text: /*
   * Copyright Nick Thompson, 2017
   * Use, modification and distribution are subject to the
   * Boost Software License, Version 1.0. (See accompanying file
   * LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
   */
  #define BOOST_TEST_MODULE catmull_rom_test
  
  #include <array>
  #include <random>
  #include <boost/cstdfloat.hpp>
  #include <boost/type_index.hpp>
  #include <boost/test/included/unit_test.hpp>
  #include <boost/test/floating_point_comparison.hpp>
  #include <boost/math/constants/constants.hpp>
  #include <boost/math/interpolators/catmull_rom.hpp>
  #include <boost/multiprecision/cpp_bin_float.hpp>
  #include <boost/multiprecision/cpp_dec_float.hpp>
  #include <boost/numeric/ublas/vector.hpp>
  
  using std::abs;
  using boost::multiprecision::cpp_bin_float_50;
  using boost::math::catmull_rom;
  
  template<class Real>
  void test_alpha_distance()
  {
      Real tol = std::numeric_limits<Real>::epsilon();
      std::array<Real, 3> v1 = {0,0,0};
      std::array<Real, 3> v2 = {1,0,0};
      Real alpha = 0.5;
      Real d = boost::math::detail::alpha_distance<std::array<Real, 3>>(v1, v2, alpha);
      BOOST_CHECK_CLOSE_FRACTION(d, 1, tol);
  
      d = boost::math::detail::alpha_distance<std::array<Real, 3>>(v1, v2, 0.0);
      BOOST_CHECK_CLOSE_FRACTION(d, 1, tol);
  
      d = boost::math::detail::alpha_distance<std::array<Real, 3>>(v1, v2, 1.0);
      BOOST_CHECK_CLOSE_FRACTION(d, 1, tol);
  
      v2[0] = 2;
      d = boost::math::detail::alpha_distance<std::array<Real, 3>>(v1, v2, alpha);
      BOOST_CHECK_CLOSE_FRACTION(d, pow(2, (Real)1/ (Real) 2), tol);
  
      d = boost::math::detail::alpha_distance<std::array<Real, 3>>(v1, v2, 0.0);
      BOOST_CHECK_CLOSE_FRACTION(d, 1, tol);
  
      d = boost::math::detail::alpha_distance<std::array<Real, 3>>(v1, v2, 1.0);
      BOOST_CHECK_CLOSE_FRACTION(d, 2, tol);
  }
  
  
  template<class Real>
  void test_linear()
  {
      std::cout << "Testing that the Catmull-Rom spline interpolates linear functions correctly on type "
                << boost::typeindex::type_id<Real>().pretty_name() << "\n";
  
      Real tol = 10*std::numeric_limits<Real>::epsilon();
      std::vector<std::array<Real, 3>> v(4);
      v[0] = {0,0,0};
      v[1] = {1,0,0};
      v[2] = {2,0,0};
      v[3] = {3,0,0};
      catmull_rom<std::array<Real, 3>> cr(std::move(v));
  
      // Test that the interpolation condition is obeyed:
      BOOST_CHECK_CLOSE_FRACTION(cr.max_parameter(), 3, tol);
      auto p0 = cr(0.0);
      BOOST_CHECK_SMALL(p0[0], tol);
      BOOST_CHECK_SMALL(p0[1], tol);
      BOOST_CHECK_SMALL(p0[2], tol);
      auto p1 = cr(1.0);
      BOOST_CHECK_CLOSE_FRACTION(p1[0], 1, tol);
      BOOST_CHECK_SMALL(p1[1], tol);
      BOOST_CHECK_SMALL(p1[2], tol);
  
      auto p2 = cr(2.0);
      BOOST_CHECK_CLOSE_FRACTION(p2[0], 2, tol);
      BOOST_CHECK_SMALL(p2[1], tol);
      BOOST_CHECK_SMALL(p2[2], tol);
  
  
      auto p3 = cr(3.0);
      BOOST_CHECK_CLOSE_FRACTION(p3[0], 3, tol);
      BOOST_CHECK_SMALL(p3[1], tol);
      BOOST_CHECK_SMALL(p3[2], tol);
  
      Real s = cr.parameter_at_point(0);
      BOOST_CHECK_SMALL(s, tol);
  
      s = cr.parameter_at_point(1);
      BOOST_CHECK_CLOSE_FRACTION(s, 1, tol);
  
      s = cr.parameter_at_point(2);
      BOOST_CHECK_CLOSE_FRACTION(s, 2, tol);
  
      s = cr.parameter_at_point(3);
      BOOST_CHECK_CLOSE_FRACTION(s, 3, tol);
  
      // Test that the function is linear on the interval [1,2]:
      for (double s = 1; s < 2; s += 0.01)
      {
          auto p = cr(s);
          BOOST_CHECK_CLOSE_FRACTION(p[0], s, tol);
          BOOST_CHECK_SMALL(p[1], tol);
          BOOST_CHECK_SMALL(p[2], tol);
  
          auto tangent = cr.prime(s);
          BOOST_CHECK_CLOSE_FRACTION(tangent[0], 1, tol);
          BOOST_CHECK_SMALL(tangent[1], tol);
          BOOST_CHECK_SMALL(tangent[2], tol);
      }
  
  }
  
  template<class Real>
  void test_circle()
  {
      using boost::math::constants::pi;
      using std::cos;
      using std::sin;
  
      std::cout << "Testing that the Catmull-Rom spline interpolates circles correctly on type "
                << boost::typeindex::type_id<Real>().pretty_name() << "\n";
  
      Real tol = 10*std::numeric_limits<Real>::epsilon();
      std::vector<std::array<Real, 2>> v(20*sizeof(Real));
      std::vector<std::array<Real, 2>> u(20*sizeof(Real));
      for (size_t i = 0; i < v.size(); ++i)
      {
          Real theta = ((Real) i/ (Real) v.size())*2*pi<Real>();
          v[i] = {cos(theta), sin(theta)};
          u[i] = v[i];
      }
      catmull_rom<std::array<Real, 2>> circle(std::move(v), true);
  
      // Interpolation condition:
      for (size_t i = 0; i < v.size(); ++i)
      {
          Real s = circle.parameter_at_point(i);
          auto p = circle(s);
          Real x = p[0];
          Real y = p[1];
          if (abs(x) < std::numeric_limits<Real>::epsilon())
          {
              BOOST_CHECK_SMALL(u[i][0], tol);
          }
          if (abs(y) < std::numeric_limits<Real>::epsilon())
          {
              BOOST_CHECK_SMALL(u[i][1], tol);
          }
          else
          {
              BOOST_CHECK_CLOSE_FRACTION(x, u[i][0], tol);
              BOOST_CHECK_CLOSE_FRACTION(y, u[i][1], tol);
          }
      }
  
      Real max_s = circle.max_parameter();
      for(Real s = 0; s < max_s; s += 0.01)
      {
          auto p = circle(s);
          Real x = p[0];
          Real y = p[1];
          BOOST_CHECK_CLOSE_FRACTION(x*x+y*y, 1, 0.001);
      }
  }
  
  
  template<class Real, size_t dimension>
  void test_affine_invariance()
  {
      std::cout << "Testing that the Catmull-Rom spline is affine invariant in dimension "
                << dimension << " on type "
                << boost::typeindex::type_id<Real>().pretty_name() << "\n";
  
      Real tol = 1000*std::numeric_limits<Real>::epsilon();
      std::vector<std::array<Real, dimension>> v(100);
      std::vector<std::array<Real, dimension>> u(100);
      std::mt19937_64 gen(438232);
      Real inv_denom = (Real) 100/( (Real) (gen.max)() + (Real) 2);
      for(size_t j = 0; j < dimension; ++j)
      {
          v[0][j] = gen()*inv_denom;
          u[0][j] = v[0][j];
      }
  
      for (size_t i = 1; i < v.size(); ++i)
      {
          for(size_t j = 0; j < dimension; ++j)
          {
              v[i][j] = v[i-1][j] + gen()*inv_denom;
              u[i][j] = v[i][j];
          }
      }
      std::array<Real, dimension> affine_shift;
      for (size_t j = 0; j < dimension; ++j)
      {
          affine_shift[j] = gen()*inv_denom;
      }
  
      catmull_rom<std::array<Real, dimension>> cr1(std::move(v));
  
      for(size_t i = 0; i< u.size(); ++i)
      {
          for(size_t j = 0; j < dimension; ++j)
          {
              u[i][j] += affine_shift[j];
          }
      }
  
      catmull_rom<std::array<Real, dimension>> cr2(std::move(u));
  
      BOOST_CHECK_CLOSE_FRACTION(cr1.max_parameter(), cr2.max_parameter(), tol);
  
      Real ds = cr1.max_parameter()/1024;
      for (Real s = 0; s < cr1.max_parameter(); s += ds)
      {
          auto p0 = cr1(s);
          auto p1 = cr2(s);
          auto tangent0 = cr1.prime(s);
          auto tangent1 = cr2.prime(s);
          for (size_t j = 0; j < dimension; ++j)
          {
              BOOST_CHECK_CLOSE_FRACTION(p0[j] + affine_shift[j], p1[j], tol);
              if (abs(tangent0[j]) > 5000*tol)
              {
                  BOOST_CHECK_CLOSE_FRACTION(tangent0[j], tangent1[j], 5000*tol);
              }
          }
      }
  }
  
  template<class Real>
  void test_helix()
  {
      using boost::math::constants::pi;
      std::cout << "Testing that the Catmull-Rom spline interpolates helices correctly on type "
                << boost::typeindex::type_id<Real>().pretty_name() << "\n";
  
      Real tol = 0.001;
      std::vector<std::array<Real, 3>> v(400*sizeof(Real));
      for (size_t i = 0; i < v.size(); ++i)
      {
          Real theta = ((Real) i/ (Real) v.size())*2*pi<Real>();
          v[i] = {cos(theta), sin(theta), theta};
      }
      catmull_rom<std::array<Real, 3>> helix(std::move(v));
  
      // Interpolation condition:
      for (size_t i = 0; i < v.size(); ++i)
      {
          Real s = helix.parameter_at_point(i);
          auto p = helix(s);
          Real t = p[2];
  
          Real x = p[0];
          Real y = p[1];
          if (abs(x) < tol)
          {
              BOOST_CHECK_SMALL(cos(t), tol);
          }
          if (abs(y) < tol)
          {
              BOOST_CHECK_SMALL(sin(t), tol);
          }
          else
          {
              BOOST_CHECK_CLOSE_FRACTION(x, cos(t), tol);
              BOOST_CHECK_CLOSE_FRACTION(y, sin(t), tol);
          }
      }
  
      Real max_s = helix.max_parameter();
      for(Real s = helix.parameter_at_point(1); s < max_s; s += 0.01)
      {
          auto p = helix(s);
          Real x = p[0];
          Real y = p[1];
          Real t = p[2];
          BOOST_CHECK_CLOSE_FRACTION(x*x+y*y, (Real) 1, (Real) 0.01);
          if (abs(x) < 0.01)
          {
              BOOST_CHECK_SMALL(cos(t),  (Real) 0.05);
          }
          if (abs(y) < 0.01)
          {
              BOOST_CHECK_SMALL(sin(t), (Real) 0.05);
          }
          else
          {
              BOOST_CHECK_CLOSE_FRACTION(x, cos(t), (Real) 0.05);
              BOOST_CHECK_CLOSE_FRACTION(y, sin(t), (Real) 0.05);
          }
      }
  }
  
  
  template<class Real>
  class mypoint3d
  {
  public:
      // Must define a value_type:
      typedef Real value_type;
  
      // Regular constructor:
      mypoint3d(Real x, Real y, Real z)
      {
          m_vec[0] = x;
          m_vec[1] = y;
          m_vec[2] = z;
      }
  
      // Must define a default constructor:
      mypoint3d() {}
  
      // Must define array access:
      Real operator[](size_t i) const
      {
          return m_vec[i];
      }
  
      // Array element assignment:
      Real& operator[](size_t i)
      {
          return m_vec[i];
      }
  
  
  private:
      std::array<Real, 3>  m_vec;
  };
  
  
  // Must define the free function "size()":
  template<class Real>
  BOOST_CONSTEXPR std::size_t size(const mypoint3d<Real>& c)
  {
      return 3;
  }
  
  template<class Real>
  void test_data_representations()
  {
      std::cout << "Testing that the Catmull-Rom spline works with multiple data representations.\n";
      mypoint3d<Real> p0(0.1, 0.2, 0.3);
      mypoint3d<Real> p1(0.2, 0.3, 0.4);
      mypoint3d<Real> p2(0.3, 0.4, 0.5);
      mypoint3d<Real> p3(0.4, 0.5, 0.6);
      mypoint3d<Real> p4(0.5, 0.6, 0.7);
      mypoint3d<Real> p5(0.6, 0.7, 0.8);
  
  
      // Tests initializer_list:
      catmull_rom<mypoint3d<Real>> cat({p0, p1, p2, p3, p4, p5});
  
      Real tol = 0.001;
      auto p = cat(cat.parameter_at_point(0));
      BOOST_CHECK_CLOSE_FRACTION(p[0], p0[0], tol);
      BOOST_CHECK_CLOSE_FRACTION(p[1], p0[1], tol);
      BOOST_CHECK_CLOSE_FRACTION(p[2], p0[2], tol);
      p = cat(cat.parameter_at_point(1));
      BOOST_CHECK_CLOSE_FRACTION(p[0], p1[0], tol);
      BOOST_CHECK_CLOSE_FRACTION(p[1], p1[1], tol);
      BOOST_CHECK_CLOSE_FRACTION(p[2], p1[2], tol);
  }
  
  template<class Real>
  void test_random_access_container()
  {
      std::cout << "Testing that the Catmull-Rom spline works with multiple data representations.\n";
      mypoint3d<Real> p0(0.1, 0.2, 0.3);
      mypoint3d<Real> p1(0.2, 0.3, 0.4);
      mypoint3d<Real> p2(0.3, 0.4, 0.5);
      mypoint3d<Real> p3(0.4, 0.5, 0.6);
      mypoint3d<Real> p4(0.5, 0.6, 0.7);
      mypoint3d<Real> p5(0.6, 0.7, 0.8);
  
      boost::numeric::ublas::vector<mypoint3d<Real>> u(6);
      u[0] = p0;
      u[1] = p1;
      u[2] = p2;
      u[3] = p3;
      u[4] = p4;
      u[5] = p5;
  
      // Tests initializer_list:
      catmull_rom<mypoint3d<Real>, decltype(u)> cat(std::move(u));
  
      Real tol = 0.001;
      auto p = cat(cat.parameter_at_point(0));
      BOOST_CHECK_CLOSE_FRACTION(p[0], p0[0], tol);
      BOOST_CHECK_CLOSE_FRACTION(p[1], p0[1], tol);
      BOOST_CHECK_CLOSE_FRACTION(p[2], p0[2], tol);
      p = cat(cat.parameter_at_point(1));
      BOOST_CHECK_CLOSE_FRACTION(p[0], p1[0], tol);
      BOOST_CHECK_CLOSE_FRACTION(p[1], p1[1], tol);
      BOOST_CHECK_CLOSE_FRACTION(p[2], p1[2], tol);
  }
  
  BOOST_AUTO_TEST_CASE(catmull_rom_test)
  {
  #if !defined(TEST) || (TEST == 1)
      test_data_representations<float>();
      test_alpha_distance<double>();
  
      test_linear<double>();
      test_linear<long double>();
  
      test_circle<float>();
      test_circle<double>();
  #endif
  #if !defined(TEST) || (TEST == 2)
      test_helix<double>();
  
      test_affine_invariance<double, 1>();
      test_affine_invariance<double, 2>();
      test_affine_invariance<double, 3>();
      test_affine_invariance<double, 4>();
  
      test_random_access_container<double>();
  #endif
  #if !defined(TEST) || (TEST == 3)
      test_affine_invariance<cpp_bin_float_50, 4>();
  #endif
  }
  