// @(#)root/physics:$Name:  $:$Id: TRotation.cxx,v 1.3 2002/05/18 08:22:00 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 socalled 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) = | 1 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,newX,newZ);

Memberfunctions 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();

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 consecutiv 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 operator *= of the
TVector3 class:

  TVector3 v;
  TRotation r;
  v.Transform(r);
  v *= r;  //Attention v = r * v */ // // #include "TRotation.h" #include "TError.h" ClassImp(TRotation) 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 indeces (%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::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); } }



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