mathcoreGenVector.C

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/// \file
/// \ingroup tutorial_math
/// \notebook -nodraw
/// Example macro testing available methods and operation of the GenVector classes.
/// The results are compared and check at the
/// numerical precision levels.
/// Some small discrepancy can appear when the macro
/// is executed on different architectures where it has been calibrated (Power PC G5)
/// The macro is divided in 4 parts:
///    - testVector3D          :  tests of the 3D Vector classes
///    - testPoint3D           :  tests of the 3D Point classes
///    - testLorentzVector     :  tests of the 4D LorentzVector classes
///    - testVectorUtil        :  tests of the utility functions of all the vector classes
///
/// To execute the macro type in:
///
/// ~~~{.cpp}
/// root[0] .x  mathcoreGenVector.C
/// ~~~
///
/// \macro_output
/// \macro_code
///
/// \author Lorenzo Moneta

#include "TMatrixD.h"
#include "TVectorD.h"
#include "TMath.h"

#include "Math/Point3D.h"
#include "Math/Vector3D.h"
#include "Math/Vector4D.h"
#include "Math/GenVector/Rotation3D.h"
#include "Math/GenVector/EulerAngles.h"
#include "Math/GenVector/AxisAngle.h"
#include "Math/GenVector/Quaternion.h"
#include "Math/GenVector/RotationX.h"
#include "Math/GenVector/RotationY.h"
#include "Math/GenVector/RotationZ.h"
#include "Math/GenVector/RotationZYX.h"
#include "Math/GenVector/LorentzRotation.h"
#include "Math/GenVector/Boost.h"
#include "Math/GenVector/BoostX.h"
#include "Math/GenVector/BoostY.h"
#include "Math/GenVector/BoostZ.h"
#include "Math/GenVector/Transform3D.h"
#include "Math/GenVector/Plane3D.h"
#include "Math/GenVector/VectorUtil.h"

using namespace ROOT::Math;

int ntest = 0;
int nfail = 0;
int ok = 0;

int compare( double v1, double v2, const char* name, double Scale = 1.0) {
   ntest = ntest + 1;

   // numerical double limit for epsilon
   double eps = Scale* 2.22044604925031308e-16;
   int iret = 0;
   double delta = v2 - v1;
   double d = 0;
   if (delta < 0 ) delta = - delta;
   if (v1 == 0 || v2 == 0) {
      if  (delta > eps ) {
         iret = 1;
      }
   }
   // skip case v1 or v2 is infinity
   else {
      d = v1;

      if ( v1 < 0) d = -d;
      // add also case when delta is small by default
      if ( delta/d  > eps && delta > eps )
      iret =  1;
   }

   if (iret == 0)
   std::cout << ".";
   else {
      int pr = std::cout.precision (18);
      int discr;
      if (d != 0)
         discr = int(delta/d/eps);
      else
         discr = int(delta/eps);

      std::cout << "\nDiscrepancy in " << name << "() : " << v1 << " != " << v2 << " discr = " << discr
              << "   (Allowed discrepancy is " << eps  << ")\n";
      std::cout.precision (pr);
      nfail = nfail + 1;
   }
   return iret;
}

int testVector3D() {
   std::cout << "\n************************************************************************\n "
             << " Vector 3D Test"
             << "\n************************************************************************\n";

   XYZVector v1(0.01, 0.02, 16);

   std::cout << "Test Cartesian-Polar :          " ;

   Polar3DVector v2(v1.R(), v1.Theta(), v1.Phi() );

   ok = 0;
   ok+= compare(v1.X(), v2.X(), "x");
   ok+= compare(v1.Y(), v2.Y(), "y");
   ok+= compare(v1.Z(), v2.Z(), "z");
   ok+= compare(v1.Phi(), v2.Phi(), "phi");
   ok+= compare(v1.Theta(), v2.Theta(), "theta");
   ok+= compare(v1.R(), v2.R(), "r");
   ok+= compare(v1.Eta(), v2.Eta(), "eta");
   ok+= compare(v1.Rho(), v2.Rho(), "rho");

   if (ok == 0) std::cout << "\t OK " << std::endl;

   std::cout << "Test Cartesian-CylindricalEta : ";

   RhoEtaPhiVector v3( v1.Rho(), v1.Eta(), v1.Phi() );

   ok = 0;
   ok+= compare(v1.X(), v3.X(), "x");
   ok+= compare(v1.Y(), v3.Y(), "y");
   ok+= compare(v1.Z(), v3.Z(), "z");
   ok+= compare(v1.Phi(), v3.Phi(), "phi");
   ok+= compare(v1.Theta(), v3.Theta(), "theta");
   ok+= compare(v1.R(), v3.R(), "r");
   ok+= compare(v1.Eta(), v3.Eta(), "eta");
   ok+= compare(v1.Rho(), v3.Rho(), "rho");

   if (ok == 0) std::cout << "\t OK " << std::endl;

   std::cout << "Test Cartesian-Cylindrical :    ";

   RhoZPhiVector v4( v1.Rho(), v1.Z(), v1.Phi() );

   ok = 0;
   ok+= compare(v1.X(), v4.X(), "x");
   ok+= compare(v1.Y(), v4.Y(), "y");
   ok+= compare(v1.Z(), v4.Z(), "z");
   ok+= compare(v1.Phi(), v4.Phi(), "phi");
   ok+= compare(v1.Theta(), v4.Theta(), "theta");
   ok+= compare(v1.R(), v4.R(), "r");
   ok+= compare(v1.Eta(), v4.Eta(), "eta");
   ok+= compare(v1.Rho(), v4.Rho(), "rho");

   if (ok == 0) std::cout << "\t OK " << std::endl;

   std::cout << "Test Operations :               " ;

   ok = 0;
   double Dot = v1.Dot(v2);
   ok+= compare( Dot, v1.Mag2(),"dot"  );
   XYZVector vcross = v1.Cross(v2);
   ok+= compare( vcross.R(), 0,"cross"  );

   XYZVector vscale1 = v1*10;
   XYZVector vscale2 = vscale1/10;
   ok+= compare( v1.R(), vscale2.R(), "scale");

   XYZVector vu = v1.Unit();
   ok+= compare(v2.Phi(),vu.Phi(),"unit Phi");
   ok+= compare(v2.Theta(),vu.Theta(),"unit Theta");
   ok+= compare(1.0,vu.R(),"unit ");

   XYZVector q1 = v1;
   RhoEtaPhiVector q2(1.0,1.0,1.0);

   XYZVector q3 = q1 + q2;
   XYZVector q4 = q3 - q2;

   ok+= compare( q4.X(), q1.X(), "op X"  );
   ok+= compare( q4.Y(), q1.Y(), "op Y" );
   ok+= compare( q4.Z(), q1.Z(), "op Z" );

   // test operator ==
   XYZVector        w1 = v1;
   Polar3DVector    w2 = v2;
   RhoEtaPhiVector  w3 = v3;
   RhoZPhiVector    w4 = v4;
   ok+= compare( w1 == v1, static_cast<double>(true), "== XYZ");
   ok+= compare( w2 == v2, static_cast<double>(true), "== Polar");
   ok+= compare( w3 == v3, static_cast<double>(true), "== RhoEtaPhi");
   ok+= compare( w4 == v4, static_cast<double>(true), "== RhoZPhi");

   if (ok == 0) std::cout << "\t OK " << std::endl;

   //test setters
   std::cout << "Test Setters :                  " ;

   q2.SetXYZ(q1.X(), q1.Y(), q1.Z() );

   ok+= compare( q2.X(), q1.X(), "setXYZ X"  );
   ok+= compare( q2.Y(), q1.Y(), "setXYZ Y" );
   ok+= compare( q2.Z(), q1.Z(), "setXYZ Z" );

   q2.SetCoordinates( 2.0*q1.Rho(), q1.Eta(), q1.Phi() );
   XYZVector q1s = 2.0*q1;
   ok+= compare( q2.X(), q1s.X(), "set X"  );
   ok+= compare( q2.Y(), q1s.Y(), "set Y" );
   ok+= compare( q2.Z(), q1s.Z(), "set Z" );

   if (ok == 0) std::cout << "\t\t OK " << std::endl;

   std::cout << "Test Linear Algebra conversion: " ;

   XYZVector vxyz1(1.,2.,3.);

   TVectorD vla1(3);
   vxyz1.Coordinates().GetCoordinates(vla1.GetMatrixArray() );

   TVectorD vla2(3);
   vla2[0] = 1.; vla2[1] = -2.; vla2[2] = 1.;

   XYZVector vxyz2;
   vxyz2.SetCoordinates(&vla2[0]);

   ok = 0;
   double prod1 =  vxyz1.Dot(vxyz2);
   double prod2 = vla1*vla2;
   ok+= compare( prod1, prod2, "la test" );

   if (ok == 0) std::cout << "\t\t OK " << std::endl;

   return ok;
}

int testPoint3D() {
   std::cout << "\n************************************************************************\n "
             << " Point 3D Tests"
             << "\n************************************************************************\n";

   XYZPoint p1(1.0, 2.0, 3.0);

   std::cout << "Test Cartesian-Polar :          ";

   Polar3DPoint p2(p1.R(), p1.Theta(), p1.Phi() );

   ok = 0;
   ok+= compare(p1.x(), p2.X(), "x");
   ok+= compare(p1.y(), p2.Y(), "y");
   ok+= compare(p1.z(), p2.Z(), "z");
   ok+= compare(p1.phi(), p2.Phi(), "phi");
   ok+= compare(p1.theta(), p2.Theta(), "theta");
   ok+= compare(p1.r(), p2.R(), "r");
   ok+= compare(p1.eta(), p2.Eta(), "eta");
   ok+= compare(p1.rho(), p2.Rho(), "rho");

   if (ok == 0) std::cout << "\t OK " << std::endl;

   std::cout << "Test Polar-CylindricalEta :     ";

   RhoEtaPhiPoint p3( p2.Rho(), p2.Eta(), p2.Phi() );

   ok = 0;
   ok+= compare(p2.X(), p3.X(), "x");
   ok+= compare(p2.Y(), p3.Y(), "y");
   ok+= compare(p2.Z(), p3.Z(), "z",3);
   ok+= compare(p2.Phi(), p3.Phi(), "phi");
   ok+= compare(p2.Theta(), p3.Theta(), "theta");
   ok+= compare(p2.R(), p3.R(), "r");
   ok+= compare(p2.Eta(), p3.Eta(), "eta");
   ok+= compare(p2.Rho(), p3.Rho(), "rho");

   if (ok == 0) std::cout << "\t OK " << std::endl;

   std::cout << "Test operations :               ";

   //std::cout << "\nTest Dot and Cross products with Vectors : ";
   Polar3DVector vperp(1.,p1.Theta() + TMath::PiOver2(),p1.Phi() );
   double Dot = p1.Dot(vperp);
   ok+= compare( Dot, 0.0,"dot", 10  );

   XYZPoint vcross = p1.Cross(vperp);
   ok+= compare( vcross.R(), p1.R(),"cross mag"  );
   ok+= compare( vcross.Dot(vperp), 0.0,"cross dir"  );

   XYZPoint pscale1 = 10*p1;
   XYZPoint pscale2 = pscale1/10;
   ok+= compare( p1.R(), pscale2.R(), "scale");

   // test operator ==
   ok+= compare( p1 == pscale2, static_cast<double>(true), "== Point");

   //RhoEtaPhiPoint q1 = p1;  ! constructor yet not working in CINT
   RhoEtaPhiPoint q1; q1 = p1;
   q1.SetCoordinates(p1.Rho(),2.0, p1.Phi() );

   Polar3DVector v2(p1.R(), p1.Theta(),p1.Phi());

   if (ok == 0) std::cout << "\t OK " << std::endl;

   return ok;
}

int testLorentzVector() {
   std::cout << "\n************************************************************************\n "
             << " Lorentz Vector Tests"
             << "\n************************************************************************\n";

   XYZTVector v1(1.0, 2.0, 3.0, 4.0);

   std::cout << "Test XYZT - PtEtaPhiE Vectors:  ";

   PtEtaPhiEVector v2( v1.Rho(), v1.Eta(), v1.Phi(), v1.E() );

   ok = 0;
   ok+= compare(v1.Px(), v2.X(), "x");
   ok+= compare(v1.Py(), v2.Y(), "y");
   ok+= compare(v1.Pz(), v2.Z(), "z", 2);
   ok+= compare(v1.E(), v2.T(), "e");
   ok+= compare(v1.Phi(), v2.Phi(), "phi");
   ok+= compare(v1.Theta(), v2.Theta(), "theta");
   ok+= compare(v1.Pt(), v2.Pt(), "pt");
   ok+= compare(v1.M(), v2.M(), "mass", 5);
   ok+= compare(v1.Et(), v2.Et(), "et");
   ok+= compare(v1.Mt(), v2.Mt(), "mt", 3);

   if (ok == 0) std::cout << "\t OK " << std::endl;

   std::cout << "Test XYZT - PtEtaPhiM Vectors:  ";

   PtEtaPhiMVector v3( v1.Rho(), v1.Eta(), v1.Phi(), v1.M() );

   ok = 0;
   ok+= compare(v1.Px(), v3.X(), "x");
   ok+= compare(v1.Py(), v3.Y(), "y");
   ok+= compare(v1.Pz(), v3.Z(), "z", 2);
   ok+= compare(v1.E(), v3.T(), "e");
   ok+= compare(v1.Phi(), v3.Phi(), "phi");
   ok+= compare(v1.Theta(), v3.Theta(), "theta");
   ok+= compare(v1.Pt(), v3.Pt(), "pt");
   ok+= compare(v1.M(), v3.M(), "mass", 5);
   ok+= compare(v1.Et(), v3.Et(), "et");
   ok+= compare(v1.Mt(), v3.Mt(), "mt", 3);

   if (ok == 0) std::cout << "\t OK " << std::endl;

   std::cout << "Test PtEtaPhiE - PxPyPzM Vect.: ";

   PxPyPzMVector v4( v3.X(), v3.Y(), v3.Z(), v3.M() );

   ok = 0;
   ok+= compare(v4.Px(), v3.X(), "x");
   ok+= compare(v4.Py(), v3.Y(), "y");
   ok+= compare(v4.Pz(), v3.Z(), "z",2);
   ok+= compare(v4.E(), v3.T(), "e");
   ok+= compare(v4.Phi(), v3.Phi(), "phi");
   ok+= compare(v4.Theta(), v3.Theta(), "theta");
   ok+= compare(v4.Pt(), v3.Pt(), "pt");
   ok+= compare(v4.M(), v3.M(), "mass",5);
   ok+= compare(v4.Et(), v3.Et(), "et");
   ok+= compare(v4.Mt(), v3.Mt(), "mt",3);

   if (ok == 0) std::cout << "\t OK " << std::endl;

   std::cout << "Test operations :               ";

   ok = 0;
   double Dot = v1.Dot(v2);
   ok+= compare( Dot, v1.M2(),"dot" , 10 );

   XYZTVector vscale1 = v1*10;
   XYZTVector vscale2 = vscale1/10;
   ok+= compare( v1.M(), vscale2.M(), "scale");

   XYZTVector q1 = v1;
   PtEtaPhiEVector  q2(1.0,1.0,1.0,5.0);

   XYZTVector q3 = q1 + q2;
   XYZTVector q4 = q3 - q2;

   ok+= compare( q4.x(), q1.X(), "op X"  );
   ok+= compare( q4.y(), q1.Y(), "op Y" );
   ok+= compare( q4.z(), q1.Z(), "op Z" );
   ok+= compare( q4.t(), q1.E(), "op E" );

   // test operator ==
   XYZTVector        w1 = v1;
   PtEtaPhiEVector   w2 = v2;
   PtEtaPhiMVector   w3 = v3;
   PxPyPzMVector     w4 = v4;
   ok+= compare( w1 == v1, static_cast<double>(true), "== PxPyPzE");
   ok+= compare( w2 == v2, static_cast<double>(true), "== PtEtaPhiE");
   ok+= compare( w3 == v3, static_cast<double>(true), "== PtEtaPhiM");
   ok+= compare( w4 == v4, static_cast<double>(true), "== PxPyPzM");

   // test gamma beta and boost
   XYZVector b = q1.BoostToCM();
   double beta = q1.Beta();
   double gamma = q1.Gamma();

   ok += compare( b.R(), beta, "beta" );
   ok += compare( gamma, 1./sqrt( 1 - beta*beta ), "gamma");

   if (ok == 0) std::cout << "\t OK " << std::endl;

   //test setters
   std::cout << "Test Setters :                  " ;

   q2.SetXYZT(q1.Px(), q1.Py(), q1.Pz(), q1.E() );

   ok+= compare( q2.X(), q1.X(), "setXYZT X"  );
   ok+= compare( q2.Y(), q1.Y(), "setXYZT Y" );
   ok+= compare( q2.Z(), q1.Z(), "setXYZT Z" ,2);
   ok+= compare( q2.T(), q1.E(), "setXYZT E" );

   q2.SetCoordinates( 2.0*q1.Rho(), q1.Eta(), q1.Phi(), 2.0*q1.E() );
   XYZTVector q1s = q1*2.0;
   ok+= compare( q2.X(), q1s.X(), "set X"  );
   ok+= compare( q2.Y(), q1s.Y(), "set Y" );
   ok+= compare( q2.Z(), q1s.Z(), "set Z" ,2);
   ok+= compare( q2.T(), q1s.T(),  "set E" );

   if (ok == 0) std::cout << "\t OK " << std::endl;

   return ok;
}

int testVectorUtil() {
   std::cout << "\n************************************************************************\n "
             << " Utility Function Tests"
             << "\n************************************************************************\n";

   std::cout << "Test Vector utility functions : ";

   XYZVector v1(1.0, 2.0, 3.0);
   Polar3DVector v2pol(v1.R(), v1.Theta()+TMath::PiOver2(), v1.Phi() + 1.0);
   // mixedmethods not yet impl.
   XYZVector v2; v2 = v2pol;

   ok = 0;
   ok += compare( VectorUtil::DeltaPhi(v1,v2), 1.0, "deltaPhi Vec");

   RhoEtaPhiVector v2cyl(v1.Rho(), v1.Eta() + 1.0, v1.Phi() + 1.0);
   v2 = v2cyl;

   ok += compare( VectorUtil::DeltaR(v1,v2), sqrt(2.0), "DeltaR Vec");

   XYZVector vperp = v1.Cross(v2);
   ok += compare( VectorUtil::CosTheta(v1,vperp), 0.0, "costheta Vec");
   ok += compare( VectorUtil::Angle(v1,vperp), TMath::PiOver2(), "angle Vec");

   if (ok == 0) std::cout << "\t\t OK " << std::endl;


   std::cout << "Test Point utility functions :  ";

   XYZPoint p1(1.0, 2.0, 3.0);
   Polar3DPoint p2pol(p1.R(), p1.Theta()+TMath::PiOver2(), p1.Phi() + 1.0);
   // mixedmethods not yet impl.
   XYZPoint p2; p2 = p2pol;

   ok = 0;
   ok += compare( VectorUtil::DeltaPhi(p1,p2), 1.0, "deltaPhi Point");

   RhoEtaPhiPoint p2cyl(p1.Rho(), p1.Eta() + 1.0, p1.Phi() + 1.0);
   p2 = p2cyl;
   ok += compare( VectorUtil::DeltaR(p1,p2), sqrt(2.0), "DeltaR Point");

   XYZPoint pperp(vperp.X(), vperp.Y(), vperp.Z());
   ok += compare( VectorUtil::CosTheta(p1,pperp), 0.0, "costheta Point");
   ok += compare( VectorUtil::Angle(p1,pperp), TMath::PiOver2(), "angle Point");

   if (ok == 0) std::cout << "\t\t OK " << std::endl;

   std::cout << "LorentzVector utility funct.:   ";

   XYZTVector q1(1.0, 2.0, 3.0,4.0);
   PtEtaPhiEVector q2cyl(q1.Pt(), q1.Eta()+1.0, q1.Phi() + 1.0, q1.E() );
   XYZTVector q2; q2 = q2cyl;

   ok = 0;
   ok += compare( VectorUtil::DeltaPhi(q1,q2), 1.0, "deltaPhi LVec");
   ok += compare( VectorUtil::DeltaR(q1,q2), sqrt(2.0), "DeltaR LVec");

   XYZTVector qsum = q1+q2;
   ok += compare( VectorUtil::InvariantMass(q1,q2), qsum.M(), "InvMass");

   if (ok == 0) std::cout << "\t\t OK " << std::endl;

   return ok;
}

int testRotation() {
   std::cout << "\n************************************************************************\n "
             << " Rotation and Transformation Tests"
             << "\n************************************************************************\n";

   std::cout << "Test Vector Rotations :         ";
   ok = 0;

   XYZPoint v(1.,2,3.);

   double pi = TMath::Pi();
   // initiate rotation with some non -trivial angles to test all matrix
   EulerAngles r1( pi/2.,pi/4., pi/3 );
   Rotation3D  r2(r1);
   // only operator= is in CINT for the other rotations
   Quaternion  r3; r3 = r2;
   AxisAngle   r4; r4 = r3;
   RotationZYX r5; r5 = r2;

   XYZPoint v1 = r1 * v;
   XYZPoint v2 = r2 * v;
   XYZPoint v3 = r3 * v;
   XYZPoint v4 = r4 * v;
   XYZPoint v5 = r5 * v;

   ok+= compare(v1.X(), v2.X(), "x",2);
   ok+= compare(v1.Y(), v2.Y(), "y",2);
   ok+= compare(v1.Z(), v2.Z(), "z",2);

   ok+= compare(v1.X(), v3.X(), "x",2);
   ok+= compare(v1.Y(), v3.Y(), "y",2);
   ok+= compare(v1.Z(), v3.Z(), "z",2);

   ok+= compare(v1.X(), v4.X(), "x",5);
   ok+= compare(v1.Y(), v4.Y(), "y",5);
   ok+= compare(v1.Z(), v4.Z(), "z",5);

   ok+= compare(v1.X(), v5.X(), "x",2);
   ok+= compare(v1.Y(), v5.Y(), "y",2);
   ok+= compare(v1.Z(), v5.Z(), "z",2);

   // test with matrix
   double rdata[9];
   r2.GetComponents(rdata, rdata+9);
   TMatrixD m(3,3,rdata);
   double vdata[3];
   v.GetCoordinates(vdata);
   TVectorD q(3,vdata);
   TVectorD q2 = m*q;

   XYZPoint v6;
   v6.SetCoordinates( q2.GetMatrixArray() );

   ok+= compare(v1.X(), v6.X(), "x");
   ok+= compare(v1.Y(), v6.Y(), "y");
   ok+= compare(v1.Z(), v6.Z(), "z");

   if (ok == 0) std::cout << "\t OK " << std::endl;
   else  std::cout << std::endl;

   std::cout << "Test Axial Rotations :          ";
   ok = 0;

   RotationX rx( pi/3);
   RotationY ry( pi/4);
   RotationZ rz( 4*pi/5);

   Rotation3D r3x(rx);
   Rotation3D r3y(ry);
   Rotation3D r3z(rz);

   Quaternion qx; qx = rx;
   Quaternion qy; qy = ry;
   Quaternion qz; qz = rz;

   RotationZYX rzyx( rz.Angle(), ry.Angle(), rx.Angle() );

   XYZPoint vrot1 = rx * ry * rz * v;
   XYZPoint vrot2 = r3x * r3y * r3z * v;

   ok+= compare(vrot1.X(), vrot2.X(), "x");
   ok+= compare(vrot1.Y(), vrot2.Y(), "y");
   ok+= compare(vrot1.Z(), vrot2.Z(), "z");

   vrot2 = qx * qy * qz * v;

   ok+= compare(vrot1.X(), vrot2.X(), "x",2);
   ok+= compare(vrot1.Y(), vrot2.Y(), "y",2);
   ok+= compare(vrot1.Z(), vrot2.Z(), "z",2);

   vrot2 = rzyx * v;

   ok+= compare(vrot1.X(), vrot2.X(), "x");
   ok+= compare(vrot1.Y(), vrot2.Y(), "y");
   ok+= compare(vrot1.Z(), vrot2.Z(), "z");

   // now inverse (first x then y then z)
   vrot1 = rz * ry * rx * v;
   vrot2 = r3z * r3y * r3x * v;

   ok+= compare(vrot1.X(), vrot2.X(), "x");
   ok+= compare(vrot1.Y(), vrot2.Y(), "y");
   ok+= compare(vrot1.Z(), vrot2.Z(), "z");

   XYZPoint vinv1 = rx.Inverse()*ry.Inverse()*rz.Inverse()*vrot1;

   ok+= compare(vinv1.X(), v.X(), "x",2);
   ok+= compare(vinv1.Y(), v.Y(), "y");
   ok+= compare(vinv1.Z(), v.Z(), "z");

   if (ok == 0) std::cout << "\t OK " << std::endl;
   else  std::cout << std::endl;


   std::cout << "Test Rotations by a PI angle :  ";
   ok = 0;

   double b[4] = { 6,8,10,3.14159265358979323 };
   AxisAngle  arPi(b,b+4 );
   Rotation3D rPi(arPi);
   AxisAngle  a1; a1 = rPi;
   ok+= compare(arPi.Axis().X(), a1.Axis().X(),"x");
   ok+= compare(arPi.Axis().Y(), a1.Axis().Y(),"y");
   ok+= compare(arPi.Axis().Z(), a1.Axis().Z(),"z");
   ok+= compare(arPi.Angle(), a1.Angle(),"angle");

   EulerAngles ePi; ePi=rPi;
   EulerAngles e1; e1=Rotation3D(a1);
   ok+= compare(ePi.Phi(), e1.Phi(),"phi");
   ok+= compare(ePi.Theta(), e1.Theta(),"theta");
   ok+= compare(ePi.Psi(), e1.Psi(),"ps1");

   if (ok == 0) std::cout << "\t\t OK " << std::endl;
   else  std::cout << std::endl;

   std::cout << "Test Inversions :               ";
   ok = 0;

   EulerAngles s1 = r1.Inverse();
   Rotation3D  s2 = r2.Inverse();
   Quaternion  s3 = r3.Inverse();
   AxisAngle   s4 = r4.Inverse();
   RotationZYX s5 = r5.Inverse();

   // Euler angles not yet impl.
   XYZPoint p = s2 * r2 * v;

   ok+= compare(p.X(), v.X(), "x",10);
   ok+= compare(p.Y(), v.Y(), "y",10);
   ok+= compare(p.Z(), v.Z(), "z",10);


   p = s3 * r3 * v;

   ok+= compare(p.X(), v.X(), "x",10);
   ok+= compare(p.Y(), v.Y(), "y",10);
   ok+= compare(p.Z(), v.Z(), "z",10);

   p = s4 * r4 * v;
   // axis angle inversion not very precise
   ok+= compare(p.X(), v.X(), "x",1E9);
   ok+= compare(p.Y(), v.Y(), "y",1E9);
   ok+= compare(p.Z(), v.Z(), "z",1E9);

   p = s5 * r5 * v;

   ok+= compare(p.X(), v.X(), "x",10);
   ok+= compare(p.Y(), v.Y(), "y",10);
   ok+= compare(p.Z(), v.Z(), "z",10);


   Rotation3D r6(r5);
   Rotation3D s6 = r6.Inverse();

   p = s6 * r6 * v;

   ok+= compare(p.X(), v.X(), "x",10);
   ok+= compare(p.Y(), v.Y(), "y",10);
   ok+= compare(p.Z(), v.Z(), "z",10);

   if (ok == 0) std::cout << "\t OK " << std::endl;
   else  std::cout << std::endl;

   // test Rectify
   std::cout << "Test rectify :                  ";
   ok = 0;

   XYZVector u1(0.999498,-0.00118212,-0.0316611);
   XYZVector u2(0,0.999304,-0.0373108);
   XYZVector u3(0.0316832,0.0372921,0.998802);
   Rotation3D rr(u1,u2,u3);
   // check orto-normality
   XYZPoint vrr = rr* v;
   ok+= compare(v.R(), vrr.R(), "R",1.E9);

   if (ok == 0) std::cout << "\t\t OK " << std::endl;
   else  std::cout << std::endl;

   std::cout << "Test Transform3D :              ";
   ok = 0;

   XYZVector d(1.,-2.,3.);
   Transform3D t(r2,d);

   XYZPoint pd = t * v;
   // apply directly rotation
   XYZPoint vd = r2 * v + d;

   ok+= compare(pd.X(), vd.X(), "x");
   ok+= compare(pd.Y(), vd.Y(), "y");
   ok+= compare(pd.Z(), vd.Z(), "z");

   // test with matrix
   double tdata[12];
   t.GetComponents(tdata);
   TMatrixD mt(3,4,tdata);
   double vData[4]; // needs a vector of dim 4
   v.GetCoordinates(vData);
   vData[3] = 1;
   TVectorD q0(4,vData);

   TVectorD qt = mt*q0;

   ok+= compare(pd.X(), qt(0), "x");
   ok+= compare(pd.Y(), qt(1), "y");
   ok+= compare(pd.Z(), qt(2), "z");

   // test inverse

   Transform3D tinv = t.Inverse();

   p = tinv * t * v;

   ok+= compare(p.X(), v.X(), "x",10);
   ok+= compare(p.Y(), v.Y(), "y",10);
   ok+= compare(p.Z(), v.Z(), "z",10);

   // test construct inverse from translation first

   Transform3D tinv2 ( r2.Inverse(), r2.Inverse() *( -d) ) ;
   p = tinv2 * t * v;

   ok+= compare(p.X(), v.X(), "x",10);
   ok+= compare(p.Y(), v.Y(), "y",10);
   ok+= compare(p.Z(), v.Z(), "z",10);

   // test from only rotation and only translation
   Transform3D ta( EulerAngles(1.,2.,3.) );
   Transform3D tb( XYZVector(1,2,3) );
   Transform3D tc(  Rotation3D(EulerAngles(1.,2.,3.)) ,  XYZVector(1,2,3) );
   Transform3D td(  ta.Rotation(), ta.Rotation()  * XYZVector(1,2,3) ) ;

   ok+= compare( tc == tb*ta, static_cast<double>(true), "== Rot*Tra");
   ok+= compare( td == ta*tb, static_cast<double>(true), "== Rot*Tra");


   if (ok == 0) std::cout << "\t OK " << std::endl;
   else  std::cout << std::endl;

   std::cout << "Test Plane3D :                  ";
   ok = 0;

   // test transfrom a 3D plane

   XYZPoint p1(1,2,3);
   XYZPoint p2(-2,-1,4);
   XYZPoint p3(-1,3,2);
   Plane3D plane(p1,p2,p3);

   XYZVector n = plane.Normal();
   // normal is perpendicular to vectors on the planes obtained from subtracting the points
   ok+= compare(n.Dot(p2-p1), 0.0, "n.v12",10);
   ok+= compare(n.Dot(p3-p1), 0.0, "n.v13",10);
   ok+= compare(n.Dot(p3-p2), 0.0, "n.v23",10);

   Plane3D plane1 = t(plane);

   // transform the points
   XYZPoint pt1 = t(p1);
   XYZPoint pt2 = t(p2);
   XYZPoint pt3 = t(p3);
   Plane3D plane2(pt1,pt2,pt3);

   XYZVector n1 = plane1.Normal();
   XYZVector n2 = plane2.Normal();

   ok+= compare(n1.X(), n2.X(), "a",10);
   ok+= compare(n1.Y(), n2.Y(), "b",10);
   ok+= compare(n1.Z(), n2.Z(), "c",10);
   ok+= compare(plane1.HesseDistance(), plane2.HesseDistance(), "d",10);

   // check distances
   ok += compare(plane1.Distance(pt1), 0.0, "distance",10);

   if (ok == 0) std::cout << "\t OK " << std::endl;
   else  std::cout << std::endl;

   std::cout << "Test LorentzRotation :          ";
   ok = 0;

   XYZTVector lv(1.,2.,3.,4.);

   // test from rotx (using boosts and 3D rotations not yet impl.)
   // rx,ry and rz already defined
   Rotation3D r3d = rx*ry*rz;

   LorentzRotation rlx(rx);
   LorentzRotation rly(ry);
   LorentzRotation rlz(rz);

   LorentzRotation rl0 = rlx*rly*rlz;
   LorentzRotation rl1( r3d);

   XYZTVector lv0 = rl0 * lv;

   XYZTVector lv1 = rl1 * lv;

   XYZTVector lv2 = r3d * lv;

   ok+= compare(lv1== lv2,true,"V0==V2");
   ok+= compare(lv1== lv2,true,"V1==V2");

   double rlData[16];
   rl0.GetComponents(rlData);
   TMatrixD ml(4,4,rlData);
   double lvData[4];
   lv.GetCoordinates(lvData);
   TVectorD ql(4,lvData);

   TVectorD qlr = ml*ql;

   ok+= compare(lv1.X(), qlr(0), "x");
   ok+= compare(lv1.Y(), qlr(1), "y");
   ok+= compare(lv1.Z(), qlr(2), "z");
   ok+= compare(lv1.E(), qlr(3), "t");

   // test inverse
   lv0 = rl0 * rl0.Inverse() * lv;

   ok+= compare(lv0.X(), lv.X(), "x");
   ok+= compare(lv0.Y(), lv.Y(), "y");
   ok+= compare(lv0.Z(), lv.Z(), "z");
   ok+= compare(lv0.E(), lv.E(), "t");

   if (ok == 0) std::cout << "\t OK " << std::endl;
   else  std::cout << std::endl;

   // test Boosts
   std::cout << "Test Boost :                    ";
   ok = 0;

   Boost bst( 0.3,0.4,0.5);   //  boost (must be <= 1)

   XYZTVector lvb = bst ( lv );

   LorentzRotation rl2 (bst);

   XYZTVector lvb2 = rl2 (lv);

   // test with lorentz rotation
   ok+= compare(lvb.X(), lvb2.X(), "x");
   ok+= compare(lvb.Y(), lvb2.Y(), "y");
   ok+= compare(lvb.Z(), lvb2.Z(), "z");
   ok+= compare(lvb.E(), lvb2.E(), "t");
   ok+= compare(lvb.M(), lv.M(), "m",50); // m must stay constant

   // test inverse
   lv0 = bst.Inverse() * lvb;

   ok+= compare(lv0.X(), lv.X(), "x",5);
   ok+= compare(lv0.Y(), lv.Y(), "y",5);
   ok+= compare(lv0.Z(), lv.Z(), "z",3);
   ok+= compare(lv0.E(), lv.E(), "t",3);

   XYZVector brest = lv.BoostToCM();
   bst.SetComponents( brest.X(), brest.Y(), brest.Z() );

   XYZTVector lvr = bst * lv;

   ok+= compare(lvr.X(), 0.0, "x",10);
   ok+= compare(lvr.Y(), 0.0, "y",10);
   ok+= compare(lvr.Z(), 0.0, "z",10);
   ok+= compare(lvr.M(), lv.M(), "m",10);

   if (ok == 0) std::cout << "\t OK " << std::endl;
   else  std::cout << std::endl;
      return ok;
}

void mathcoreGenVector() {


   testVector3D();
   testPoint3D();
   testLorentzVector();
   testVectorUtil();
   testRotation();

   std::cout << "\n\nNumber of tests " << ntest << " failed = " << nfail << std::endl;
}