// Boost.Geometry
// Copyright (c) 2016-2019 Oracle and/or its affiliates.
// 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_SJOBERG_INTERSECTION_HPP
#define BOOST_GEOMETRY_FORMULAS_SJOBERG_INTERSECTION_HPP
#include <boost/math/constants/constants.hpp>
#include <boost/geometry/core/radius.hpp>
#include <boost/geometry/util/condition.hpp>
#include <boost/geometry/util/math.hpp>
#include <boost/geometry/util/normalize_spheroidal_coordinates.hpp>
#include <boost/geometry/formulas/flattening.hpp>
#include <boost/geometry/formulas/spherical.hpp>
namespace boost { namespace geometry { namespace formula
{
/*!
\brief The intersection of two great circles as proposed by Sjoberg.
\see See
- [Sjoberg02] Lars E. Sjoberg, Intersections on the sphere and ellipsoid, 2002
http://link.springer.com/article/10.1007/s00190-001-0230-9
*/
template <typename CT>
struct sjoberg_intersection_spherical_02
{
// TODO: if it will be used as standalone formula
// support segments on equator and endpoints on poles
static inline bool apply(CT const& lon1, CT const& lat1, CT const& lon_a2, CT const& lat_a2,
CT const& lon2, CT const& lat2, CT const& lon_b2, CT const& lat_b2,
CT & lon, CT & lat)
{
CT tan_lat = 0;
bool res = apply_alt(lon1, lat1, lon_a2, lat_a2,
lon2, lat2, lon_b2, lat_b2,
lon, tan_lat);
if (res)
{
lat = atan(tan_lat);
}
return res;
}
static inline bool apply_alt(CT const& lon1, CT const& lat1, CT const& lon_a2, CT const& lat_a2,
CT const& lon2, CT const& lat2, CT const& lon_b2, CT const& lat_b2,
CT & lon, CT & tan_lat)
{
CT const cos_lon1 = cos(lon1);
CT const sin_lon1 = sin(lon1);
CT const cos_lon2 = cos(lon2);
CT const sin_lon2 = sin(lon2);
CT const sin_lat1 = sin(lat1);
CT const sin_lat2 = sin(lat2);
CT const cos_lat1 = cos(lat1);
CT const cos_lat2 = cos(lat2);
CT const tan_lat_a2 = tan(lat_a2);
CT const tan_lat_b2 = tan(lat_b2);
return apply(lon1, lon_a2, lon2, lon_b2,
sin_lon1, cos_lon1, sin_lat1, cos_lat1,
sin_lon2, cos_lon2, sin_lat2, cos_lat2,
tan_lat_a2, tan_lat_b2,
lon, tan_lat);
}
private:
static inline bool apply(CT const& lon1, CT const& lon_a2, CT const& lon2, CT const& lon_b2,
CT const& sin_lon1, CT const& cos_lon1, CT const& sin_lat1, CT const& cos_lat1,
CT const& sin_lon2, CT const& cos_lon2, CT const& sin_lat2, CT const& cos_lat2,
CT const& tan_lat_a2, CT const& tan_lat_b2,
CT & lon, CT & tan_lat)
{
// NOTE:
// cos_lat_ = 0 <=> segment on equator
// tan_alpha_ = 0 <=> segment vertical
CT const tan_lat1 = sin_lat1 / cos_lat1; //tan(lat1);
CT const tan_lat2 = sin_lat2 / cos_lat2; //tan(lat2);
CT const dlon1 = lon_a2 - lon1;
CT const sin_dlon1 = sin(dlon1);
CT const dlon2 = lon_b2 - lon2;
CT const sin_dlon2 = sin(dlon2);
CT const cos_dlon1 = cos(dlon1);
CT const cos_dlon2 = cos(dlon2);
CT const tan_alpha1_x = cos_lat1 * tan_lat_a2 - sin_lat1 * cos_dlon1;
CT const tan_alpha2_x = cos_lat2 * tan_lat_b2 - sin_lat2 * cos_dlon2;
CT const c0 = 0;
bool const is_vertical1 = math::equals(sin_dlon1, c0) || math::equals(tan_alpha1_x, c0);
bool const is_vertical2 = math::equals(sin_dlon2, c0) || math::equals(tan_alpha2_x, c0);
CT tan_alpha1 = 0;
CT tan_alpha2 = 0;
if (is_vertical1 && is_vertical2)
{
// circles intersect at one of the poles or are collinear
return false;
}
else if (is_vertical1)
{
tan_alpha2 = sin_dlon2 / tan_alpha2_x;
lon = lon1;
}
else if (is_vertical2)
{
tan_alpha1 = sin_dlon1 / tan_alpha1_x;
lon = lon2;
}
else
{
tan_alpha1 = sin_dlon1 / tan_alpha1_x;
tan_alpha2 = sin_dlon2 / tan_alpha2_x;
CT const T1 = tan_alpha1 * cos_lat1;
CT const T2 = tan_alpha2 * cos_lat2;
CT const T1T2 = T1*T2;
CT const tan_lon_y = T1 * sin_lon2 - T2 * sin_lon1 + T1T2 * (tan_lat1 * cos_lon1 - tan_lat2 * cos_lon2);
CT const tan_lon_x = T1 * cos_lon2 - T2 * cos_lon1 - T1T2 * (tan_lat1 * sin_lon1 - tan_lat2 * sin_lon2);
lon = atan2(tan_lon_y, tan_lon_x);
}
// choose closer result
CT const pi = math::pi<CT>();
CT const lon_2 = lon > c0 ? lon - pi : lon + pi;
CT const lon_dist1 = (std::max)((std::min)(math::longitude_difference<radian>(lon1, lon),
math::longitude_difference<radian>(lon_a2, lon)),
(std::min)(math::longitude_difference<radian>(lon2, lon),
math::longitude_difference<radian>(lon_b2, lon)));
CT const lon_dist2 = (std::max)((std::min)(math::longitude_difference<radian>(lon1, lon_2),
math::longitude_difference<radian>(lon_a2, lon_2)),
(std::min)(math::longitude_difference<radian>(lon2, lon_2),
math::longitude_difference<radian>(lon_b2, lon_2)));
if (lon_dist2 < lon_dist1)
{
lon = lon_2;
}
CT const sin_lon = sin(lon);
CT const cos_lon = cos(lon);
if (math::abs(tan_alpha1) >= math::abs(tan_alpha2)) // pick less vertical segment
{
CT const sin_dlon_1 = sin_lon * cos_lon1 - cos_lon * sin_lon1;
CT const cos_dlon_1 = cos_lon * cos_lon1 + sin_lon * sin_lon1;
CT const lat_y_1 = sin_dlon_1 + tan_alpha1 * sin_lat1 * cos_dlon_1;
CT const lat_x_1 = tan_alpha1 * cos_lat1;
tan_lat = lat_y_1 / lat_x_1;
}
else
{
CT const sin_dlon_2 = sin_lon * cos_lon2 - cos_lon * sin_lon2;
CT const cos_dlon_2 = cos_lon * cos_lon2 + sin_lon * sin_lon2;
CT const lat_y_2 = sin_dlon_2 + tan_alpha2 * sin_lat2 * cos_dlon_2;
CT const lat_x_2 = tan_alpha2 * cos_lat2;
tan_lat = lat_y_2 / lat_x_2;
}
return true;
}
};
/*! Approximation of dLambda_j [Sjoberg07], expanded into taylor series in e^2
Maxima script:
dLI_j(c_j, sinB_j, sinB) := integrate(1 / (sqrt(1 - c_j ^ 2 - x ^ 2)*(1 + sqrt(1 - e2*(1 - x ^ 2)))), x, sinB_j, sinB);
dL_j(c_j, B_j, B) := -e2 * c_j * dLI_j(c_j, B_j, B);
S: taylor(dLI_j(c_j, sinB_j, sinB), e2, 0, 3);
assume(c_j < 1);
assume(c_j > 0);
L1: factor(integrate(sqrt(-x ^ 2 - c_j ^ 2 + 1) / (x ^ 2 + c_j ^ 2 - 1), x));
L2: factor(integrate(((x ^ 2 - 1)*sqrt(-x ^ 2 - c_j ^ 2 + 1)) / (x ^ 2 + c_j ^ 2 - 1), x));
L3: factor(integrate(((x ^ 4 - 2 * x ^ 2 + 1)*sqrt(-x ^ 2 - c_j ^ 2 + 1)) / (x ^ 2 + c_j ^ 2 - 1), x));
L4: factor(integrate(((x ^ 6 - 3 * x ^ 4 + 3 * x ^ 2 - 1)*sqrt(-x ^ 2 - c_j ^ 2 + 1)) / (x ^ 2 + c_j ^ 2 - 1), x));
\see See
- [Sjoberg07] Lars E. Sjoberg, Geodetic intersection on the ellipsoid, 2007
http://link.springer.com/article/10.1007/s00190-007-0204-7
*/
template <unsigned int Order, typename CT>
inline CT sjoberg_d_lambda_e_sqr(CT const& sin_betaj, CT const& sin_beta,
CT const& Cj, CT const& sqrt_1_Cj_sqr,
CT const& e_sqr)
{
using math::detail::bounded;
if (Order == 0)
{
return 0;
}
CT const c1 = 1;
CT const c2 = 2;
CT const asin_B = asin(bounded(sin_beta / sqrt_1_Cj_sqr, -c1, c1));
CT const asin_Bj = asin(sin_betaj / sqrt_1_Cj_sqr);
CT const L0 = (asin_B - asin_Bj) / c2;
if (Order == 1)
{
return -Cj * e_sqr * L0;
}
CT const c0 = 0;
CT const c16 = 16;
CT const X = sin_beta;
CT const Xj = sin_betaj;
CT const X_sqr = math::sqr(X);
CT const Xj_sqr = math::sqr(Xj);
CT const Cj_sqr = math::sqr(Cj);
CT const Cj_sqr_plus_one = Cj_sqr + c1;
CT const one_minus_Cj_sqr = c1 - Cj_sqr;
CT const sqrt_Y = math::sqrt(bounded(-X_sqr + one_minus_Cj_sqr, c0));
CT const sqrt_Yj = math::sqrt(-Xj_sqr + one_minus_Cj_sqr);
CT const L1 = (Cj_sqr_plus_one * (asin_B - asin_Bj) + X * sqrt_Y - Xj * sqrt_Yj) / c16;
if (Order == 2)
{
return -Cj * e_sqr * (L0 + e_sqr * L1);
}
CT const c3 = 3;
CT const c5 = 5;
CT const c128 = 128;
CT const E = Cj_sqr * (c3 * Cj_sqr + c2) + c3;
CT const F = X * (-c2 * X_sqr + c3 * Cj_sqr + c5);
CT const Fj = Xj * (-c2 * Xj_sqr + c3 * Cj_sqr + c5);
CT const L2 = (E * (asin_B - asin_Bj) + F * sqrt_Y - Fj * sqrt_Yj) / c128;
if (Order == 3)
{
return -Cj * e_sqr * (L0 + e_sqr * (L1 + e_sqr * L2));
}
CT const c8 = 8;
CT const c9 = 9;
CT const c10 = 10;
CT const c15 = 15;
CT const c24 = 24;
CT const c26 = 26;
CT const c33 = 33;
CT const c6144 = 6144;
CT const G = Cj_sqr * (Cj_sqr * (Cj_sqr * c15 + c9) + c9) + c15;
CT const H = -c10 * Cj_sqr - c26;
CT const I = Cj_sqr * (Cj_sqr * c15 + c24) + c33;
CT const J = X_sqr * (X * (c8 * X_sqr + H)) + X * I;
CT const Jj = Xj_sqr * (Xj * (c8 * Xj_sqr + H)) + Xj * I;
CT const L3 = (G * (asin_B - asin_Bj) + J * sqrt_Y - Jj * sqrt_Yj) / c6144;
// Order 4 and higher
return -Cj * e_sqr * (L0 + e_sqr * (L1 + e_sqr * (L2 + e_sqr * L3)));
}
/*!
\brief The representation of geodesic as proposed by Sjoberg.
\see See
- [Sjoberg07] Lars E. Sjoberg, Geodetic intersection on the ellipsoid, 2007
http://link.springer.com/article/10.1007/s00190-007-0204-7
- [Sjoberg12] Lars E. Sjoberg, Solutions to the ellipsoidal Clairaut constant
and the inverse geodetic problem by numerical integration, 2012
https://www.degruyter.com/view/j/jogs.2012.2.issue-3/v10156-011-0037-4/v10156-011-0037-4.xml
*/
template <typename CT, unsigned int Order>
class sjoberg_geodesic
{
sjoberg_geodesic() {}
static int sign_C(CT const& alphaj)
{
CT const c0 = 0;
CT const c2 = 2;
CT const pi = math::pi<CT>();
CT const pi_half = pi / c2;
return (pi_half < alphaj && alphaj < pi) || (-pi_half < alphaj && alphaj < c0) ? -1 : 1;
}
public:
sjoberg_geodesic(CT const& lon, CT const& lat, CT const& alpha, CT const& f)
: lonj(lon)
, latj(lat)
, alphaj(alpha)
{
CT const c0 = 0;
CT const c1 = 1;
CT const c2 = 2;
//CT const pi = math::pi<CT>();
//CT const pi_half = pi / c2;
one_minus_f = c1 - f;
e_sqr = f * (c2 - f);
tan_latj = tan(lat);
tan_betaj = one_minus_f * tan_latj;
betaj = atan(tan_betaj);
sin_betaj = sin(betaj);
cos_betaj = cos(betaj);
sin_alphaj = sin(alphaj);
// Clairaut constant (lower-case in the paper)
Cj = sign_C(alphaj) * cos_betaj * sin_alphaj;
Cj_sqr = math::sqr(Cj);
sqrt_1_Cj_sqr = math::sqrt(c1 - Cj_sqr);
sign_lon_diff = alphaj >= 0 ? 1 : -1; // || alphaj == -pi ?
//sign_lon_diff = 1;
is_on_equator = math::equals(sqrt_1_Cj_sqr, c0);
is_Cj_zero = math::equals(Cj, c0);
t0j = c0;
asin_tj_t0j = c0;
if (! is_Cj_zero)
{
t0j = sqrt_1_Cj_sqr / Cj;
}
if (! is_on_equator)
{
//asin_tj_t0j = asin(tan_betaj / t0j);
asin_tj_t0j = asin(tan_betaj * Cj / sqrt_1_Cj_sqr);
}
}
struct vertex_data
{
//CT beta0j;
CT sin_beta0j;
CT dL0j;
CT lon0j;
};
vertex_data get_vertex_data() const
{
CT const c2 = 2;
CT const pi = math::pi<CT>();
CT const pi_half = pi / c2;
vertex_data res;
if (! is_Cj_zero)
{
//res.beta0j = atan(t0j);
//res.sin_beta0j = sin(res.beta0j);
res.sin_beta0j = math::sign(t0j) * sqrt_1_Cj_sqr;
res.dL0j = d_lambda(res.sin_beta0j);
res.lon0j = lonj + sign_lon_diff * (pi_half - asin_tj_t0j + res.dL0j);
}
else
{
//res.beta0j = pi_half;
//res.sin_beta0j = betaj >= 0 ? 1 : -1;
res.sin_beta0j = 1;
res.dL0j = 0;
res.lon0j = lonj;
}
return res;
}
bool is_sin_beta_ok(CT const& sin_beta) const
{
CT const c1 = 1;
return math::abs(sin_beta / sqrt_1_Cj_sqr) <= c1;
}
bool k_diff(CT const& sin_beta,
CT & delta_k) const
{
if (is_Cj_zero)
{
delta_k = 0;
return true;
}
// beta out of bounds and not close
if (! (is_sin_beta_ok(sin_beta)
|| math::equals(math::abs(sin_beta), sqrt_1_Cj_sqr)) )
{
return false;
}
// NOTE: beta may be slightly out of bounds here but d_lambda handles that
CT const dLj = d_lambda(sin_beta);
delta_k = sign_lon_diff * (/*asin_t_t0j*/ - asin_tj_t0j + dLj);
return true;
}
bool lon_diff(CT const& sin_beta, CT const& t,
CT & delta_lon) const
{
using math::detail::bounded;
CT const c1 = 1;
if (is_Cj_zero)
{
delta_lon = 0;
return true;
}
CT delta_k = 0;
if (! k_diff(sin_beta, delta_k))
{
return false;
}
CT const t_t0j = t / t0j;
// NOTE: t may be slightly out of bounds here
CT const asin_t_t0j = asin(bounded(t_t0j, -c1, c1));
delta_lon = sign_lon_diff * asin_t_t0j + delta_k;
return true;
}
bool k_diffs(CT const& sin_beta, vertex_data const& vd,
CT & delta_k_before, CT & delta_k_behind,
bool check_sin_beta = true) const
{
CT const pi = math::pi<CT>();
if (is_Cj_zero)
{
delta_k_before = 0;
delta_k_behind = sign_lon_diff * pi;
return true;
}
// beta out of bounds and not close
if (check_sin_beta
&& ! (is_sin_beta_ok(sin_beta)
|| math::equals(math::abs(sin_beta), sqrt_1_Cj_sqr)) )
{
return false;
}
// NOTE: beta may be slightly out of bounds here but d_lambda handles that
CT const dLj = d_lambda(sin_beta);
delta_k_before = sign_lon_diff * (/*asin_t_t0j*/ - asin_tj_t0j + dLj);
// This version require no additional dLj calculation
delta_k_behind = sign_lon_diff * (pi /*- asin_t_t0j*/ - asin_tj_t0j + vd.dL0j + (vd.dL0j - dLj));
// [Sjoberg12]
//CT const dL101 = d_lambda(sin_betaj, vd.sin_beta0j);
// WARNING: the following call might not work if beta was OoB because only the second argument is bounded
//CT const dL_01 = d_lambda(sin_beta, vd.sin_beta0j);
//delta_k_behind = sign_lon_diff * (pi /*- asin_t_t0j*/ - asin_tj_t0j + dL101 + dL_01);
return true;
}
bool lon_diffs(CT const& sin_beta, CT const& t, vertex_data const& vd,
CT & delta_lon_before, CT & delta_lon_behind) const
{
using math::detail::bounded;
CT const c1 = 1;
CT const pi = math::pi<CT>();
if (is_Cj_zero)
{
delta_lon_before = 0;
delta_lon_behind = sign_lon_diff * pi;
return true;
}
CT delta_k_before = 0, delta_k_behind = 0;
if (! k_diffs(sin_beta, vd, delta_k_before, delta_k_behind))
{
return false;
}
CT const t_t0j = t / t0j;
// NOTE: t may be slightly out of bounds here
CT const asin_t_t0j = asin(bounded(t_t0j, -c1, c1));
CT const sign_asin_t_t0j = sign_lon_diff * asin_t_t0j;
delta_lon_before = sign_asin_t_t0j + delta_k_before;
delta_lon_behind = -sign_asin_t_t0j + delta_k_behind;
return true;
}
bool lon(CT const& sin_beta, CT const& t, vertex_data const& vd,
CT & lon_before, CT & lon_behind) const
{
using math::detail::bounded;
CT const c1 = 1;
CT const pi = math::pi<CT>();
if (is_Cj_zero)
{
lon_before = lonj;
lon_behind = lonj + sign_lon_diff * pi;
return true;
}
if (! (is_sin_beta_ok(sin_beta)
|| math::equals(math::abs(sin_beta), sqrt_1_Cj_sqr)) )
{
return false;
}
CT const t_t0j = t / t0j;
CT const asin_t_t0j = asin(bounded(t_t0j, -c1, c1));
CT const dLj = d_lambda(sin_beta);
lon_before = lonj + sign_lon_diff * (asin_t_t0j - asin_tj_t0j + dLj);
lon_behind = vd.lon0j + (vd.lon0j - lon_before);
return true;
}
CT lon(CT const& delta_lon) const
{
return lonj + delta_lon;
}
CT lat(CT const& t) const
{
// t = tan(beta) = (1-f)tan(lat)
return atan(t / one_minus_f);
}
void vertex(CT & lon, CT & lat) const
{
lon = get_vertex_data().lon0j;
if (! is_Cj_zero)
{
lat = sjoberg_geodesic::lat(t0j);
}
else
{
CT const c2 = 2;
lat = math::pi<CT>() / c2;
}
}
CT lon_of_equator_intersection() const
{
CT const c0 = 0;
CT const dLj = d_lambda(c0);
CT const asin_tj_t0j = asin(Cj * tan_betaj / sqrt_1_Cj_sqr);
return lonj - asin_tj_t0j + dLj;
}
CT d_lambda(CT const& sin_beta) const
{
return sjoberg_d_lambda_e_sqr<Order>(sin_betaj, sin_beta, Cj, sqrt_1_Cj_sqr, e_sqr);
}
// [Sjoberg12]
/*CT d_lambda(CT const& sin_beta1, CT const& sin_beta2) const
{
return sjoberg_d_lambda_e_sqr<Order>(sin_beta1, sin_beta2, Cj, sqrt_1_Cj_sqr, e_sqr);
}*/
CT lonj;
CT latj;
CT alphaj;
CT one_minus_f;
CT e_sqr;
CT tan_latj;
CT tan_betaj;
CT betaj;
CT sin_betaj;
CT cos_betaj;
CT sin_alphaj;
CT Cj;
CT Cj_sqr;
CT sqrt_1_Cj_sqr;
int sign_lon_diff;
bool is_on_equator;
bool is_Cj_zero;
CT t0j;
CT asin_tj_t0j;
};
/*!
\brief The intersection of two geodesics as proposed by Sjoberg.
\see See
- [Sjoberg02] Lars E. Sjoberg, Intersections on the sphere and ellipsoid, 2002
http://link.springer.com/article/10.1007/s00190-001-0230-9
- [Sjoberg07] Lars E. Sjoberg, Geodetic intersection on the ellipsoid, 2007
http://link.springer.com/article/10.1007/s00190-007-0204-7
- [Sjoberg12] Lars E. Sjoberg, Solutions to the ellipsoidal Clairaut constant
and the inverse geodetic problem by numerical integration, 2012
https://www.degruyter.com/view/j/jogs.2012.2.issue-3/v10156-011-0037-4/v10156-011-0037-4.xml
*/
template
<
typename CT,
template <typename, bool, bool, bool, bool, bool> class Inverse,
unsigned int Order = 4
>
class sjoberg_intersection
{
typedef sjoberg_geodesic<CT, Order> geodesic_type;
typedef Inverse<CT, false, true, false, false, false> inverse_type;
typedef typename inverse_type::result_type inverse_result;
static bool const enable_02 = true;
static int const max_iterations_02 = 10;
static int const max_iterations_07 = 20;
public:
template <typename T1, typename T2, typename Spheroid>
static inline bool apply(T1 const& lona1, T1 const& lata1,
T1 const& lona2, T1 const& lata2,
T2 const& lonb1, T2 const& latb1,
T2 const& lonb2, T2 const& latb2,
CT & lon, CT & lat,
Spheroid const& spheroid)
{
CT const lon_a1 = lona1;
CT const lat_a1 = lata1;
CT const lon_a2 = lona2;
CT const lat_a2 = lata2;
CT const lon_b1 = lonb1;
CT const lat_b1 = latb1;
CT const lon_b2 = lonb2;
CT const lat_b2 = latb2;
inverse_result const res1 = inverse_type::apply(lon_a1, lat_a1, lon_a2, lat_a2, spheroid);
inverse_result const res2 = inverse_type::apply(lon_b1, lat_b1, lon_b2, lat_b2, spheroid);
return apply(lon_a1, lat_a1, lon_a2, lat_a2, res1.azimuth,
lon_b1, lat_b1, lon_b2, lat_b2, res2.azimuth,
lon, lat, spheroid);
}
// TODO: Currently may not work correctly if one of the endpoints is the pole
template <typename Spheroid>
static inline bool apply(CT const& lon_a1, CT const& lat_a1, CT const& lon_a2, CT const& lat_a2, CT const& alpha_a1,
CT const& lon_b1, CT const& lat_b1, CT const& lon_b2, CT const& lat_b2, CT const& alpha_b1,
CT & lon, CT & lat,
Spheroid const& spheroid)
{
// coordinates in radians
CT const c0 = 0;
CT const c1 = 1;
CT const f = formula::flattening<CT>(spheroid);
CT const one_minus_f = c1 - f;
geodesic_type geod1(lon_a1, lat_a1, alpha_a1, f);
geodesic_type geod2(lon_b1, lat_b1, alpha_b1, f);
// Cj = 1 if on equator <=> sqrt_1_Cj_sqr = 0
// Cj = 0 if vertical <=> sqrt_1_Cj_sqr = 1
if (geod1.is_on_equator && geod2.is_on_equator)
{
return false;
}
else if (geod1.is_on_equator)
{
lon = geod2.lon_of_equator_intersection();
lat = c0;
return true;
}
else if (geod2.is_on_equator)
{
lon = geod1.lon_of_equator_intersection();
lat = c0;
return true;
}
// (lon1 - lon2) normalized to (-180, 180]
CT const lon1_minus_lon2 = math::longitude_distance_signed<radian>(geod2.lonj, geod1.lonj);
// vertical segments
if (geod1.is_Cj_zero && geod2.is_Cj_zero)
{
CT const pi = math::pi<CT>();
// the geodesics are parallel, the intersection point cannot be calculated
if ( math::equals(lon1_minus_lon2, c0)
|| math::equals(lon1_minus_lon2 + (lon1_minus_lon2 < c0 ? pi : -pi), c0) )
{
return false;
}
lon = c0;
// the geodesics intersect at one of the poles
CT const pi_half = pi / CT(2);
CT const abs_lat_a1 = math::abs(lat_a1);
CT const abs_lat_a2 = math::abs(lat_a2);
if (math::equals(abs_lat_a1, abs_lat_a2))
{
lat = pi_half;
}
else
{
// pick the pole closest to one of the points of the first segment
CT const& closer_lat = abs_lat_a1 > abs_lat_a2 ? lat_a1 : lat_a2;
lat = closer_lat >= 0 ? pi_half : -pi_half;
}
return true;
}
CT lon_sph = 0;
// Starting tan(beta)
CT t = 0;
/*if (geod1.is_Cj_zero)
{
CT const k_base = lon1_minus_lon2 + geod2.sign_lon_diff * geod2.asin_tj_t0j;
t = sin(k_base) * geod2.t0j;
lon_sph = vertical_intersection_longitude(geod1.lonj, lon_b1, lon_b2);
}
else if (geod2.is_Cj_zero)
{
CT const k_base = lon1_minus_lon2 - geod1.sign_lon_diff * geod1.asin_tj_t0j;
t = sin(-k_base) * geod1.t0j;
lon_sph = vertical_intersection_longitude(geod2.lonj, lon_a1, lon_a2);
}
else*/
{
// TODO: Consider using betas instead of latitudes.
// Some function calls might be saved this way.
CT tan_lat_sph = 0;
sjoberg_intersection_spherical_02<CT>::apply_alt(lon_a1, lat_a1, lon_a2, lat_a2,
lon_b1, lat_b1, lon_b2, lat_b2,
lon_sph, tan_lat_sph);
// Return for sphere
if (math::equals(f, c0))
{
lon = lon_sph;
lat = atan(tan_lat_sph);
return true;
}
t = one_minus_f * tan_lat_sph; // tan(beta)
}
// TODO: no need to calculate atan here if reduced latitudes were used
// instead of latitudes above, in sjoberg_intersection_spherical_02
CT const beta = atan(t);
if (enable_02 && newton_method(geod1, geod2, beta, t, lon1_minus_lon2, lon_sph, lon, lat))
{
// TODO: Newton's method may return wrong result in some specific cases
// Detected for sphere and nearly sphere, e.g. A=6371228, B=6371227
// and segments s1=(-121 -19,37 8) and s2=(-19 -15,-104 -58)
// It's unclear if this is a bug or a characteristic of this method
// so until this is investigated check if the resulting longitude is
// between endpoints of the segments. It should be since before calling
// this formula sides of endpoints WRT other segments are checked.
if ( is_result_longitude_ok(geod1, lon_a1, lon_a2, lon)
&& is_result_longitude_ok(geod2, lon_b1, lon_b2, lon) )
{
return true;
}
}
return converge_07(geod1, geod2, beta, t, lon1_minus_lon2, lon_sph, lon, lat);
}
private:
static inline bool newton_method(geodesic_type const& geod1, geodesic_type const& geod2, // in
CT beta, CT t, CT const& lon1_minus_lon2, CT const& lon_sph, // in
CT & lon, CT & lat) // out
{
CT const c0 = 0;
CT const c1 = 1;
CT const e_sqr = geod1.e_sqr;
CT lon1_diff = 0;
CT lon2_diff = 0;
// The segment is vertical and intersection point is behind the vertex
// this method is unable to calculate correct result
if (geod1.is_Cj_zero && math::abs(geod1.lonj - lon_sph) > math::half_pi<CT>())
return false;
if (geod2.is_Cj_zero && math::abs(geod2.lonj - lon_sph) > math::half_pi<CT>())
return false;
CT abs_dbeta_last = 0;
// [Sjoberg02] converges faster than solution in [Sjoberg07]
// Newton-Raphson method
for (int i = 0; i < max_iterations_02; ++i)
{
CT const cos_beta = cos(beta);
if (math::equals(cos_beta, c0))
{
return false;
}
CT const sin_beta = sin(beta);
CT const cos_beta_sqr = math::sqr(cos_beta);
CT const G = c1 - e_sqr * cos_beta_sqr;
CT f1 = 0;
CT f2 = 0;
if (!geod1.is_Cj_zero)
{
bool is_beta_ok = geod1.lon_diff(sin_beta, t, lon1_diff);
if (is_beta_ok)
{
CT const H = cos_beta_sqr - geod1.Cj_sqr;
if (math::equals(H, c0))
{
return false;
}
f1 = geod1.Cj / cos_beta * math::sqrt(G / H);
}
else
{
return false;
}
}
if (!geod2.is_Cj_zero)
{
bool is_beta_ok = geod2.lon_diff(sin_beta, t, lon2_diff);
if (is_beta_ok)
{
CT const H = cos_beta_sqr - geod2.Cj_sqr;
if (math::equals(H, c0))
{
// NOTE: This may happen for segment nearly
// at the equator. Detected for (radian):
// (-0.0872665 -0.0872665, -0.0872665 0.0872665)
// x
// (0 1.57e-07, -0.392699 1.57e-07)
return false;
}
f2 = geod2.Cj / cos_beta * math::sqrt(G / H);
}
else
{
return false;
}
}
// NOTE: Things may go wrong if the IP is near the vertex
// 1. May converge into the wrong direction (from the other way around).
// This happens when the starting point is on the other side than the vertex
// 2. During converging may "jump" into the other side of the vertex.
// In this case sin_beta/sqrt_1_Cj_sqr and t/t0j is not in [-1, 1]
// 3. f1-f2 may be 0 which means that the intermediate point is on the vertex
// In this case it's not possible to check if this is the correct result
// 4. f1-f2 may also be 0 in other cases, e.g.
// geodesics are symmetrical wrt equator and longitude directions are different
CT const dbeta_denom = f1 - f2;
//CT const dbeta_denom = math::abs(f1) + math::abs(f2);
if (math::equals(dbeta_denom, c0))
{
return false;
}
// The sign of dbeta is changed WRT [Sjoberg02]
CT const dbeta = (lon1_minus_lon2 + lon1_diff - lon2_diff) / dbeta_denom;
CT const abs_dbeta = math::abs(dbeta);
if (i > 0 && abs_dbeta > abs_dbeta_last)
{
// The algorithm is not converging
// The intersection may be on the other side of the vertex
return false;
}
abs_dbeta_last = abs_dbeta;
if (math::equals(dbeta, c0))
{
// Result found
break;
}
// Because the sign of dbeta is changed WRT [Sjoberg02] dbeta is subtracted here
beta = beta - dbeta;
t = tan(beta);
}
lat = geod1.lat(t);
// NOTE: if Cj is 0 then the result is lonj or lonj+180
lon = ! geod1.is_Cj_zero
? geod1.lon(lon1_diff)
: geod2.lon(lon2_diff);
return true;
}
static inline bool is_result_longitude_ok(geodesic_type const& geod,
CT const& lon1, CT const& lon2, CT const& lon)
{
CT const c0 = 0;
if (geod.is_Cj_zero)
return true; // don't check vertical segment
CT dist1p = math::longitude_distance_signed<radian>(lon1, lon);
CT dist12 = math::longitude_distance_signed<radian>(lon1, lon2);
if (dist12 < c0)
{
dist1p = -dist1p;
dist12 = -dist12;
}
return (c0 <= dist1p && dist1p <= dist12)
|| math::equals(dist1p, c0)
|| math::equals(dist1p, dist12);
}
struct geodesics_type
{
geodesics_type(geodesic_type const& g1, geodesic_type const& g2)
: geod1(g1)
, geod2(g2)
, vertex1(geod1.get_vertex_data())
, vertex2(geod2.get_vertex_data())
{}
geodesic_type const& geod1;
geodesic_type const& geod2;
typename geodesic_type::vertex_data vertex1;
typename geodesic_type::vertex_data vertex2;
};
struct converge_07_result
{
converge_07_result()
: lon1(0), lon2(0), k1_diff(0), k2_diff(0), t1(0), t2(0)
{}
CT lon1, lon2;
CT k1_diff, k2_diff;
CT t1, t2;
};
static inline bool converge_07(geodesic_type const& geod1, geodesic_type const& geod2,
CT beta, CT t,
CT const& lon1_minus_lon2, CT const& lon_sph,
CT & lon, CT & lat)
{
//CT const c0 = 0;
//CT const c1 = 1;
//CT const c2 = 2;
//CT const pi = math::pi<CT>();
geodesics_type geodesics(geod1, geod2);
converge_07_result result;
// calculate first pair of longitudes
if (!converge_07_step_one(CT(sin(beta)), t, lon1_minus_lon2, geodesics, lon_sph, result, false))
{
return false;
}
int t_direction = 0;
CT lon_diff_prev = math::longitude_difference<radian>(result.lon1, result.lon2);
// [Sjoberg07]
for (int i = 2; i < max_iterations_07; ++i)
{
// pick t candidates from previous result based on dir
CT t_cand1 = result.t1;
CT t_cand2 = result.t2;
// if direction is 0 the closer one is the first
if (t_direction < 0)
{
t_cand1 = (std::min)(result.t1, result.t2);
t_cand2 = (std::max)(result.t1, result.t2);
}
else if (t_direction > 0)
{
t_cand1 = (std::max)(result.t1, result.t2);
t_cand2 = (std::min)(result.t1, result.t2);
}
else
{
t_direction = t_cand1 < t_cand2 ? -1 : 1;
}
CT t1 = t;
CT beta1 = beta;
// check if the further calculation is needed
if (converge_07_update(t1, beta1, t_cand1))
{
break;
}
bool try_t2 = false;
converge_07_result result_curr;
if (converge_07_step_one(CT(sin(beta1)), t1, lon1_minus_lon2, geodesics, lon_sph, result_curr))
{
CT const lon_diff1 = math::longitude_difference<radian>(result_curr.lon1, result_curr.lon2);
if (lon_diff_prev > lon_diff1)
{
t = t1;
beta = beta1;
lon_diff_prev = lon_diff1;
result = result_curr;
}
else if (t_cand1 != t_cand2)
{
try_t2 = true;
}
else
{
// the result is not fully correct but it won't be more accurate
break;
}
}
// ! converge_07_step_one
else
{
if (t_cand1 != t_cand2)
{
try_t2 = true;
}
else
{
return false;
}
}
if (try_t2)
{
CT t2 = t;
CT beta2 = beta;
// check if the further calculation is needed
if (converge_07_update(t2, beta2, t_cand2))
{
break;
}
if (! converge_07_step_one(CT(sin(beta2)), t2, lon1_minus_lon2, geodesics, lon_sph, result_curr))
{
return false;
}
CT const lon_diff2 = math::longitude_difference<radian>(result_curr.lon1, result_curr.lon2);
if (lon_diff_prev > lon_diff2)
{
t_direction *= -1;
t = t2;
beta = beta2;
lon_diff_prev = lon_diff2;
result = result_curr;
}
else
{
// the result is not fully correct but it won't be more accurate
break;
}
}
}
lat = geod1.lat(t);
lon = ! geod1.is_Cj_zero ? result.lon1 : result.lon2;
math::normalize_longitude<radian>(lon);
return true;
}
static inline bool converge_07_update(CT & t, CT & beta, CT const& t_new)
{
CT const c0 = 0;
CT const beta_new = atan(t_new);
CT const dbeta = beta_new - beta;
beta = beta_new;
t = t_new;
return math::equals(dbeta, c0);
}
static inline CT const& pick_t(CT const& t1, CT const& t2, int direction)
{
return direction < 0 ? (std::min)(t1, t2) : (std::max)(t1, t2);
}
static inline bool converge_07_step_one(CT const& sin_beta,
CT const& t,
CT const& lon1_minus_lon2,
geodesics_type const& geodesics,
CT const& lon_sph,
converge_07_result & result,
bool check_sin_beta = true)
{
bool ok = converge_07_one_geod(sin_beta, t, geodesics.geod1, geodesics.vertex1, lon_sph,
result.lon1, result.k1_diff, check_sin_beta)
&& converge_07_one_geod(sin_beta, t, geodesics.geod2, geodesics.vertex2, lon_sph,
result.lon2, result.k2_diff, check_sin_beta);
if (!ok)
{
return false;
}
CT const k = lon1_minus_lon2 + result.k1_diff - result.k2_diff;
// get 2 possible ts one lesser and one greater than t
// t1 is the closer one
calc_ts(t, k, geodesics.geod1, geodesics.geod2, result.t1, result.t2);
return true;
}
static inline bool converge_07_one_geod(CT const& sin_beta, CT const& t,
geodesic_type const& geod,
typename geodesic_type::vertex_data const& vertex,
CT const& lon_sph,
CT & lon, CT & k_diff,
bool check_sin_beta)
{
using math::detail::bounded;
CT const c1 = 1;
CT k_diff_before = 0;
CT k_diff_behind = 0;
bool is_beta_ok = geod.k_diffs(sin_beta, vertex, k_diff_before, k_diff_behind, check_sin_beta);
if (! is_beta_ok)
{
return false;
}
CT const asin_t_t0j = ! geod.is_Cj_zero ? asin(bounded(t / geod.t0j, -c1, c1)) : 0;
CT const sign_asin_t_t0j = geod.sign_lon_diff * asin_t_t0j;
CT const lon_before = geod.lonj + sign_asin_t_t0j + k_diff_before;
CT const lon_behind = geod.lonj - sign_asin_t_t0j + k_diff_behind;
CT const lon_dist_before = math::longitude_distance_signed<radian>(lon_before, lon_sph);
CT const lon_dist_behind = math::longitude_distance_signed<radian>(lon_behind, lon_sph);
if (math::abs(lon_dist_before) <= math::abs(lon_dist_behind))
{
k_diff = k_diff_before;
lon = lon_before;
}
else
{
k_diff = k_diff_behind;
lon = lon_behind;
}
return true;
}
static inline void calc_ts(CT const& t, CT const& k,
geodesic_type const& geod1, geodesic_type const& geod2,
CT & t1, CT& t2)
{
CT const c0 = 0;
CT const c1 = 1;
CT const c2 = 2;
CT const K = sin(k);
BOOST_GEOMETRY_ASSERT(!geod1.is_Cj_zero || !geod2.is_Cj_zero);
if (geod1.is_Cj_zero)
{
t1 = K * geod2.t0j;
t2 = -t1;
}
else if (geod2.is_Cj_zero)
{
t1 = -K * geod1.t0j;
t2 = -t1;
}
else
{
CT const A = math::sqr(geod1.t0j) + math::sqr(geod2.t0j);
CT const B = c2 * geod1.t0j * geod2.t0j * math::sqrt(c1 - math::sqr(K));
CT const K_t01_t02 = K * geod1.t0j * geod2.t0j;
CT const D1 = math::sqrt(A + B);
CT const D2 = math::sqrt(A - B);
CT const t_new1 = math::equals(D1, c0) ? c0 : K_t01_t02 / D1;
CT const t_new2 = math::equals(D2, c0) ? c0 : K_t01_t02 / D2;
CT const t_new3 = -t_new1;
CT const t_new4 = -t_new2;
// Pick 2 nearest t_new, one greater and one lesser than current t
CT const abs_t_new1 = math::abs(t_new1);
CT const abs_t_new2 = math::abs(t_new2);
CT const abs_t_max = (std::max)(abs_t_new1, abs_t_new2);
t1 = -abs_t_max; // lesser
t2 = abs_t_max; // greater
if (t1 < t)
{
if (t_new1 < t && t_new1 > t1)
t1 = t_new1;
if (t_new2 < t && t_new2 > t1)
t1 = t_new2;
if (t_new3 < t && t_new3 > t1)
t1 = t_new3;
if (t_new4 < t && t_new4 > t1)
t1 = t_new4;
}
if (t2 > t)
{
if (t_new1 > t && t_new1 < t2)
t2 = t_new1;
if (t_new2 > t && t_new2 < t2)
t2 = t_new2;
if (t_new3 > t && t_new3 < t2)
t2 = t_new3;
if (t_new4 > t && t_new4 < t2)
t2 = t_new4;
}
}
// the first one is the closer one
if (math::abs(t - t2) < math::abs(t - t1))
{
std::swap(t2, t1);
}
}
static inline CT fj(CT const& cos_beta, CT const& cos2_beta, CT const& Cj, CT const& e_sqr)
{
CT const c1 = 1;
CT const Cj_sqr = math::sqr(Cj);
return Cj / cos_beta * math::sqrt((c1 - e_sqr * cos2_beta) / (cos2_beta - Cj_sqr));
}
/*static inline CT vertical_intersection_longitude(CT const& ip_lon, CT const& seg_lon1, CT const& seg_lon2)
{
CT const c0 = 0;
CT const lon_2 = ip_lon > c0 ? ip_lon - pi : ip_lon + pi;
return (std::min)(math::longitude_difference<radian>(ip_lon, seg_lon1),
math::longitude_difference<radian>(ip_lon, seg_lon2))
<=
(std::min)(math::longitude_difference<radian>(lon_2, seg_lon1),
math::longitude_difference<radian>(lon_2, seg_lon2))
? ip_lon : lon_2;
}*/
};
}}} // namespace boost::geometry::formula
#endif // BOOST_GEOMETRY_FORMULAS_SJOBERG_INTERSECTION_HPP