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libcarla/include/system/boost/geometry/formulas/area_formulas.hpp

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2024-10-18 13:19:59 +08:00
// 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
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