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| 29 | |
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| 30 | <h2>Four-Vectors</h2> |
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| 31 | |
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| 32 | The <code>Vec4</code> class gives a simple implementation of four-vectors. |
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| 33 | The member function names are based on the assumption that these |
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| 34 | represent four-momentum vectors. Thus one can get or set |
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| 35 | <i>p_x, p_y, p_z</i> and <i>e</i>, but not <i>x, y, z</i> |
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| 36 | or <i>t</i>. This is only a matter of naming, however; a |
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| 37 | <code>Vec4</code> can equally well be used to store a space-time |
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| 38 | four-vector. |
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| 39 | |
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| 40 | <p/> |
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| 41 | The <code>Particle</code> object contains a <code>Vec4 p</code> that |
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| 42 | stores the particle four-momentum, and another <code>Vec4 vProd</code> |
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| 43 | for the production vertex. For the latter the input/output method |
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| 44 | names are adapted to the space-time character rather than the normal |
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| 45 | energy-momentum one. Thus a user would not normally access the |
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| 46 | <code>Vec4</code> classes directly, but only via the methods of the |
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| 47 | <code>Particle</code> class, |
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| 48 | see <?php $filepath = $_GET["filepath"]; |
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| 49 | echo "<a href='ParticleProperties.php?filepath=".$filepath."' target='page'>";?>Particle Properties</a>. |
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| 50 | |
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| 51 | <p/> |
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| 52 | Nevertheless you are free to use the PYTHIA four-vectors, e.g. as |
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| 53 | part of some simple analysis code based directly on the PYTHIA output, |
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| 54 | say to define the four-vector sum of a set of particles. But note that |
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| 55 | this class was never set up to allow complete generality, only to |
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| 56 | provide the operations that are of use inside PYTHIA. There is no |
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| 57 | separate class for three-vectors, since such can easily be represented |
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| 58 | by four-vectors where the fourth component is not used. |
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| 59 | |
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| 60 | <p/> |
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| 61 | Four-vectors have the expected functionality: they can be created, |
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| 62 | copied, added, multiplied, rotated, boosted, and manipulated in other |
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| 63 | ways. Operator overloading is implemented where reasonable. Properties |
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| 64 | can be read out, not only the components themselves but also for derived |
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| 65 | quantities such as absolute momentum and direction angles. |
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| 66 | |
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| 67 | <h3>Constructors and basic operators</h3> |
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| 68 | |
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| 69 | A few methods are available to create or copy a four-vector: |
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| 70 | |
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| 71 | <a name="method1"></a> |
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| 72 | <p/><strong>Vec4::Vec4() </strong> <br/> |
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| 73 | creates a four-vector with all components set to 0. |
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| 74 | |
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| 75 | |
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| 76 | <a name="method2"></a> |
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| 77 | <p/><strong>Vec4::Vec4(const Vec4& v) </strong> <br/> |
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| 78 | creates a four-vector copy of the input four-vector. |
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| 79 | |
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| 80 | |
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| 81 | <a name="method3"></a> |
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| 82 | <p/><strong>Vec4& Vec4::operator=(const Vec4& v) </strong> <br/> |
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| 83 | copies the input four-vector. |
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| 84 | |
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| 85 | |
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| 86 | <a name="method4"></a> |
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| 87 | <p/><strong>Vec4& Vec4::operator=(double value) </strong> <br/> |
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| 88 | gives a four-vector with all components set to <i>value</i>. |
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| 89 | |
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| 90 | |
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| 91 | <h3>Member methods for input</h3> |
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| 92 | |
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| 93 | The values stored in a four-vector can be modified in a few different |
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| 94 | ways: |
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| 95 | |
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| 96 | <a name="method5"></a> |
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| 97 | <p/><strong>void Vec4::reset() </strong> <br/> |
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| 98 | sets all components to 0. |
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| 99 | |
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| 100 | |
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| 101 | <a name="method6"></a> |
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| 102 | <p/><strong>void Vec4::p(double pxIn, double pyIn, double pzIn, double eIn) </strong> <br/> |
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| 103 | sets all components to their input values. |
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| 104 | |
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| 105 | |
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| 106 | <a name="method7"></a> |
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| 107 | <p/><strong>void Vec4::p(Vec4 pIn) </strong> <br/> |
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| 108 | sets all components equal to those of the input four-vector. |
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| 109 | |
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| 110 | |
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| 111 | <a name="method8"></a> |
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| 112 | <p/><strong>void Vec4::px(double pxIn) </strong> <br/> |
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| 113 | |
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| 114 | <strong>void Vec4::py(double pyIn) </strong> <br/> |
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| 115 | |
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| 116 | <strong>void Vec4::pz(double pzIn) </strong> <br/> |
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| 117 | |
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| 118 | <strong>void Vec4::e(double eIn) </strong> <br/> |
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| 119 | sets the respective component to the input value. |
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| 120 | |
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| 121 | |
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| 122 | <h3>Member methods for output</h3> |
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| 123 | |
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| 124 | A number of methods provides output of basic or derived quantities: |
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| 125 | |
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| 126 | <a name="method9"></a> |
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| 127 | <p/><strong>double Vec4::px() </strong> <br/> |
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| 128 | |
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| 129 | <strong>double Vec4::py() </strong> <br/> |
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| 130 | |
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| 131 | <strong>double Vec4::pz() </strong> <br/> |
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| 132 | |
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| 133 | <strong>double Vec4::e() </strong> <br/> |
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| 134 | gets the respective component. |
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| 135 | |
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| 136 | |
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| 137 | <a name="method10"></a> |
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| 138 | <p/><strong>double Vec4::mCalc() </strong> <br/> |
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| 139 | |
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| 140 | <strong>double Vec4::m2Calc() </strong> <br/> |
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| 141 | the (squared) mass, calculated from the four-vectors. |
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| 142 | If <i>m^2 < 0</i> the mass is given with a minus sign, |
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| 143 | <i>-sqrt(-m^2)</i>. Note the possible loss of precision |
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| 144 | in the calculation of <i>E^2 - p^2</i>; for particles the |
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| 145 | correct mass is stored separately to avoid such problems. |
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| 146 | |
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| 147 | |
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| 148 | <a name="method11"></a> |
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| 149 | <p/><strong>double Vec4::pT() </strong> <br/> |
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| 150 | |
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| 151 | <strong>double Vec4::pT2() </strong> <br/> |
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| 152 | the (squared) transverse momentum. |
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| 153 | |
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| 154 | |
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| 155 | <a name="method12"></a> |
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| 156 | <p/><strong>double Vec4::pAbs() </strong> <br/> |
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| 157 | |
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| 158 | <strong>double Vec4::pAbs2() </strong> <br/> |
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| 159 | the (squared) absolute momentum. |
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| 160 | |
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| 161 | |
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| 162 | <a name="method13"></a> |
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| 163 | <p/><strong>double Vec4::eT() </strong> <br/> |
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| 164 | |
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| 165 | <strong>double Vec4::eT2() </strong> <br/> |
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| 166 | the (squared) transverse energy, |
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| 167 | <i>eT = e * sin(theta) = e * pT / pAbs</i>. |
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| 168 | |
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| 169 | |
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| 170 | <a name="method14"></a> |
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| 171 | <p/><strong>double Vec4::theta() </strong> <br/> |
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| 172 | the polar angle, in the range 0 through |
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| 173 | <i>pi</i>. |
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| 174 | |
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| 175 | |
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| 176 | <a name="method15"></a> |
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| 177 | <p/><strong>double Vec4::phi() </strong> <br/> |
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| 178 | the azimuthal angle, in the range <i>-pi</i> through <i>pi</i>. |
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| 179 | |
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| 180 | |
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| 181 | <a name="method16"></a> |
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| 182 | <p/><strong>double Vec4::thetaXZ() </strong> <br/> |
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| 183 | the angle in the <i>xz</i> plane, in the range <i>-pi</i> through |
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| 184 | <i>pi</i>, with 0 along the <i>+z</i> axis. |
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| 185 | |
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| 186 | |
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| 187 | <a name="method17"></a> |
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| 188 | <p/><strong>double Vec4::pPos() </strong> <br/> |
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| 189 | |
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| 190 | <strong>double Vec4::pNeg() </strong> <br/> |
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| 191 | the combinations <i>E+-p_z</i>. |
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| 192 | |
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| 193 | <h3>Friend methods for output</h3> |
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| 194 | |
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| 195 | There are also some <code>friend</code> methods that take one, two |
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| 196 | or three four-vectors as argument. Several of them only use the |
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| 197 | three-vector part of the four-vector. |
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| 198 | |
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| 199 | <a name="method18"></a> |
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| 200 | <p/><strong>friend ostream& operator<<(ostream&, const Vec4& v) </strong> <br/> |
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| 201 | writes out the values of the four components of a <code>Vec4</code> and, |
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| 202 | within brackets, a fifth component being the invariant length of the |
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| 203 | four-vector, as provided by <code>mCalc()</code> above, and it all |
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| 204 | ended with a newline. |
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| 205 | |
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| 206 | |
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| 207 | <a name="method19"></a> |
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| 208 | <p/><strong>friend double m(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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| 209 | |
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| 210 | <strong>friend double m2(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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| 211 | the (squared) invariant mass. |
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| 212 | |
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| 213 | |
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| 214 | <a name="method20"></a> |
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| 215 | <p/><strong>friend double dot3(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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| 216 | the three-product. |
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| 217 | |
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| 218 | |
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| 219 | <a name="method21"></a> |
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| 220 | <p/><strong>friend double cross3(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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| 221 | the cross-product. |
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| 222 | |
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| 223 | |
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| 224 | <a name="method22"></a> |
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| 225 | <p/><strong>friend double theta(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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| 226 | |
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| 227 | <strong>friend double costheta(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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| 228 | the (cosine) of the opening angle between the vectors, |
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| 229 | in the range 0 through <i>pi</i>. |
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| 230 | |
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| 231 | |
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| 232 | <a name="method23"></a> |
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| 233 | <p/><strong>friend double phi(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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| 234 | |
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| 235 | <strong>friend double cosphi(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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| 236 | the (cosine) of the azimuthal angle between the vectors around the |
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| 237 | <i>z</i> axis, in the range 0 through <i>pi</i>. |
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| 238 | |
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| 239 | |
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| 240 | <a name="method24"></a> |
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| 241 | <p/><strong>friend double phi(const Vec4& v1, const Vec4& v2, const Vec4& v3) </strong> <br/> |
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| 242 | |
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| 243 | <strong>friend double cosphi(const Vec4& v1, const Vec4& v2, const Vec4& v3) </strong> <br/> |
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| 244 | the (cosine) of the azimuthal angle between the first two vectors |
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| 245 | around the direction of the third, in the range 0 through <i>pi</i>. |
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| 246 | |
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| 247 | |
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| 248 | <h3>Operations with four-vectors</h3> |
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| 249 | |
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| 250 | Of course one should be able to add, subtract and scale four-vectors, |
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| 251 | and more: |
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| 252 | |
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| 253 | <a name="method25"></a> |
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| 254 | <p/><strong>Vec4 Vec4::operator-() </strong> <br/> |
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| 255 | return a vector with flipped sign for all components, while leaving |
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| 256 | the original vector unchanged. |
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| 257 | |
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| 258 | |
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| 259 | <a name="method26"></a> |
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| 260 | <p/><strong>Vec4& Vec4::operator+=(const Vec4& v) </strong> <br/> |
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| 261 | add a four-vector to an existing one. |
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| 262 | |
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| 263 | |
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| 264 | <a name="method27"></a> |
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| 265 | <p/><strong>Vec4& Vec4::operator-=(const Vec4& v) </strong> <br/> |
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| 266 | subtract a four-vector from an existing one. |
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| 267 | |
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| 268 | |
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| 269 | <a name="method28"></a> |
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| 270 | <p/><strong>Vec4& Vec4::operator*=(double f) </strong> <br/> |
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| 271 | multiply all four-vector components by a real number. |
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| 272 | |
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| 273 | |
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| 274 | <a name="method29"></a> |
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| 275 | <p/><strong>Vec4& Vec4::operator/=(double f) </strong> <br/> |
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| 276 | divide all four-vector components by a real number. |
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| 277 | |
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| 278 | |
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| 279 | <a name="method30"></a> |
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| 280 | <p/><strong>friend Vec4 operator+(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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| 281 | add two four-vectors. |
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| 282 | |
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| 283 | |
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| 284 | <a name="method31"></a> |
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| 285 | <p/><strong>friend Vec4 operator-(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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| 286 | subtract two four-vectors. |
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| 287 | |
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| 288 | |
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| 289 | <a name="method32"></a> |
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| 290 | <p/><strong>friend Vec4 operator*(double f, const Vec4& v) </strong> <br/> |
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| 291 | |
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| 292 | <strong>friend Vec4 operator*(const Vec4& v, double f) </strong> <br/> |
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| 293 | multiply a four-vector by a real number. |
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| 294 | |
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| 295 | |
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| 296 | <a name="method33"></a> |
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| 297 | <p/><strong>friend Vec4 operator/(const Vec4& v, double f) </strong> <br/> |
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| 298 | divide a four-vector by a real number. |
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| 299 | |
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| 300 | |
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| 301 | <a name="method34"></a> |
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| 302 | <p/><strong>friend double operator*(const Vec4& v1, const Vec4 v2) </strong> <br/> |
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| 303 | four-vector product. |
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| 304 | |
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| 305 | |
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| 306 | <p/> |
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| 307 | There are also a few related operations that are normal member methods: |
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| 308 | |
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| 309 | <a name="method35"></a> |
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| 310 | <p/><strong>void Vec4::rescale3(double f) </strong> <br/> |
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| 311 | multiply the three-vector components by <i>f</i>, but keep the |
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| 312 | fourth component unchanged. |
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| 313 | |
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| 314 | |
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| 315 | <a name="method36"></a> |
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| 316 | <p/><strong>void Vec4::rescale4(double f) </strong> <br/> |
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| 317 | multiply all four-vector components by <i>f</i>. |
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| 318 | |
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| 319 | |
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| 320 | <a name="method37"></a> |
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| 321 | <p/><strong>void Vec4::flip3() </strong> <br/> |
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| 322 | flip the sign of the three-vector components, but keep the |
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| 323 | fourth component unchanged. |
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| 324 | |
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| 325 | |
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| 326 | <a name="method38"></a> |
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| 327 | <p/><strong>void Vec4::flip4() </strong> <br/> |
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| 328 | flip the sign of all four-vector components. |
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| 329 | |
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| 330 | |
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| 331 | <h3>Rotations and boosts</h3> |
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| 332 | |
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| 333 | A common task is to rotate or boost four-vectors. In case only one |
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| 334 | four-vector is affected the operation may be performed directly on it. |
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| 335 | However, in case many particles are affected, the helper class |
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| 336 | <code>RotBstMatrix</code> can be used to speed up operations. |
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| 337 | |
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| 338 | <a name="method39"></a> |
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| 339 | <p/><strong>void Vec4::rot(double theta, double phi) </strong> <br/> |
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| 340 | rotate the three-momentum with the polar angle <i>theta</i> |
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| 341 | and the azimuthal angle <i>phi</i>. |
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| 342 | |
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| 343 | |
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| 344 | <a name="method40"></a> |
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| 345 | <p/><strong>void Vec4::rotaxis(double phi, double nx, double ny, double nz) </strong> <br/> |
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| 346 | rotate the three-momentum with the azimuthal angle <i>phi</i> |
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| 347 | around the direction defined by the <i>(n_x, n_y, n_z)</i> |
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| 348 | three-vector. |
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| 349 | |
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| 350 | |
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| 351 | <a name="method41"></a> |
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| 352 | <p/><strong>void Vec4::rotaxis(double phi, Vec4& n) </strong> <br/> |
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| 353 | rotate the three-momentum with the azimuthal angle <i>phi</i> |
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| 354 | around the direction defined by the three-vector part of <i>n</i>. |
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| 355 | |
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| 356 | |
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| 357 | <a name="method42"></a> |
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| 358 | <p/><strong>void Vec4::bst(double betaX, double betaY, double betaZ) </strong> <br/> |
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| 359 | boost the four-momentum by <i>beta = (beta_x, beta_y, beta_z)</i>. |
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| 360 | |
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| 361 | |
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| 362 | <a name="method43"></a> |
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| 363 | <p/><strong>void Vec4::bst(double betaX, double betaY, double betaZ,double gamma) </strong> <br/> |
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| 364 | boost the four-momentum by <i>beta = (beta_x, beta_y, beta_z)</i>, |
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| 365 | where the <i>gamma = 1/sqrt(1 - beta^2)</i> is also input to allow |
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| 366 | better precision when <i>beta</i> is close to unity. |
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| 367 | |
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| 368 | |
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| 369 | <a name="method44"></a> |
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| 370 | <p/><strong>void Vec4::bst(const Vec4& p) </strong> <br/> |
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| 371 | boost the four-momentum by <i>beta = (p_x/E, p_y/E, p_z/E)</i>. |
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| 372 | |
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| 373 | |
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| 374 | <a name="method45"></a> |
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| 375 | <p/><strong>void Vec4::bst(const Vec4& p, double m) </strong> <br/> |
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| 376 | boost the four-momentum by <i>beta = (p_x/E, p_y/E, p_z/E)</i>, |
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| 377 | where the <i>gamma = E/m</i> is also calculated from input to allow |
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| 378 | better precision when <i>beta</i> is close to unity. |
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| 379 | |
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| 380 | |
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| 381 | <a name="method46"></a> |
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| 382 | <p/><strong>void Vec4::bstback(const Vec4& p) </strong> <br/> |
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| 383 | boost the four-momentum by <i>beta = (-p_x/E, -p_y/E, -p_z/E)</i>. |
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| 384 | |
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| 385 | |
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| 386 | <a name="method47"></a> |
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| 387 | <p/><strong>void Vec4::bstback(const Vec4& p, double m) </strong> <br/> |
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| 388 | boost the four-momentum by <i>beta = (-p_x/E, -p_y/E, -p_z/E)</i>, |
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| 389 | where the <i>gamma = E/m</i> is also calculated from input to allow |
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| 390 | better precision when <i>beta</i> is close to unity. |
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| 391 | |
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| 392 | |
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| 393 | <a name="method48"></a> |
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| 394 | <p/><strong>void Vec4::rotbst(const RotBstMatrix& M) </strong> <br/> |
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| 395 | perform a combined rotation and boost; see below for a description |
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| 396 | of the <code>RotBstMatrix</code>. |
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| 397 | |
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| 398 | |
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| 399 | <p/> |
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| 400 | For a longer sequence of rotations and boosts, and where several |
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| 401 | <code>Vec4</code> are to be rotated and boosted in the same way, |
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| 402 | a more efficient approach is to define a <code>RotBstMatrix</code>, |
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| 403 | which forms a separate auxiliary class. You can build up this |
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| 404 | 4-by-4 matrix by successive calls to the methods of the class, |
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| 405 | such that the matrix encodes the full sequence of operations. |
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| 406 | The order in which you do these calls must agree with the imagined |
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| 407 | order in which the rotations/boosts should be applied to a |
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| 408 | four-momentum, since in general the operations do not commute. |
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| 409 | |
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| 410 | <a name="method49"></a> |
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| 411 | <p/><strong>RotBstMatrix::RotBstMatrix() </strong> <br/> |
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| 412 | creates a diagonal unit matrix, i.e. one that leaves a four-vector |
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| 413 | unchanged. |
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| 414 | |
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| 415 | |
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| 416 | <a name="method50"></a> |
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| 417 | <p/><strong>RotBstMatrix::RotBstMatrix(const RotBstMatrix& Min) </strong> <br/> |
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| 418 | creates a copy of the input matrix. |
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| 419 | |
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| 420 | |
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| 421 | <a name="method51"></a> |
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| 422 | <p/><strong>RotBstMatrix& RotBstMatrix::operator=(const RotBstMatrix4& Min) </strong> <br/> |
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| 423 | copies the input matrix. |
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| 424 | |
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| 425 | |
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| 426 | <a name="method52"></a> |
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| 427 | <p/><strong>void RotBstMatrix::rot(double theta = 0., double phi = 0.) </strong> <br/> |
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| 428 | rotate by this polar and azimuthal angle. |
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| 429 | |
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| 430 | |
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| 431 | <a name="method53"></a> |
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| 432 | <p/><strong>void RotBstMatrix::rot(const Vec4& p) </strong> <br/> |
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| 433 | rotate so that a vector originally along the <i>+z</i> axis becomes |
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| 434 | parallel with <i>p</i>. More specifically, rotate by <i>-phi</i>, |
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| 435 | <i>theta</i> and <i>phi</i>, with angles defined by <i>p</i>. |
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| 436 | |
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| 437 | |
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| 438 | <a name="method54"></a> |
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| 439 | <p/><strong>void RotBstMatrix::bst(double betaX = 0., double betaY = 0., double betaZ = 0.) </strong> <br/> |
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| 440 | boost by this <i>beta</i> vector. |
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| 441 | |
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| 442 | |
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| 443 | <a name="method55"></a> |
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| 444 | <p/><strong>void RotBstMatrix::bst(const Vec4&) </strong> <br/> |
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| 445 | |
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| 446 | <strong>void RotBstMatrix::bstback(const Vec4&) </strong> <br/> |
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| 447 | boost with a <i>beta = p/E</i> or <i>beta = -p/E</i>, respectively. |
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| 448 | |
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| 449 | |
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| 450 | <a name="method56"></a> |
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| 451 | <p/><strong>void RotBstMatrix::bst(const Vec4& p1, const Vec4& p2) </strong> <br/> |
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| 452 | boost so that <i>p_1</i> is transformed to <i>p_2</i>. It is assumed |
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| 453 | that the two vectors obey <i>p_1^2 = p_2^2</i>. |
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| 454 | |
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| 455 | |
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| 456 | <a name="method57"></a> |
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| 457 | <p/><strong>void RotBstMatrix::toCMframe(const Vec4& p1, const Vec4& p2) </strong> <br/> |
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| 458 | boost and rotate to the rest frame of <i>p_1</i> and <i>p_2</i>, |
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| 459 | with <i>p_1</i> along the <i>+z</i> axis. |
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| 460 | |
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| 461 | |
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| 462 | <a name="method58"></a> |
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| 463 | <p/><strong>void RotBstMatrix::fromCMframe(const Vec4& p1, const Vec4& p2) </strong> <br/> |
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| 464 | rotate and boost from the rest frame of <i>p_1</i> and <i>p_2</i>, |
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| 465 | with <i>p_1</i> along the <i>+z</i> axis, to the actual frame of |
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| 466 | <i>p_1</i> and <i>p_2</i>, i.e. the inverse of the above. |
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| 467 | |
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| 468 | |
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| 469 | <a name="method59"></a> |
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| 470 | <p/><strong>void RotBstMatrix::rotbst(const RotBstMatrix& Min); </strong> <br/> |
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| 471 | combine the current matrix with another one. |
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| 472 | |
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| 473 | |
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| 474 | <a name="method60"></a> |
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| 475 | <p/><strong>void RotBstMatrix::invert() </strong> <br/> |
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| 476 | invert the matrix, which corresponds to an opposite sequence and sign |
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| 477 | of rotations and boosts. |
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| 478 | |
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| 479 | |
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| 480 | <a name="method61"></a> |
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| 481 | <p/><strong>void RotBstMatrix::reset() </strong> <br/> |
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| 482 | reset to no rotation/boost; i.e. the default at creation. |
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| 483 | |
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| 484 | |
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| 485 | <a name="method62"></a> |
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| 486 | <p/><strong>double RotBstMatrix::deviation() </strong> <br/> |
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| 487 | crude estimate how much a matrix deviates from the unit matrix: |
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| 488 | the sum of the absolute values of all non-diagonal matrix elements |
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| 489 | plus the sum of the absolute deviation of the diagonal matrix |
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| 490 | elements from unity. |
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| 491 | |
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| 492 | |
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| 493 | <a name="method63"></a> |
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| 494 | <p/><strong>friend ostream& operator<<(ostream&, const RotBstMatrix& M) </strong> <br/> |
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| 495 | writes out the values of the sixteen components of a |
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| 496 | <code>RotBstMatrix</code>, on four consecutive lines and |
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| 497 | ended with a newline. |
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| 498 | |
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| 499 | |
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| 500 | </body> |
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| 501 | </html> |
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| 502 | |
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| 503 | <!-- Copyright (C) 2012 Torbjorn Sjostrand --> |
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