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<H2>
TVector3</H2>
<TT>TVector3</TT> is a general three vector class, which can be used for
the description of different vectors in 3D.
<H3>
Declaration / Access to the components</H3>
<TT>TVector3</TT> has been implemented as a vector of three <TT>Double_t</TT>
variables, representing the cartesian coordinates. By default all components
are initialized to zero:
<P><TT> TVector3 v1; //
v1 = (0,0,0)</TT>
<BR><TT> TVector3 v2(1); // v2 = (1,0,0)</TT>
<BR><TT> TVector3 v3(1,2,3); // v3 = (1,2,3)</TT>
<BR><TT> TVector3 v4(v2); // v4 = v2</TT>
<P>It is also possible (but not recommended) to initialize a <TT>TVector3</TT>
with a <TT>Double_t</TT> or <TT>Float_t</TT> C array.
<P>You can get the basic components either by name or by index using <TT>operator()</TT>:
<P><TT> xx = v1.X(); or xx =
v1(0);</TT>
<BR><TT> yy = v1.Y();
yy = v1(1);</TT>
<BR><TT> zz = v1.Z();
zz = v1(2);</TT>
<P>The memberfunctions <TT>SetX()</TT>, <TT>SetY()</TT>, <TT>SetZ()</TT>
and<TT> SetXYZ()</TT> allow to set the components:
<P><TT> v1.SetX(1.); v1.SetY(2.); v1.SetZ(3.);</TT>
<BR><TT> v1.SetXYZ(1.,2.,3.);</TT>
<BR>
<H3>
Noncartesian coordinates</H3>
To get information on the <TT>TVector3</TT> in spherical (rho,phi,theta)
or cylindrical (z,r,theta) coordinates, the
<BR>the member functions <TT>Mag()</TT> (=magnitude=rho in spherical coordinates),
<TT>Mag2()</TT>, <TT>Theta()</TT>, <TT>CosTheta()</TT>, <TT>Phi()</TT>,
<TT>Perp()</TT> (the transverse component=r in cylindrical coordinates),
<TT>Perp2()</TT> can be used:
<P><TT> Double_t m = v.Mag(); // get magnitude
(=rho=Sqrt(x*x+y*y+z*z)))</TT>
<BR><TT> Double_t m2 = v.Mag2(); // get magnitude squared</TT>
<BR><TT> Double_t t = v.Theta(); // get polar angle</TT>
<BR><TT> Double_t ct = v.CosTheta();// get cos of theta</TT>
<BR><TT> Double_t p = v.Phi(); // get azimuth
angle</TT>
<BR><TT> Double_t pp = v.Perp(); // get transverse component</TT>
<BR><TT> Double_t pp2= v.Perp2(); // get transvers component
squared</TT>
<P>It is also possible to get the transverse component with respect to
another vector:
<P><TT> Double_t ppv1 = v.Perp(v1);</TT>
<BR><TT> Double_t pp2v1 = v.Perp2(v1);</TT>
<P>The pseudorapiditiy ( eta=-ln (tan (phi/2)) ) can be get by <TT>Eta()</TT>
or <TT>PseudoRapidity()</TT>:
<BR>
<BR><TT> Double_t eta = v.PseudoRapidity();</TT>
<P>There are set functions to change one of the noncartesian coordinates:
<P><TT> v.SetTheta(.5); // keeping rho and phi</TT>
<BR><TT> v.SetPhi(.8); // keeping rho and theta</TT>
<BR><TT> v.SetMag(10.); // keeping theta and phi</TT>
<BR><TT> v.SetPerp(3.); // keeping z and phi</TT>
<BR>
<H3>
Arithmetic / Comparison</H3>
The <TT>TVector3</TT> class provides the operators to add, subtract, scale and compare
vectors:
<P><TT> v3 = -v1;</TT>
<BR><TT> v1 = v2+v3;</TT>
<BR><TT> v1 += v3;</TT>
<BR><TT> v1 = v1 - v3</TT>
<BR><TT> v1 -= v3;</TT>
<BR><TT> v1 *= 10;</TT>
<BR><TT> v1 = 5*v2;</TT>
<P><TT> if(v1==v2) {...}</TT>
<BR><TT> if(v1!=v2) {...}</TT>
<BR>
<H3>
Related Vectors</H3>
<TT> v2 = v1.Unit(); // get unit
vector parallel to v1</TT>
<BR><TT> v2 = v1.Orthogonal(); // get vector orthogonal to v1</TT>
<H3>
Scalar and vector products</H3>
<TT> s = v1.Dot(v2); // scalar product</TT>
<BR><TT> s = v1 * v2; // scalar product</TT>
<BR><TT> v = v1.Cross(v2); // vector product</TT>
<H3>
Angle between two vectors</H3>
<TT> Double_t a = v1.Angle(v2);</TT>
<H3>
Rotations</H3>
<H5>
Rotation around axes</H5>
<TT> v.RotateX(.5);</TT>
<BR><TT> v.RotateY(TMath::Pi());</TT>
<BR><TT> v.RotateZ(angle);</TT>
<H5>
Rotation around a vector</H5>
<TT> v1.Rotate(TMath::Pi()/4, v2); // rotation around v2</TT>
<H5>
Rotation by TRotation</H5>
<TT>TVector3</TT> objects can be rotated by objects of the <TT>TRotation</TT>
class using the <TT>Transform()</TT> member functions,
<BR>the <TT>operator *=</TT> or the <TT>operator *</TT> of the TRotation
class:
<P><TT> TRotation m;</TT>
<BR><TT> ...</TT>
<BR><TT> v1.transform(m);</TT>
<BR><TT> v1 = m*v1;</TT>
<BR><TT> v1 *= m; // Attention v1 = m*v1</TT>
<H5>
Transformation from rotated frame</H5>
<TT> TVector3 direction = v.Unit()</TT>
<BR><TT> v1.RotateUz(direction); // direction must be TVector3 of
unit length</TT>
<P>transforms v1 from the rotated frame (z' parallel to direction, x' in
the theta plane and y' in the xy plane as well as perpendicular to the
theta plane) to the (x,y,z) frame.
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#include "TVector3.h"
#include "TRotation.h"
#include "TClass.h"
ClassImp(TVector3)
TVector3::TVector3(const TVector3 & p) : TObject(p),
fX(p.fX), fY(p.fY), fZ(p.fZ) {}
TVector3::TVector3(Double_t x, Double_t y, Double_t z)
: fX(x), fY(y), fZ(z) {}
TVector3::TVector3(const Double_t * x0)
: fX(x0[0]), fY(x0[1]), fZ(x0[2]) {}
TVector3::TVector3(const Float_t * x0)
: fX(x0[0]), fY(x0[1]), fZ(x0[2]) {}
TVector3::~TVector3() {}
Double_t TVector3::operator () (int i) const {
switch(i) {
case 0:
return fX;
case 1:
return fY;
case 2:
return fZ;
default:
Error("operator()(i)", "bad index (%d) returning 0",i);
}
return 0.;
}
Double_t & TVector3::operator () (int i) {
switch(i) {
case 0:
return fX;
case 1:
return fY;
case 2:
return fZ;
default:
Error("operator()(i)", "bad index (%d) returning &fX",i);
}
return fX;
}
TVector3 & TVector3::operator *= (const TRotation & m){
return *this = m * (*this);
}
TVector3 & TVector3::Transform(const TRotation & m) {
return *this = m * (*this);
}
void TVector3::RotateX(Double_t angle) {
Double_t s = TMath::Sin(angle);
Double_t c = TMath::Cos(angle);
Double_t yy = fY;
fY = c*yy - s*fZ;
fZ = s*yy + c*fZ;
}
void TVector3::RotateY(Double_t angle) {
Double_t s = TMath::Sin(angle);
Double_t c = TMath::Cos(angle);
Double_t zz = fZ;
fZ = c*zz - s*fX;
fX = s*zz + c*fX;
}
void TVector3::RotateZ(Double_t angle) {
Double_t s = TMath::Sin(angle);
Double_t c = TMath::Cos(angle);
Double_t xx = fX;
fX = c*xx - s*fY;
fY = s*xx + c*fY;
}
void TVector3::Rotate(Double_t angle, const TVector3 & axis){
TRotation trans;
trans.Rotate(angle, axis);
operator*=(trans);
}
void TVector3::RotateUz(const TVector3& NewUzVector) {
Double_t u1 = NewUzVector.fX;
Double_t u2 = NewUzVector.fY;
Double_t u3 = NewUzVector.fZ;
Double_t up = u1*u1 + u2*u2;
if (up) {
up = TMath::Sqrt(up);
Double_t px = fX, py = fY, pz = fZ;
fX = (u1*u3*px - u2*py + u1*up*pz)/up;
fY = (u2*u3*px + u1*py + u2*up*pz)/up;
fZ = (u3*u3*px - px + u3*up*pz)/up;
} else if (u3 < 0.) { fX = -fX; fZ = -fZ; }
else {};
}
Double_t TVector3::PseudoRapidity() const {
double cosTheta = CosTheta();
if (cosTheta*cosTheta < 1) return -0.5* TMath::Log( (1.0-cosTheta)/(1.0+cosTheta) );
Warning("PseudoRapidity","transvers momentum = 0! return +/- 10e10");
if (fZ > 0) return 10e10;
else return -10e10;
}
void TVector3::SetPtEtaPhi(Double_t pt, Double_t eta, Double_t phi) {
Double_t apt = TMath::Abs(pt);
SetXYZ(apt*TMath::Cos(phi), apt*TMath::Sin(phi), apt/TMath::Tan(2.0*TMath::ATan(TMath::Exp(-eta))) );
}
void TVector3::SetPtThetaPhi(Double_t pt, Double_t theta, Double_t phi) {
fX = pt * TMath::Cos(phi);
fY = pt * TMath::Sin(phi);
Double_t tanTheta = TMath::Tan(theta);
fZ = tanTheta ? pt / tanTheta : 0;
}
void TVector3::Streamer(TBuffer &R__b)
{
if (R__b.IsReading()) {
UInt_t R__s, R__c;
Version_t R__v = R__b.ReadVersion(&R__s, &R__c);
if (R__v > 2) {
TVector3::Class()->ReadBuffer(R__b, this, R__v, R__s, R__c);
return;
}
if (R__v < 2) TObject::Streamer(R__b);
R__b >> fX;
R__b >> fY;
R__b >> fZ;
R__b.CheckByteCount(R__s, R__c, TVector3::IsA());
} else {
TVector3::Class()->WriteBuffer(R__b,this);
}
}
TVector3 operator + (const TVector3 & a, const TVector3 & b) {
return TVector3(a.X() + b.X(), a.Y() + b.Y(), a.Z() + b.Z());
}
TVector3 operator - (const TVector3 & a, const TVector3 & b) {
return TVector3(a.X() - b.X(), a.Y() - b.Y(), a.Z() - b.Z());
}
TVector3 operator * (const TVector3 & p, Double_t a) {
return TVector3(a*p.X(), a*p.Y(), a*p.Z());
}
TVector3 operator * (Double_t a, const TVector3 & p) {
return TVector3(a*p.X(), a*p.Y(), a*p.Z());
}
Double_t operator * (const TVector3 & a, const TVector3 & b) {
return a.Dot(b);
}
TVector3 operator * (const TMatrix & m, const TVector3 & v ) {
return TVector3( m(0,0)*v.X()+m(0,1)*v.Y()+m(0,2)*v.Z(),
m(1,0)*v.X()+m(1,1)*v.Y()+m(1,2)*v.Z(),
m(2,0)*v.X()+m(2,1)*v.Y()+m(2,2)*v.Z());
}
void TVector3::Print(Option_t*)const
{
Printf("%s %s (x,y,z)=(%f,%f,%f) (rho,theta,phi)=(%f,%f,%f)",GetName(),GetTitle(),X(),Y(),Z(),
Mag(),Theta()*TMath::RadToDeg(),Phi()*TMath::RadToDeg());
}
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