Utilities/detail/RealQuadraticEquation.hpp

0001 // This file is part of the ACTS project.
0002 //
0003 // Copyright (C) 2016 CERN for the benefit of the ACTS project
0004 //
0005 // This Source Code Form is subject to the terms of the Mozilla Public
0006 // License, v. 2.0. If a copy of the MPL was not distributed with this
0007 // file, You can obtain one at https://mozilla.org/MPL/2.0/.
0008
0009 ///////////////////////////////////////////////////////////////////
0010 // RealQuadraticEquation.h, Acts project
0011 ///////////////////////////////////////////////////////////////////
0012
0013 #pragma once
0014
0015 #include <cmath>
0016 #include <utility>
0017
0018 namespace Acts::detail {
0019
0020 /// @struct RealQuadradicEquation
0021 ///   Mathematic struct for solving real quadratic equations
0022 ///
0023 ///  <b>Mathematical motivation</b>:<br>
0024 ///  The equation is given by:<br>
0025 ///  @f$ \alpha x^{2} + \beta x + \gamma = 0  @f$
0026 ///  and has therefore the analytical solution:<br>
0027 ///  @f$ x_{1, 2} = - \frac{\beta \pm
0028 ///  \sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha}@f$
0029 /// <br>
0030 /// <br>
0031 ///  - case @f$ \beta > 0 @f$:<br>
0032 ///  @f$ x_{1} = - \frac{\beta + \sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha}  :=
0033 /// \frac{q}{\alpha}@f$, <br>
0034 ///  so that @f$ q= -\frac{1}{2}(\beta+sqrt{\beta^{2}-4\alpha\gamma})@f$.
0035 ///  @f$ x_{2} @f$ can now be written as: @f$ x_{2} = \frac{\gamma}{q} =
0036 /// -\frac{2\gamma}{\beta+sqrt{\beta^{2}-4\alpha\gamma}}@f$, since: <br>
0037 ///  @f$ -\frac{2\gamma}{\beta+sqrt{\beta^{2}-4\alpha\gamma}} =
0038 /// -\frac{2\gamma}{\beta}\frac{1}{1+\sqrt{1-4\alpha\gamma/\beta^{2}}}@f$, and
0039 /// <br>
0040 ///  @f$ x_{2}\frac{1}{1-\sqrt{1-4\alpha\gamma/\beta^{2}}} =
0041 /// -\frac{2\gamma}{\beta}\frac{1}{1-1+4\alpha\gamma/\beta^{2}}=-\frac{\beta}{2\alpha}.@f$<br>
0042 ///  Hence,@f$ -\frac{\beta(1-\sqrt{1-4\alpha\gamma/\beta^{2}}}{2\alpha} = -
0043 /// \frac{\beta - \sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha} @f$.<br>
0044 ///  - case @f$ \beta > 0 @f$ is similar.
0045 ///
0046
0047 struct RealQuadraticEquation {
0048   double first = 0;
0049   double second = 0;
0050   int solutions = 0;
0051
0052   /// Constructor
0053   ///
0054   /// @param alpha is the first parameter of the quad equation
0055   /// @param beta is the second parameter of the quad equation
0056   /// @param gamma is the third parameter of the quad equation
0057   RealQuadraticEquation(double alpha, double beta, double gamma) {
0058     double discriminant = beta * beta - 4 * alpha * gamma;
0059     if (discriminant >= 0) {
0060       solutions = (discriminant == 0) ? 1 : 2;
0061       double q = -0.5 * (beta + (beta > 0 ? std::sqrt(discriminant)
0062                                           : -std::sqrt(discriminant)));
0063       first = q / alpha;
0064       second = gamma / q;
0065     }
0066   }
0067 };
0068
0069 }  // namespace Acts::detail