//*CMZ : 2.22/06 19/06/99 09.10.34 by Peter Malzacher //*-- 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/ * //*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-* // /*
| 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.
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
| 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) = | cos(a) -sin(a) 0 |
| 0
0 1 |
and are implemented as member functions RotateX(), RotateY()
and RotateZ():
r.RotateX(TMath::Pi()); // rotation around the x-axis
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);
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();
r = r2 * r1;
| 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
*/
//
//
//*KEEP,TRotation,T=C++.
#include "TRotation.h"
//*KEEP,TError.
#include "TError.h"
//*KEND.
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)
: 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);
}
}