `RotateY()` and `RotateZ()`:

`r.RotateX(TMath::Pi()); // rotation around the x-axis`

### Rotation around arbitary axis

The member function `Rotate()` allows to rotate around an arbitary vector (not neccessary a unit one) and returns the result.

`r.Rotate(TMath::Pi()/3,TVector3(3,4,5));`

It is possible to find a unit vector and an angle, which describe the same rotation as the current one:

`Double_t angle;`
`TVector3 axis;`
`r.GetAngleAxis(angle,axis);`

### Rotation of local axes

Member function `RotateAxes()` adds a rotation of local axes to the current rotation and returns the result:

`TVector3 newX(0,1,0);`
`TVector3 newY(0,0,1);`
`TVector3 newZ(1,0,0);`
`a.RotateAxes(newX,newY,newZ);`

Member functions `ThetaX()`, `ThetaY()`, `ThetaZ()`, `PhiX()`, `PhiY()`,`PhiZ()` return azimuth and polar angles of the rotated axes:

`Double_t tx,ty,tz,px,py,pz;`
`tx= a.ThetaX();`
`...`
`pz= a.PhiZ();`

### Setting The Rotations

The member function `SetToIdentity()` will set the rotation object to the identity (no rotation). With a minor caveat, the Euler angles of the rotation may be set using `SetXEulerAngles()` or individually set with `SetXPhi()`, `SetXTheta()`, and `SetXPsi()`. These routines set the Euler angles using the X-convention which is defined by a rotation about the Z-axis, about the new X-axis, and about the new Z-axis. This is the convention used in Landau and Lifshitz, Goldstein and other common physics texts. The Y-convention euler angles can be set with `SetYEulerAngles()`, `SetYPhi()`, `SetYTheta()`, and `SetYPsi()`. The caveat is that Euler angles usually define the rotation of the new coordinate system with respect to the original system, however, the TRotation class specifies the rotation of the object in the original system (an active rotation). To recover the usual Euler rotations (ie. rotate the system not the object), you must take the inverse of the rotation. The member functions `SetXAxis()`, `SetYAxis()`, and `SetZAxis()` will create a rotation which rotates the requested axis of the object to be parallel to a vector. If used with one argument, the rotation about that axis is arbitrary. If used with two arguments, the second variable defines the `XY`, `YZ`, or `ZX` respectively.

### Inverse rotation

`TRotation a,b;`
`...`
`b = a.Inverse(); // b is inverse of a, a is unchanged`
`b = a.Invert(); // invert a and set b = a`

### Compound Rotations

The `operator *` has been implemented in a way that follows the mathematical notation of a product of the two matrices which describe the two consecutive rotations. Therefore the second rotation should be placed first:

`r = r2 * r1;`

### Rotation of TVector3

The TRotation class provides an `operator *` which allows to express a rotation of a `TVector3` analog to the mathematical notation

`| x' | | xx xy xz | | x |`
`| y' | = | yx yy yz | | y |`
`| z' | | zx zy zz | | z |```

e.g.:

`TVector3 v(1,1,1);`
`v = r * v;```

You can also use the `Transform()` member function or the o`perator *=` of the
TVector3 class:``

`TVector3 v;`
`TRotation r;`
`v.Transform(r);`
`v *= r; //Attention v = r * v`

    //

    #include "TRotation.h"
    #include "TQuaternion.h"
    #include "TError.h"

    ClassImp(TRotation)

    #define TOLERANCE (1.0E-6)

     TRotation::TRotation()
    : fxx(1.0), fxy(0.0), fxz(0.0), fyx(0.0), fyy(1.0), fyz(0.0),
      fzx(0.0), fzy(0.0), fzz(1.0) {}

     TRotation::TRotation(const TRotation & m) : TObject(m),
      fxx(m.fxx), fxy(m.fxy), fxz(m.fxz), fyx(m.fyx), fyy(m.fyy), fyz(m.fyz),
      fzx(m.fzx), fzy(m.fzy), fzz(m.fzz) {}

     TRotation::TRotation(Double_t mxx, Double_t mxy, Double_t mxz,
    			 Double_t myx, Double_t myy, Double_t myz,
    			 Double_t mzx, Double_t mzy, Double_t mzz)
    : fxx(mxx), fxy(mxy), fxz(mxz), fyx(myx), fyy(myy), fyz(myz),
      fzx(mzx), fzy(mzy), fzz(mzz) {}

    Double_t TRotation::operator() (int i, int j) const {
      if (i == 0) {
        if (j == 0) { return fxx; }
        if (j == 1) { return fxy; }
        if (j == 2) { return fxz; }
      } else if (i == 1) {
        if (j == 0) { return fyx; }
        if (j == 1) { return fyy; }
        if (j == 2) { return fyz; }
      } else if (i == 2) {
        if (j == 0) { return fzx; }
        if (j == 1) { return fzy; }
        if (j == 2) { return fzz; }
      }

      Warning("operator()(i,j)", "bad indices (%d , %d)",i,j);

      return 0.0;
    }

    TRotation TRotation::operator* (const TRotation & b) const {
      return TRotation(fxx*b.fxx + fxy*b.fyx + fxz*b.fzx,
    		     fxx*b.fxy + fxy*b.fyy + fxz*b.fzy,
    		     fxx*b.fxz + fxy*b.fyz + fxz*b.fzz,
    		     fyx*b.fxx + fyy*b.fyx + fyz*b.fzx,
    		     fyx*b.fxy + fyy*b.fyy + fyz*b.fzy,
    		     fyx*b.fxz + fyy*b.fyz + fyz*b.fzz,
    		     fzx*b.fxx + fzy*b.fyx + fzz*b.fzx,
    		     fzx*b.fxy + fzy*b.fyy + fzz*b.fzy,
    		     fzx*b.fxz + fzy*b.fyz + fzz*b.fzz);
    }

    //_____________________________________
     TRotation::TRotation(const TQuaternion & Q) {
    	// Constructor for a rotation based on a Quaternion
    	// if magnitude of quaternion is null, creates identity rotation
    	// if quaternion is non-unit, creates rotation corresponding to the normalized (unit) quaternion

    	double two_r2 = 2 * Q.fRealPart * Q.fRealPart;
    	double two_x2 = 2 * Q.fVectorPart.X() * Q.fVectorPart.X();
    	double two_y2 = 2 * Q.fVectorPart.Y() * Q.fVectorPart.Y();
    	double two_z2 = 2 * Q.fVectorPart.Z() * Q.fVectorPart.Z();
    	double two_xy = 2 * Q.fVectorPart.X() * Q.fVectorPart.Y();
    	double two_xz = 2 * Q.fVectorPart.X() * Q.fVectorPart.Z();
    	double two_xr = 2 * Q.fVectorPart.X() * Q.fRealPart;
    	double two_yz = 2 * Q.fVectorPart.Y() * Q.fVectorPart.Z();
    	double two_yr = 2 * Q.fVectorPart.Y() * Q.fRealPart;
    	double two_zr = 2 * Q.fVectorPart.Z() * Q.fRealPart;

    	// protect agains zero quaternion
    	double mag2 = Q.QMag2();
    	if (mag2 > 0) {

    		// diago + identity
    		fxx = two_r2 + two_x2;
    		fyy = two_r2 + two_y2;
    		fzz = two_r2 + two_z2;

    		//	line 0 column 1 and conjugate
    		fxy = two_xy - two_zr;
    		fyx = two_xy + two_zr;

    		//	line 0 column 2 and conjugate
    		fxz = two_xz + two_yr;
    		fzx = two_xz - two_yr;

    		//	line 1 column 2 and conjugate
    		fyz = two_yz - two_xr;
    		fzy = two_yz + two_xr;

    		// protect agains non-unit quaternion
    		if (TMath::Abs(mag2-1) > 1e-10) {
    			fxx /= mag2;
    			fyy /= mag2;
    			fzz /= mag2;
    			fxy /= mag2;
    			fyx /= mag2;
    			fxz /= mag2;
    			fzx /= mag2;
    			fyz /= mag2;
    			fzy /= mag2;
    		}

    		// diago : remove identity
    		fxx -= 1;
    		fyy -= 1;
    		fzz -= 1;

    	} else {
    		// Identity

    		fxx = fyy = fzz = 1;
    		fxy = fyx = fxz = fzx = fyz = fzy = 0;

    	}

    }

     TRotation & TRotation::Rotate(Double_t a, const TVector3& axis) {
      if (a != 0.0) {
        Double_t ll = axis.Mag();
        if (ll == 0.0) {
          Warning("Rotate(angle,axis)"," zero axis");
        }else{
          Double_t sa = TMath::Sin(a), ca = TMath::Cos(a);
          Double_t dx = axis.X()/ll, dy = axis.Y()/ll, dz = axis.Z()/ll;
          TRotation m(
    	ca+(1-ca)*dx*dx,          (1-ca)*dx*dy-sa*dz,    (1-ca)*dx*dz+sa*dy,
    	   (1-ca)*dy*dx+sa*dz, ca+(1-ca)*dy*dy,          (1-ca)*dy*dz-sa*dx,
    	   (1-ca)*dz*dx-sa*dy,    (1-ca)*dz*dy+sa*dx, ca+(1-ca)*dz*dz );
          Transform(m);
        }
      }
      return *this;
    }

     TRotation & TRotation::RotateX(Double_t a) {
      Double_t c = TMath::Cos(a);
      Double_t s = TMath::Sin(a);
      Double_t x = fyx, y = fyy, z = fyz;
      fyx = c*x - s*fzx;
      fyy = c*y - s*fzy;
      fyz = c*z - s*fzz;
      fzx = s*x + c*fzx;
      fzy = s*y + c*fzy;
      fzz = s*z + c*fzz;
      return *this;
    }

     TRotation & TRotation::RotateY(Double_t a){
      Double_t c = TMath::Cos(a);
      Double_t s = TMath::Sin(a);
      Double_t x = fzx, y = fzy, z = fzz;
      fzx = c*x - s*fxx;
      fzy = c*y - s*fxy;
      fzz = c*z - s*fxz;
      fxx = s*x + c*fxx;
      fxy = s*y + c*fxy;
      fxz = s*z + c*fxz;
      return *this;
    }

     TRotation & TRotation::RotateZ(Double_t a) {
      Double_t c = TMath::Cos(a);
      Double_t s = TMath::Sin(a);
      Double_t x = fxx, y = fxy, z = fxz;
      fxx = c*x - s*fyx;
      fxy = c*y - s*fyy;
      fxz = c*z - s*fyz;
      fyx = s*x + c*fyx;
      fyy = s*y + c*fyy;
      fyz = s*z + c*fyz;
      return *this;
    }

     TRotation & TRotation::RotateAxes(const TVector3 &newX,
                                      const TVector3 &newY,
                                      const TVector3 &newZ) {
      Double_t del = 0.001;
      TVector3 w = newX.Cross(newY);

      if (TMath::Abs(newZ.X()-w.X()) > del ||
          TMath::Abs(newZ.Y()-w.Y()) > del ||
          TMath::Abs(newZ.Z()-w.Z()) > del ||
          TMath::Abs(newX.Mag2()-1.) > del ||
          TMath::Abs(newY.Mag2()-1.) > del ||
          TMath::Abs(newZ.Mag2()-1.) > del ||
          TMath::Abs(newX.Dot(newY)) > del ||
          TMath::Abs(newY.Dot(newZ)) > del ||
          TMath::Abs(newZ.Dot(newX)) > del) {
        Warning("RotateAxes","bad axis vectors");
        return *this;
      }else{
        return Transform(TRotation(newX.X(), newY.X(), newZ.X(),
                                     newX.Y(), newY.Y(), newZ.Y(),
                                     newX.Z(), newY.Z(), newZ.Z()));
      }
    }

     Double_t TRotation::PhiX() const {
      return (fyx == 0.0 && fxx == 0.0) ? 0.0 : TMath::ATan2(fyx,fxx);
    }

     Double_t TRotation::PhiY() const {
      return (fyy == 0.0 && fxy == 0.0) ? 0.0 : TMath::ATan2(fyy,fxy);
    }

     Double_t TRotation::PhiZ() const {
      return (fyz == 0.0 && fxz == 0.0) ? 0.0 : TMath::ATan2(fyz,fxz);
    }

     Double_t TRotation::ThetaX() const {
      return TMath::ACos(fzx);
    }

     Double_t TRotation::ThetaY() const {
      return TMath::ACos(fzy);
    }

     Double_t TRotation::ThetaZ() const {
      return TMath::ACos(fzz);
    }

     void TRotation::AngleAxis(Double_t &angle, TVector3 &axis) const {
      Double_t cosa  = 0.5*(fxx+fyy+fzz-1);
      Double_t cosa1 = 1-cosa;
      if (cosa1 <= 0) {
        angle = 0;
        axis  = TVector3(0,0,1);
      }else{
        Double_t x=0, y=0, z=0;
        if (fxx > cosa) x = TMath::Sqrt((fxx-cosa)/cosa1);
        if (fyy > cosa) y = TMath::Sqrt((fyy-cosa)/cosa1);
        if (fzz > cosa) z = TMath::Sqrt((fzz-cosa)/cosa1);
        if (fzy < fyz)  x = -x;
        if (fxz < fzx)  y = -y;
        if (fyx < fxy)  z = -z;
        angle = TMath::ACos(cosa);
        axis  = TVector3(x,y,z);
      }
    }

     TRotation & TRotation::SetXEulerAngles(Double_t phi,
                                          Double_t theta,
                                          Double_t psi) {
      // Rotate using the x-convention (Landau and Lifshitz, Goldstein, &c) by
      // doing the explicit rotations.  This is slightly less efficient than
      // directly applying the rotation, but makes the code much clearer.  My
      // presumption is that this code is not going to be a speed bottle neck.

      SetToIdentity();
      RotateZ(phi);
      RotateX(theta);
      RotateZ(psi);

      return *this;
    }

     TRotation & TRotation::SetYEulerAngles(Double_t phi,
                                           Double_t theta,
                                           Double_t psi) {
      // Rotate using the y-convention.

      SetToIdentity();
      RotateZ(phi);
      RotateY(theta);
      RotateZ(psi);
      return *this;
    }

     TRotation & TRotation::RotateXEulerAngles(Double_t phi,
                                             Double_t theta,
                                             Double_t psi) {
      TRotation euler;
      euler.SetXEulerAngles(phi,theta,psi);
      return Transform(euler);
    }

     TRotation & TRotation::RotateYEulerAngles(Double_t phi,
                                              Double_t theta,
                                              Double_t psi) {
      TRotation euler;
      euler.SetYEulerAngles(phi,theta,psi);
      return Transform(euler);
    }

     void TRotation::SetXPhi(Double_t phi) {
      SetXEulerAngles(phi,GetXTheta(),GetXPsi());
    }

     void TRotation::SetXTheta(Double_t theta) {
      SetXEulerAngles(GetXPhi(),theta,GetXPsi());
    }

     void TRotation::SetXPsi(Double_t psi) {
      SetXEulerAngles(GetXPhi(),GetXTheta(),psi);
    }

     void TRotation::SetYPhi(Double_t phi) {
      SetYEulerAngles(phi,GetYTheta(),GetYPsi());
    }

     void TRotation::SetYTheta(Double_t theta) {
      SetYEulerAngles(GetYPhi(),theta,GetYPsi());
    }

     void TRotation::SetYPsi(Double_t psi) {
      SetYEulerAngles(GetYPhi(),GetYTheta(),psi);
    }

     Double_t TRotation::GetXPhi(void) const {
      Double_t finalPhi;

      Double_t s2 =  1.0 - fzz*fzz;
      if (s2 < 0) {
        Warning("GetPhi()"," |fzz| > 1 ");
        s2 = 0;
      }
      const Double_t sinTheta = TMath::Sqrt(s2);

      if (sinTheta != 0) {
        const Double_t cscTheta = 1/sinTheta;
        Double_t cosAbsPhi =  fzy * cscTheta;
        if ( TMath::Abs(cosAbsPhi) > 1 ) {	// NaN-proofing
            Warning("GetPhi()","finds | cos phi | > 1");
            cosAbsPhi = 1;
        }
        const Double_t absPhi = TMath::ACos(cosAbsPhi);
        if (fzx > 0) {
            finalPhi = absPhi;
        } else if (fzx < 0) {
            finalPhi = -absPhi;
        } else if (fzy > 0) {
            finalPhi = 0.0;
        } else {
            finalPhi = TMath::Pi();
        }
      }
      else {              // sinTheta == 0 so |Fzz| = 1
        const Double_t absPhi = .5 * TMath::ACos (fxx);
        if (fxy > 0) {
          finalPhi =  -absPhi;
        } else if (fxy < 0) {
          finalPhi =   absPhi;
        } else if (fxx>0) {
          finalPhi = 0.0;
        } else {
          finalPhi = fzz * TMath::PiOver2();
        }
      }
      return finalPhi;
    }

     Double_t TRotation::GetYPhi(void) const {
      return GetXPhi() + TMath::Pi()/2.0;
    }

     Double_t TRotation::GetXTheta(void) const {
      return  ThetaZ();
    }

     Double_t TRotation::GetYTheta(void) const {
      return  ThetaZ();
    }

     Double_t TRotation::GetXPsi(void) const {
      double finalPsi = 0.0;

      Double_t s2 =  1.0 - fzz*fzz;
      if (s2 < 0) {
        Warning("GetPsi()"," |fzz| > 1 ");
        s2 = 0;
      }
      const Double_t sinTheta = TMath::Sqrt(s2);

      if (sinTheta != 0) {
        const Double_t cscTheta = 1/sinTheta;
        Double_t cosAbsPsi =  - fyz * cscTheta;
        if ( TMath::Abs(cosAbsPsi) > 1 ) {	// NaN-proofing
          Warning("GetPsi()","| cos psi | > 1 ");
          cosAbsPsi = 1;
        }
        const Double_t absPsi = TMath::ACos(cosAbsPsi);
        if (fxz > 0) {
          finalPsi = absPsi;
        } else if (fxz < 0) {
          finalPsi = -absPsi;
        } else {
          finalPsi = (fyz < 0) ? 0 : TMath::Pi();
        }
      }
      else {              // sinTheta == 0 so |Fzz| = 1
        Double_t absPsi = fxx;
        if ( TMath::Abs(fxx) > 1 ) {	// NaN-proofing
          Warning("GetPsi()","| fxx | > 1 ");
          absPsi = 1;
        }
        absPsi = .5 * TMath::ACos (absPsi);
        if (fyx > 0) {
          finalPsi = absPsi;
        } else if (fyx < 0) {
          finalPsi = -absPsi;
        } else {
          finalPsi = (fxx > 0) ? 0 : TMath::PiOver2();
        }
      }
      return finalPsi;
    }

     Double_t TRotation::GetYPsi(void) const {
      return GetXPsi() - TMath::Pi()/2;
    }

     TRotation & TRotation::SetXAxis(const TVector3& axis,
                                    const TVector3& xyPlane) {
      TVector3 xAxis(xyPlane);
      TVector3 yAxis;
      TVector3 zAxis(axis);
      MakeBasis(xAxis,yAxis,zAxis);
      fxx = zAxis.X();  fyx = zAxis.Y();  fzx = zAxis.Z();
      fxy = xAxis.X();  fyy = xAxis.Y();  fzy = xAxis.Z();
      fxz = yAxis.X();  fyz = yAxis.Y();  fzz = yAxis.Z();
      return *this;
    }

     TRotation & TRotation::SetXAxis(const TVector3& axis) {
      TVector3 xyPlane(0.0,1.0,0.0);
      return SetXAxis(axis,xyPlane);
    }

     TRotation & TRotation::SetYAxis(const TVector3& axis,
                                    const TVector3& yzPlane) {
      TVector3 xAxis(yzPlane);
      TVector3 yAxis;
      TVector3 zAxis(axis);
      MakeBasis(xAxis,yAxis,zAxis);
      fxx = yAxis.X();  fyx = yAxis.Y();  fzx = yAxis.Z();
      fxy = zAxis.X();  fyy = zAxis.Y();  fzy = zAxis.Z();
      fxz = xAxis.X();  fyz = xAxis.Y();  fzz = xAxis.Z();
      return *this;
    }

     TRotation & TRotation::SetYAxis(const TVector3& axis) {
      TVector3 yzPlane(0.0,0.0,1.0);
      return SetYAxis(axis,yzPlane);
    }

     TRotation & TRotation::SetZAxis(const TVector3& axis,
                                    const TVector3& zxPlane) {
      TVector3 xAxis(zxPlane);
      TVector3 yAxis;
      TVector3 zAxis(axis);
      MakeBasis(xAxis,yAxis,zAxis);
      fxx = xAxis.X();  fyx = xAxis.Y();  fzx = xAxis.Z();
      fxy = yAxis.X();  fyy = yAxis.Y();  fzy = yAxis.Z();
      fxz = zAxis.X();  fyz = zAxis.Y();  fzz = zAxis.Z();
      return *this;
    }

     TRotation & TRotation::SetZAxis(const TVector3& axis) {
      TVector3 zxPlane(1.0,0.0,0.0);
      return SetZAxis(axis,zxPlane);
    }

     void TRotation::MakeBasis(TVector3& xAxis,
                              TVector3& yAxis,
                              TVector3& zAxis) const {
      // Make the zAxis into a unit variable.
      Double_t zmag = zAxis.Mag();
      if (zmag<TOLERANCE) {
          Warning("MakeBasis(X,Y,Z)","non-zero Z Axis is required");
      }
      zAxis *= (1.0/zmag);

      Double_t xmag = xAxis.Mag();
      if (xmag<TOLERANCE*zmag) {
          xAxis = zAxis.Orthogonal();
          xmag = 1.0;
      }

      // Find the yAxis
      yAxis = zAxis.Cross(xAxis)*(1.0/xmag);
      Double_t ymag = yAxis.Mag();
      if (ymag<TOLERANCE*zmag) {
          yAxis = zAxis.Orthogonal();
      }

      else {
          yAxis *= (1.0/ymag);
      }

      xAxis = yAxis.Cross(zAxis);
    }