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+// Boost.Geometry
+
+// Copyright (c) 2015-2016 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Vissarion Fysikopoulos, 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/formulas/flattening.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 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 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 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 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[]
+ */
+ static inline void evaluate_coeffs_var2(CT var2,
+ CT 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) : 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 AzimuthStrategy,
+ typename PointOfSegment,
+ typename SpheroidConst
+ >
+ static inline return_type_ellipsoidal ellipsoidal(PointOfSegment const& p1,
+ PointOfSegment const& p2,
+ SpheroidConst spheroid_const)
+ {
+ return_type_ellipsoidal result;
+
+ // Azimuth Approximation
+
+ typedef Inverse<CT, false, true, true, false, false> inverse_type;
+ typedef typename inverse_type::result_type inverse_result;
+
+ inverse_result i_res = inverse_type::apply(get_as_radian<0>(p1),
+ get_as_radian<1>(p1),
+ get_as_radian<0>(p2),
+ get_as_radian<1>(p2),
+ spheroid_const.m_spheroid);
+
+ CT alp1 = i_res.azimuth;
+ CT alp2 = i_res.reverse_azimuth;
+
+ // Constants
+
+ CT const ep = spheroid_const.m_ep;
+ CT const f = formula::flattening<CT>(spheroid_const.m_spheroid);
+ CT const one_minus_f = CT(1) - f;
+ std::size_t const series_order_plus_one = SeriesOrder + 1;
+ std::size_t const series_order_plus_two = SeriesOrder + 2;
+
+ // Basic trigonometric computations
+
+ CT tan_bet1 = tan(get_as_radian<1>(p1)) * one_minus_f;
+ CT tan_bet2 = tan(get_as_radian<1>(p2)) * one_minus_f;
+ CT cos_bet1 = cos(atan(tan_bet1));
+ CT cos_bet2 = cos(atan(tan_bet2));
+ CT sin_bet1 = tan_bet1 * cos_bet1;
+ CT sin_bet2 = tan_bet2 * cos_bet2;
+ CT sin_alp1 = sin(alp1);
+ CT cos_alp1 = cos(alp1);
+ CT cos_alp2 = cos(alp2);
+ CT sin_alp0 = sin_alp1 * cos_bet1;
+
+ // Spherical term computation
+
+ CT sin_omg1 = sin_alp0 * sin_bet1;
+ CT cos_omg1 = cos_alp1 * cos_bet1;
+ CT sin_omg2 = sin_alp0 * sin_bet2;
+ CT cos_omg2 = cos_alp2 * cos_bet2;
+ CT cos_omg12 = cos_omg1 * cos_omg2 + sin_omg1 * sin_omg2;
+ CT excess;
+
+ bool meridian = get<0>(p2) - get<0>(p1) == CT(0)
+ || get<1>(p1) == CT(90) || get<1>(p1) == -CT(90)
+ || get<1>(p2) == CT(90) || get<1>(p2) == -CT(90);
+
+ if (!meridian && cos_omg12 > -CT(0.7)
+ && sin_bet2 - sin_bet1 < CT(1.75)) // short segment
+ {
+ 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));
+ }
+ else
+ {
+ /*
+ CT sin_alp2 = sin(alp2);
+ CT sin_alp12 = sin_alp2 * cos_alp1 - cos_alp2 * sin_alp1;
+ CT cos_alp12 = cos_alp2 * cos_alp1 + sin_alp2 * sin_alp1;
+ excess = atan2(sin_alp12, cos_alp12);
+ */
+ excess = alp2 - alp1;
+ }
+
+ result.spherical_term = excess;
+
+ // Ellipsoidal term computation (uses integral approximation)
+
+ CT cos_alp0 = math::sqrt(CT(1) - math::sqr(sin_alp0));
+ 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];
+ const std::size_t coeffs_var_size = (series_order_plus_two
+ * series_order_plus_one) / 2;
+ CT coeffs_var[coeffs_var_size];
+
+ if(ExpandEpsN){ // expand by eps and n
+
+ CT k2 = math::sqr(ep * cos_alp0);
+ CT sqrt_k2_plus_one = math::sqrt(CT(1) + k2);
+ CT eps = (sqrt_k2_plus_one - CT(1)) / (sqrt_k2_plus_one + CT(1));
+ CT n = f / (CT(2) - f);
+
+ // Generate and evaluate the polynomials on n
+ // to get the series coefficients (that depend on eps)
+ evaluate_coeffs_n(n, coeffs_var);
+
+ // Generate and evaluate the polynomials on eps (i.e. var2 = eps)
+ // to get the final series coefficients
+ evaluate_coeffs_var2(eps, coeffs_var, coeffs);
+
+ }else{ // expand by k2 and ep
+
+ CT k2 = math::sqr(ep * cos_alp0);
+ CT ep2 = math::sqr(ep);
+
+ // 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
+ CT 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;
+ }
+
+ // Keep track whenever a segment crosses the prime meridian
+ // First normalize to [0,360)
+ template <typename PointOfSegment, typename StateType>
+ static inline int crosses_prime_meridian(PointOfSegment const& p1,
+ PointOfSegment const& p2,
+ StateType& state)
+ {
+ 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);
+
+ if(max_lon > pi && min_lon < pi && max_lon - min_lon > pi)
+ {
+ state.m_crosses_prime_meridian++;
+ }
+
+ return state.m_crosses_prime_meridian;
+ }
+
+};
+
+}}} // namespace boost::geometry::formula
+
+
+#endif // BOOST_GEOMETRY_FORMULAS_AREA_FORMULAS_HPP