text: // Boost.Geometry
  
  // Copyright (c) 2015-2021 Oracle and/or its affiliates.
  
  // Contributed and/or modified by Vissarion Fysikopoulos, on behalf of Oracle
  // Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle
  
  // Use, modification and distribution is 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)
  
  #ifndef BOOST_GEOMETRY_FORMULAS_AREA_FORMULAS_HPP
  #define BOOST_GEOMETRY_FORMULAS_AREA_FORMULAS_HPP
  
  #include <boost/geometry/core/radian_access.hpp>
  #include <boost/geometry/formulas/flattening.hpp>
  #include <boost/geometry/formulas/mean_radius.hpp>
  #include <boost/geometry/formulas/karney_inverse.hpp>
  #include <boost/geometry/util/math.hpp>
  #include <boost/math/special_functions/hypot.hpp>
  
  namespace boost { namespace geometry { namespace formula
  {
  
  /*!
  \brief Formulas for computing spherical and ellipsoidal polygon area.
   The current class computes the area of the trapezoid defined by a segment
   the two meridians passing by the endpoints and the equator.
  \author See
  - Danielsen JS, The area under the geodesic. Surv Rev 30(232):
  61–66, 1989
  - Charles F.F Karney, Algorithms for geodesics, 2011
  https://arxiv.org/pdf/1109.4448.pdf
  */
  
  template
  <
      typename CT,
      std::size_t SeriesOrder = 2,
      bool ExpandEpsN = true
  >
  class area_formulas
  {
  
  public:
  
      //TODO: move the following to a more general space to be used by other
      //      classes as well
      /*
          Evaluate the polynomial in x using Horner's method.
      */
      template <typename NT, typename IteratorType>
      static inline NT horner_evaluate(NT const& x,
                                       IteratorType begin,
                                       IteratorType end)
      {
          NT result(0);
          IteratorType it = end;
          do
          {
              result = result * x + *--it;
          }
          while (it != begin);
          return result;
      }
  
      /*
          Clenshaw algorithm for summing trigonometric series
          https://en.wikipedia.org/wiki/Clenshaw_algorithm
      */
      template <typename NT, typename IteratorType>
      static inline NT clenshaw_sum(NT const& cosx,
                                    IteratorType begin,
                                    IteratorType end)
      {
          IteratorType it = end;
          bool odd = true;
          CT b_k, b_k1(0), b_k2(0);
          do
          {
              CT c_k = odd ? *--it : NT(0);
              b_k = c_k + NT(2) * cosx * b_k1 - b_k2;
              b_k2 = b_k1;
              b_k1 = b_k;
              odd = !odd;
          }
          while (it != begin);
  
          return *begin + b_k1 * cosx - b_k2;
      }
  
      template<typename T>
      static inline void normalize(T& x, T& y)
      {
          T h = boost::math::hypot(x, y);
          x /= h;
          y /= h;
      }
  
      /*
       Generate and evaluate the series expansion of the following integral
  
          I4 = -integrate( (t(ep2) - t(k2*sin(sigma1)^2)) / (ep2 - k2*sin(sigma1)^2)
             * sin(sigma1)/2, sigma1, pi/2, sigma )
       where
  
          t(x) = sqrt(1+1/x)*asinh(sqrt(x)) + x
  
       valid for ep2 and k2 small.  We substitute k2 = 4 * eps / (1 - eps)^2
       and ep2 = 4 * n / (1 - n)^2 and expand in eps and n.
  
       The resulting sum of the series is of the form
  
          sum(C4[l] * cos((2*l+1)*sigma), l, 0, maxpow-1) )
  
       The above expansion is performed in Computer Algebra System Maxima.
       The C++ code (that yields the function evaluate_coeffs_n below) is generated
       by the following Maxima script and is based on script:
       http://geographiclib.sourceforge.net/html/geod.mac
  
          // Maxima script begin
          taylordepth:5$
          ataylor(expr,var,ord):=expand(ratdisrep(taylor(expr,var,0,ord)))$
          jtaylor(expr,var1,var2,ord):=block([zz],expand(subst([zz=1],
          ratdisrep(taylor(subst([var1=zz*var1,var2=zz*var2],expr),zz,0,ord)))))$
  
          compute(maxpow):=block([int,t,intexp,area, x,ep2,k2],
          maxpow:maxpow-1,
          t : sqrt(1+1/x) * asinh(sqrt(x)) + x,
          int:-(tf(ep2) - tf(k2*sin(sigma)^2)) / (ep2 - k2*sin(sigma)^2)
          * sin(sigma)/2,
          int:subst([tf(ep2)=subst([x=ep2],t),
          tf(k2*sin(sigma)^2)=subst([x=k2*sin(sigma)^2],t)],
          int),
          int:subst([abs(sin(sigma))=sin(sigma)],int),
          int:subst([k2=4*eps/(1-eps)^2,ep2=4*n/(1-n)^2],int),
          intexp:jtaylor(int,n,eps,maxpow),
          area:trigreduce(integrate(intexp,sigma)),
          area:expand(area-subst(sigma=%pi/2,area)),
          for i:0 thru maxpow do C4[i]:coeff(area,cos((2*i+1)*sigma)),
          if expand(area-sum(C4[i]*cos((2*i+1)*sigma),i,0,maxpow)) # 0
          then error("left over terms in I4"),
          'done)$
  
          printcode(maxpow):=
          block([tab2:"    ",tab3:"      "],
          print(" switch (SeriesOrder) {"),
          for nn:1 thru maxpow do block([c],
          print(concat(tab2,"case ",string(nn-1),":")),
          c:0,
          for m:0 thru nn-1 do block(
            [q:jtaylor(subst([n=n],C4[m]),n,eps,nn-1),
            linel:1200],
            for j:m thru nn-1 do (
              print(concat(tab3,"coeffs_n[",c,"] = ",
                  string(horner(coeff(q,eps,j))),";")),
              c:c+1)
          ),
          print(concat(tab3,"break;"))),
          print("    }"),
          'done)$
  
          maxpow:6$
          compute(maxpow)$
          printcode(maxpow)$
          // Maxima script end
  
       In the resulting code we should replace each number x by CT(x)
       e.g. using the following scirpt:
         sed -e 's/[0-9]\+/CT(&)/g; s/\[CT(/\[/g; s/)\]/\]/g;
                 s/case\sCT(/case /g; s/):/:/g'
      */
  
      static inline void evaluate_coeffs_n(CT const& n, CT coeffs_n[])
      {
          switch (SeriesOrder) {
          case 0:
              coeffs_n[0] = CT(2)/CT(3);
              break;
          case 1:
              coeffs_n[0] = (CT(10)-CT(4)*n)/CT(15);
              coeffs_n[1] = -CT(1)/CT(5);
              coeffs_n[2] = CT(1)/CT(45);
              break;
          case 2:
              coeffs_n[0] = (n*(CT(8)*n-CT(28))+CT(70))/CT(105);
              coeffs_n[1] = (CT(16)*n-CT(7))/CT(35);
              coeffs_n[2] = -CT(2)/CT(105);
              coeffs_n[3] = (CT(7)-CT(16)*n)/CT(315);
              coeffs_n[4] = -CT(2)/CT(105);
              coeffs_n[5] = CT(4)/CT(525);
              break;
          case 3:
              coeffs_n[0] = (n*(n*(CT(4)*n+CT(24))-CT(84))+CT(210))/CT(315);
              coeffs_n[1] = ((CT(48)-CT(32)*n)*n-CT(21))/CT(105);
              coeffs_n[2] = (-CT(32)*n-CT(6))/CT(315);
              coeffs_n[3] = CT(11)/CT(315);
              coeffs_n[4] = (n*(CT(32)*n-CT(48))+CT(21))/CT(945);
              coeffs_n[5] = (CT(64)*n-CT(18))/CT(945);
              coeffs_n[6] = -CT(1)/CT(105);
              coeffs_n[7] = (CT(12)-CT(32)*n)/CT(1575);
              coeffs_n[8] = -CT(8)/CT(1575);
              coeffs_n[9] = CT(8)/CT(2205);
              break;
          case 4:
              coeffs_n[0] = (n*(n*(n*(CT(16)*n+CT(44))+CT(264))-CT(924))+CT(2310))/CT(3465);
              coeffs_n[1] = (n*(n*(CT(48)*n-CT(352))+CT(528))-CT(231))/CT(1155);
              coeffs_n[2] = (n*(CT(1088)*n-CT(352))-CT(66))/CT(3465);
              coeffs_n[3] = (CT(121)-CT(368)*n)/CT(3465);
              coeffs_n[4] = CT(4)/CT(1155);
              coeffs_n[5] = (n*((CT(352)-CT(48)*n)*n-CT(528))+CT(231))/CT(10395);
              coeffs_n[6] = ((CT(704)-CT(896)*n)*n-CT(198))/CT(10395);
              coeffs_n[7] = (CT(80)*n-CT(99))/CT(10395);
              coeffs_n[8] = CT(4)/CT(1155);
              coeffs_n[9] = (n*(CT(320)*n-CT(352))+CT(132))/CT(17325);
              coeffs_n[10] = (CT(384)*n-CT(88))/CT(17325);
              coeffs_n[11] = -CT(8)/CT(1925);
              coeffs_n[12] = (CT(88)-CT(256)*n)/CT(24255);
              coeffs_n[13] = -CT(16)/CT(8085);
              coeffs_n[14] = CT(64)/CT(31185);
              break;
          case 5:
              coeffs_n[0] = (n*(n*(n*(n*(CT(100)*n+CT(208))+CT(572))+CT(3432))-CT(12012))+CT(30030))
                            /CT(45045);
              coeffs_n[1] = (n*(n*(n*(CT(64)*n+CT(624))-CT(4576))+CT(6864))-CT(3003))/CT(15015);
              coeffs_n[2] = (n*((CT(14144)-CT(10656)*n)*n-CT(4576))-CT(858))/CT(45045);
              coeffs_n[3] = ((-CT(224)*n-CT(4784))*n+CT(1573))/CT(45045);
              coeffs_n[4] = (CT(1088)*n+CT(156))/CT(45045);
              coeffs_n[5] = CT(97)/CT(15015);
              coeffs_n[6] = (n*(n*((-CT(64)*n-CT(624))*n+CT(4576))-CT(6864))+CT(3003))/CT(135135);
              coeffs_n[7] = (n*(n*(CT(5952)*n-CT(11648))+CT(9152))-CT(2574))/CT(135135);
              coeffs_n[8] = (n*(CT(5792)*n+CT(1040))-CT(1287))/CT(135135);
              coeffs_n[9] = (CT(468)-CT(2944)*n)/CT(135135);
              coeffs_n[10] = CT(1)/CT(9009);
              coeffs_n[11] = (n*((CT(4160)-CT(1440)*n)*n-CT(4576))+CT(1716))/CT(225225);
              coeffs_n[12] = ((CT(4992)-CT(8448)*n)*n-CT(1144))/CT(225225);
              coeffs_n[13] = (CT(1856)*n-CT(936))/CT(225225);
              coeffs_n[14] = CT(8)/CT(10725);
              coeffs_n[15] = (n*(CT(3584)*n-CT(3328))+CT(1144))/CT(315315);
              coeffs_n[16] = (CT(1024)*n-CT(208))/CT(105105);
              coeffs_n[17] = -CT(136)/CT(63063);
              coeffs_n[18] = (CT(832)-CT(2560)*n)/CT(405405);
              coeffs_n[19] = -CT(128)/CT(135135);
              coeffs_n[20] = CT(128)/CT(99099);
              break;
          }
      }
  
      /*
         Expand in k2 and ep2.
      */
      static inline void evaluate_coeffs_ep(CT const& ep, CT coeffs_n[])
      {
          switch (SeriesOrder) {
          case 0:
              coeffs_n[0] = CT(2)/CT(3);
              break;
          case 1:
              coeffs_n[0] = (CT(10)-ep)/CT(15);
              coeffs_n[1] = -CT(1)/CT(20);
              coeffs_n[2] = CT(1)/CT(180);
              break;
          case 2:
              coeffs_n[0] = (ep*(CT(4)*ep-CT(7))+CT(70))/CT(105);
              coeffs_n[1] = (CT(4)*ep-CT(7))/CT(140);
              coeffs_n[2] = CT(1)/CT(42);
              coeffs_n[3] = (CT(7)-CT(4)*ep)/CT(1260);
              coeffs_n[4] = -CT(1)/CT(252);
              coeffs_n[5] = CT(1)/CT(2100);
              break;
          case 3:
              coeffs_n[0] = (ep*((CT(12)-CT(8)*ep)*ep-CT(21))+CT(210))/CT(315);
              coeffs_n[1] = ((CT(12)-CT(8)*ep)*ep-CT(21))/CT(420);
              coeffs_n[2] = (CT(3)-CT(2)*ep)/CT(126);
              coeffs_n[3] = -CT(1)/CT(72);
              coeffs_n[4] = (ep*(CT(8)*ep-CT(12))+CT(21))/CT(3780);
              coeffs_n[5] = (CT(2)*ep-CT(3))/CT(756);
              coeffs_n[6] = CT(1)/CT(360);
              coeffs_n[7] = (CT(3)-CT(2)*ep)/CT(6300);
              coeffs_n[8] = -CT(1)/CT(1800);
              coeffs_n[9] = CT(1)/CT(17640);
              break;
          case 4:
              coeffs_n[0] = (ep*(ep*(ep*(CT(64)*ep-CT(88))+CT(132))-CT(231))+CT(2310))/CT(3465);
              coeffs_n[1] = (ep*(ep*(CT(64)*ep-CT(88))+CT(132))-CT(231))/CT(4620);
              coeffs_n[2] = (ep*(CT(16)*ep-CT(22))+CT(33))/CT(1386);
              coeffs_n[3] = (CT(8)*ep-CT(11))/CT(792);
              coeffs_n[4] = CT(1)/CT(110);
              coeffs_n[5] = (ep*((CT(88)-CT(64)*ep)*ep-CT(132))+CT(231))/CT(41580);
              coeffs_n[6] = ((CT(22)-CT(16)*ep)*ep-CT(33))/CT(8316);
              coeffs_n[7] = (CT(11)-CT(8)*ep)/CT(3960);
              coeffs_n[8] = -CT(1)/CT(495);
              coeffs_n[9] = (ep*(CT(16)*ep-CT(22))+CT(33))/CT(69300);
              coeffs_n[10] = (CT(8)*ep-CT(11))/CT(19800);
              coeffs_n[11] = CT(1)/CT(1925);
              coeffs_n[12] = (CT(11)-CT(8)*ep)/CT(194040);
              coeffs_n[13] = -CT(1)/CT(10780);
              coeffs_n[14] = CT(1)/CT(124740);
              break;
          case 5:
              coeffs_n[0] = (ep*(ep*(ep*((CT(832)-CT(640)*ep)*ep-CT(1144))+CT(1716))-CT(3003))+CT(30030))/CT(45045);
              coeffs_n[1] = (ep*(ep*((CT(832)-CT(640)*ep)*ep-CT(1144))+CT(1716))-CT(3003))/CT(60060);
              coeffs_n[2] = (ep*((CT(208)-CT(160)*ep)*ep-CT(286))+CT(429))/CT(18018);
              coeffs_n[3] = ((CT(104)-CT(80)*ep)*ep-CT(143))/CT(10296);
              coeffs_n[4] = (CT(13)-CT(10)*ep)/CT(1430);
              coeffs_n[5] = -CT(1)/CT(156);
              coeffs_n[6] = (ep*(ep*(ep*(CT(640)*ep-CT(832))+CT(1144))-CT(1716))+CT(3003))/CT(540540);
              coeffs_n[7] = (ep*(ep*(CT(160)*ep-CT(208))+CT(286))-CT(429))/CT(108108);
              coeffs_n[8] = (ep*(CT(80)*ep-CT(104))+CT(143))/CT(51480);
              coeffs_n[9] = (CT(10)*ep-CT(13))/CT(6435);
              coeffs_n[10] = CT(5)/CT(3276);
              coeffs_n[11] = (ep*((CT(208)-CT(160)*ep)*ep-CT(286))+CT(429))/CT(900900);
              coeffs_n[12] = ((CT(104)-CT(80)*ep)*ep-CT(143))/CT(257400);
              coeffs_n[13] = (CT(13)-CT(10)*ep)/CT(25025);
              coeffs_n[14] = -CT(1)/CT(2184);
              coeffs_n[15] = (ep*(CT(80)*ep-CT(104))+CT(143))/CT(2522520);
              coeffs_n[16] = (CT(10)*ep-CT(13))/CT(140140);
              coeffs_n[17] = CT(5)/CT(45864);
              coeffs_n[18] = (CT(13)-CT(10)*ep)/CT(1621620);
              coeffs_n[19] = -CT(1)/CT(58968);
              coeffs_n[20] = CT(1)/CT(792792);
              break;
          }
      }
  
      /*
          Given the set of coefficients coeffs1[] evaluate on var2 and return
          the set of coefficients coeffs2[]
      */
      template <typename CoeffsType>
      static inline void evaluate_coeffs_var2(CT const& var2,
                                              CoeffsType const coeffs1[],
                                              CT coeffs2[])
      {
          std::size_t begin(0), end(0);
          for(std::size_t i = 0; i <= SeriesOrder; i++)
          {
              end = begin + SeriesOrder + 1 - i;
              coeffs2[i] = ((i==0) ? CT(1) : math::pow(var2, int(i)))
                          * horner_evaluate(var2, coeffs1 + begin, coeffs1 + end);
              begin = end;
          }
      }
  
  
      /*
          Compute the spherical excess of a geodesic (or shperical) segment
      */
      template
      <
          bool LongSegment,
          typename PointOfSegment
      >
      static inline CT spherical(PointOfSegment const& p1,
                                 PointOfSegment const& p2)
      {
          CT excess;
  
          if (LongSegment) // not for segments parallel to equator
          {
              CT cbet1 = cos(geometry::get_as_radian<1>(p1));
              CT sbet1 = sin(geometry::get_as_radian<1>(p1));
              CT cbet2 = cos(geometry::get_as_radian<1>(p2));
              CT sbet2 = sin(geometry::get_as_radian<1>(p2));
  
              CT omg12 = geometry::get_as_radian<0>(p1)
                       - geometry::get_as_radian<0>(p2);
              CT comg12 = cos(omg12);
              CT somg12 = sin(omg12);
  
              CT alp1 = atan2(cbet1 * sbet2
                              - sbet1 * cbet2 * comg12,
                              cbet2 * somg12);
  
              CT alp2 = atan2(cbet1 * sbet2 * comg12
                              - sbet1 * cbet2,
                              cbet1 * somg12);
  
              excess = alp2 - alp1;
  
          } else {
  
              // Trapezoidal formula
  
              CT tan_lat1 =
                      tan(geometry::get_as_radian<1>(p1) / 2.0);
              CT tan_lat2 =
                      tan(geometry::get_as_radian<1>(p2) / 2.0);
  
              excess = CT(2.0)
                      * atan(((tan_lat1 + tan_lat2) / (CT(1) + tan_lat1 * tan_lat2))
                             * tan((geometry::get_as_radian<0>(p2)
                                  - geometry::get_as_radian<0>(p1)) / 2));
          }
  
          return excess;
      }
  
      struct return_type_ellipsoidal
      {
          return_type_ellipsoidal()
              :   spherical_term(0),
                  ellipsoidal_term(0)
          {}
  
          CT spherical_term;
          CT ellipsoidal_term;
      };
  
      /*
          Compute the ellipsoidal correction of a geodesic (or shperical) segment
      */
      template
      <
          template <typename, bool, bool, bool, bool, bool> class Inverse,
          typename PointOfSegment,
          typename SpheroidConst
      >
      static inline auto ellipsoidal(PointOfSegment const& p1,
                                     PointOfSegment const& p2,
                                     SpheroidConst const& spheroid_const)
      {
          return_type_ellipsoidal result;
  
          CT const lon1r = get_as_radian<0>(p1);
          CT const lat1r = get_as_radian<1>(p1);
          CT const lon2r = get_as_radian<0>(p2);
          CT const lat2r = get_as_radian<1>(p2);
  
          // Azimuth Approximation
  
          using inverse_type = Inverse<CT, true, true, true, false, false>;
          auto i_res = inverse_type::apply(lon1r, lat1r, lon2r, lat2r, spheroid_const.m_spheroid);
  
          CT const alp1 = i_res.azimuth;
          CT const alp2 = i_res.reverse_azimuth;
  
          // Constants
  
          CT const ep = spheroid_const.m_ep;
          CT const one_minus_f = CT(1) - spheroid_const.m_f;
  
          // Basic trigonometric computations
          // the compiler could optimize here using sincos function
          // TODO: optimization: those quantities are already computed in inverse formula
          // at least in some inverse formulas, so do not compute them again here
          /*
          CT sin_bet1 = sin(lat1r);
          CT cos_bet1 = cos(lat1r);
          CT sin_bet2 = sin(lat2r);
          CT cos_bet2 = cos(lat2r);
  
          sin_bet1 *= one_minus_f;
          sin_bet2 *= one_minus_f;
          normalize(sin_bet1, cos_bet1);
          normalize(sin_bet2, cos_bet2);
          */
  
          CT const tan_bet1 = tan(lat1r) * one_minus_f;
          CT const tan_bet2 = tan(lat2r) * one_minus_f;
          CT const cos_bet1 = cos(atan(tan_bet1));
          CT const cos_bet2 = cos(atan(tan_bet2));
          CT const sin_bet1 = tan_bet1 * cos_bet1;
          CT const sin_bet2 = tan_bet2 * cos_bet2;
  
          CT const sin_alp1 = sin(alp1);
          CT const cos_alp1 = cos(alp1);
          CT const cos_alp2 = cos(alp2);
          CT const sin_alp0 = sin_alp1 * cos_bet1;
  
          // Spherical term computation
  
          CT excess;
  
          auto const half_pi = math::pi<CT>() / 2;
          bool meridian = lon2r - lon1r == CT(0)
              || lat1r == half_pi || lat1r == -half_pi
              || lat2r == half_pi || lat2r == -half_pi;
  
          if (!meridian && (i_res.distance)
              < mean_radius<CT>(spheroid_const.m_spheroid) / CT(638))  // short segment
          {
              CT tan_lat1 = tan(lat1r / 2.0);
              CT tan_lat2 = tan(lat2r / 2.0);
  
              excess = CT(2.0)
                  * atan(((tan_lat1 + tan_lat2) / (CT(1) + tan_lat1 * tan_lat2))
                  * tan((lon2r - lon1r) / 2));
          }
          else
          {
              /* in some cases this formula gives more accurate results
               *
               *             CT sin_omg12 =  cos_omg1 * sin_omg2 - sin_omg1 * cos_omg2;
              normalize(sin_omg12, cos_omg12);
  
              CT cos_omg12p1 = CT(1) + cos_omg12;
              CT cos_bet1p1 = CT(1) + cos_bet1;
              CT cos_bet2p1 = CT(1) + cos_bet2;
              excess = CT(2) * atan2(sin_omg12 * (sin_bet1 * cos_bet2p1 + sin_bet2 * cos_bet1p1),
                                     cos_omg12p1 * (sin_bet1 * sin_bet2 + cos_bet1p1 * cos_bet2p1));
              */
  
              excess = alp2 - alp1;
          }
  
          result.spherical_term = excess;
  
          // Ellipsoidal term computation (uses integral approximation)
  
          CT const cos_alp0 = math::sqrt(CT(1) - math::sqr(sin_alp0));
          //CT const cos_alp0 = hypot(cos_alp1, sin_alp1 * sin_bet1);
          CT cos_sig1 = cos_alp1 * cos_bet1;
          CT cos_sig2 = cos_alp2 * cos_bet2;
          CT sin_sig1 = sin_bet1;
          CT sin_sig2 = sin_bet2;
  
          normalize(sin_sig1, cos_sig1);
          normalize(sin_sig2, cos_sig2);
  
          CT coeffs[SeriesOrder + 1];
  
          if (ExpandEpsN) // expand by eps and n
          {
              CT const k2 = math::sqr(ep * cos_alp0);
              CT const sqrt_k2_plus_one = math::sqrt(CT(1) + k2);
              CT const eps = (sqrt_k2_plus_one - CT(1)) / (sqrt_k2_plus_one + CT(1));
  
              // Generate and evaluate the polynomials on eps (i.e. var2 = eps)
              // to get the final series coefficients
              evaluate_coeffs_var2(eps, spheroid_const.m_coeffs_var, coeffs);
          }
          else
          { // expand by k2 and ep
  
              CT const k2 = math::sqr(ep * cos_alp0);
              CT const ep2 = math::sqr(ep);
  
              CT coeffs_var[((SeriesOrder+2)*(SeriesOrder+1))/2];
  
              // Generate and evaluate the polynomials on ep2
              evaluate_coeffs_ep(ep2, coeffs_var);
  
              // Generate and evaluate the polynomials on k2 (i.e. var2 = k2)
              evaluate_coeffs_var2(k2, coeffs_var, coeffs);
          }
  
          // Evaluate the trigonometric sum
          constexpr auto series_order_plus_one = SeriesOrder + 1;
          CT const I12 = clenshaw_sum(cos_sig2, coeffs, coeffs + series_order_plus_one)
              - clenshaw_sum(cos_sig1, coeffs, coeffs + series_order_plus_one);
  
          // The part of the ellipsodal correction that depends on
          // point coordinates
          result.ellipsoidal_term = cos_alp0 * sin_alp0 * I12;
  
          return result;
      }
  
      // Check whenever a segment crosses the prime meridian
      // First normalize to [0,360)
      template <typename PointOfSegment>
      static inline bool crosses_prime_meridian(PointOfSegment const& p1,
                                                PointOfSegment const& p2)
      {
          CT const pi
              = geometry::math::pi<CT>();
          CT const two_pi
              = geometry::math::two_pi<CT>();
  
          CT p1_lon = get_as_radian<0>(p1)
                                  - ( floor( get_as_radian<0>(p1) / two_pi )
                                    * two_pi );
          CT p2_lon = get_as_radian<0>(p2)
                                  - ( floor( get_as_radian<0>(p2) / two_pi )
                                    * two_pi );
  
          CT max_lon = (std::max)(p1_lon, p2_lon);
          CT min_lon = (std::min)(p1_lon, p2_lon);
  
          return max_lon > pi && min_lon < pi && max_lon - min_lon > pi;
      }
  
  };
  
  }}} // namespace boost::geometry::formula
  
  
  #endif // BOOST_GEOMETRY_FORMULAS_AREA_FORMULAS_HPP
  