LiXiaoqi/libcarla
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
master
libcarla/include/system/boost/geometry/formulas/karney_inverse.hpp

992 lines
35 KiB
C++
Raw Permalink Normal View History

init
2024-10-18 13:19:59 +08:00
// Boost.Geometry
// Copyright (c) 2018 Adeel Ahmad, Islamabad, Pakistan.
// Contributed and/or modified by Adeel Ahmad, as part of Google Summer of Code 2018 program.
// This file was modified by Oracle on 2019-2021.
// Modifications copyright (c) 2019-2021 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)
// This file is converted from GeographicLib, https://geographiclib.sourceforge.io
// GeographicLib is originally written by Charles Karney.
// Author: Charles Karney (2008-2017)
// Last updated version of GeographicLib: 1.49
// Original copyright notice:
// Copyright (c) Charles Karney (2008-2017) <charles@karney.com> and licensed
// under the MIT/X11 License. For more information, see
// https://geographiclib.sourceforge.io
#ifndef BOOST_GEOMETRY_FORMULAS_KARNEY_INVERSE_HPP
#define BOOST_GEOMETRY_FORMULAS_KARNEY_INVERSE_HPP
#include <boost/math/constants/constants.hpp>
#include <boost/math/special_functions/hypot.hpp>
#include <boost/geometry/util/condition.hpp>
#include <boost/geometry/util/math.hpp>
#include <boost/geometry/util/precise_math.hpp>
#include <boost/geometry/util/series_expansion.hpp>
#include <boost/geometry/util/normalize_spheroidal_coordinates.hpp>
#include <boost/geometry/formulas/flattening.hpp>
#include <boost/geometry/formulas/result_inverse.hpp>
namespace boost { namespace geometry { namespace math {
/*!
\brief The exact difference of two angles reduced to (-180deg, 180deg].
*/
template<typename T>
inline T difference_angle(T const& x, T const& y, T& e)
{
auto res1 = boost::geometry::detail::precise_math::two_sum(
std::remainder(-x, T(360)), std::remainder(y, T(360)));
normalize_azimuth<degree, T>(res1[0]);
// Here y - x = d + t (mod 360), exactly, where d is in (-180,180] and
// abs(t) <= eps (eps = 2^-45 for doubles). The only case where the
// addition of t takes the result outside the range (-180,180] is d = 180
// and t > 0. The case, d = -180 + eps, t = -eps, can't happen, since
// sum_error would have returned the exact result in such a case (i.e., given t = 0).
auto res2 = boost::geometry::detail::precise_math::two_sum(
res1[0] == 180 && res1[1] > 0 ? -180 : res1[0], res1[1]);
e = res2[1];
return res2[0];
}
}}} // namespace boost::geometry::math
namespace boost { namespace geometry { namespace formula
{
namespace se = series_expansion;
namespace detail
{
template <
typename CT,
bool EnableDistance,
bool EnableAzimuth,
bool EnableReverseAzimuth = false,
bool EnableReducedLength = false,
bool EnableGeodesicScale = false,
size_t SeriesOrder = 8
>
class karney_inverse
{
static const bool CalcQuantities = EnableReducedLength || EnableGeodesicScale;
static const bool CalcAzimuths = EnableAzimuth || EnableReverseAzimuth || CalcQuantities;
static const bool CalcFwdAzimuth = EnableAzimuth || CalcQuantities;
static const bool CalcRevAzimuth = EnableReverseAzimuth || CalcQuantities;
public:
typedef result_inverse<CT> result_type;
template <typename T1, typename T2, typename Spheroid>
static inline result_type apply(T1 const& lo1,
T1 const& la1,
T2 const& lo2,
T2 const& la2,
Spheroid const& spheroid)
{
static CT const c0 = 0;
static CT const c0_001 = 0.001;
static CT const c0_1 = 0.1;
static CT const c1 = 1;
static CT const c2 = 2;
static CT const c3 = 3;
static CT const c8 = 8;
static CT const c16 = 16;
static CT const c90 = 90;
static CT const c180 = 180;
static CT const c200 = 200;
static CT const pi = math::pi<CT>();
static CT const d2r = math::d2r<CT>();
static CT const r2d = math::r2d<CT>();
result_type result;
CT lat1 = la1 * r2d;
CT lat2 = la2 * r2d;
CT lon1 = lo1 * r2d;
CT lon2 = lo2 * r2d;
CT const a = CT(get_radius<0>(spheroid));
CT const b = CT(get_radius<2>(spheroid));
CT const f = formula::flattening<CT>(spheroid);
CT const one_minus_f = c1 - f;
CT const two_minus_f = c2 - f;
CT const tol0 = std::numeric_limits<CT>::epsilon();
CT const tol1 = c200 * tol0;
CT const tol2 = sqrt(tol0);
// Check on bisection interval.
CT const tol_bisection = tol0 * tol2;
CT const etol2 = c0_1 * tol2 /
sqrt((std::max)(c0_001, std::abs(f)) * (std::min)(c1, c1 - f / c2) / c2);
CT tiny = std::sqrt((std::numeric_limits<CT>::min)());
CT const n = f / two_minus_f;
CT const e2 = f * two_minus_f;
CT const ep2 = e2 / math::sqr(one_minus_f);
// Compute the longitudinal difference.
CT lon12_error;
CT lon12 = math::difference_angle(lon1, lon2, lon12_error);
int lon12_sign = lon12 >= 0 ? 1 : -1;
// Make points close to the meridian to lie on it.
lon12 = lon12_sign * lon12;
lon12_error = (c180 - lon12) - lon12_sign * lon12_error;
// Convert to radians.
CT lam12 = lon12 * d2r;
CT sin_lam12;
CT cos_lam12;
if (lon12 > c90)
{
math::sin_cos_degrees(lon12_error, sin_lam12, cos_lam12);
cos_lam12 *= -c1;
}
else
{
math::sin_cos_degrees(lon12, sin_lam12, cos_lam12);
}
// Make points close to the equator to lie on it.
lat1 = math::round_angle(std::abs(lat1) > c90 ? c90 : lat1);
lat2 = math::round_angle(std::abs(lat2) > c90 ? c90 : lat2);
// Arrange points in a canonical form, as explained in
// paper, Algorithms for geodesics, Eq. (44):
//
// 0 <= lon12 <= 180
// -90 <= lat1 <= 0
// lat1 <= lat2 <= -lat1
int swap_point = std::abs(lat1) < std::abs(lat2) ? -1 : 1;
if (swap_point < 0)
{
lon12_sign *= -1;
swap(lat1, lat2);
}
// Enforce lat1 to be <= 0.
int lat_sign = lat1 < 0 ? 1 : -1;
lat1 *= lat_sign;
lat2 *= lat_sign;
CT sin_beta1, cos_beta1;
math::sin_cos_degrees(lat1, sin_beta1, cos_beta1);
sin_beta1 *= one_minus_f;
math::normalize_unit_vector<CT>(sin_beta1, cos_beta1);
cos_beta1 = (std::max)(tiny, cos_beta1);
CT sin_beta2, cos_beta2;
math::sin_cos_degrees(lat2, sin_beta2, cos_beta2);
sin_beta2 *= one_minus_f;
math::normalize_unit_vector<CT>(sin_beta2, cos_beta2);
cos_beta2 = (std::max)(tiny, cos_beta2);
// If cos_beta1 < -sin_beta1, then cos_beta2 - cos_beta1 is a
// sensitive measure of the |beta1| - |beta2|. Alternatively,
// (cos_beta1 >= -sin_beta1), abs(sin_beta2) + sin_beta1 is
// a better measure.
// Sometimes these quantities vanish and in that case we
// force beta2 = +/- bet1a exactly.
if (cos_beta1 < -sin_beta1)
{
if (cos_beta1 == cos_beta2)
{
sin_beta2 = sin_beta2 < 0 ? sin_beta1 : -sin_beta1;
}
}
else
{
if (std::abs(sin_beta2) == -sin_beta1)
{
cos_beta2 = cos_beta1;
}
}
CT const dn1 = sqrt(c1 + ep2 * math::sqr(sin_beta1));
CT const dn2 = sqrt(c1 + ep2 * math::sqr(sin_beta2));
CT sigma12;
CT m12x = c0;
CT s12x;
CT M21;
// Index zero element of coeffs_C1 is unused.
se::coeffs_C1<SeriesOrder, CT> const coeffs_C1(n);
bool meridian = lat1 == -90 || sin_lam12 == 0;
CT cos_alpha1, sin_alpha1;
CT cos_alpha2, sin_alpha2;
if (meridian)
{
// Endpoints lie on a single full meridian.
// Point to the target latitude.
cos_alpha1 = cos_lam12;
sin_alpha1 = sin_lam12;
// Heading north at the target.
cos_alpha2 = c1;
sin_alpha2 = c0;
CT sin_sigma1 = sin_beta1;
CT cos_sigma1 = cos_alpha1 * cos_beta1;
CT sin_sigma2 = sin_beta2;
CT cos_sigma2 = cos_alpha2 * cos_beta2;
CT sigma12 = std::atan2((std::max)(c0, cos_sigma1 * sin_sigma2 - sin_sigma1 * cos_sigma2),
cos_sigma1 * cos_sigma2 + sin_sigma1 * sin_sigma2);
CT dummy;
meridian_length(n, ep2, sigma12, sin_sigma1, cos_sigma1, dn1,
sin_sigma2, cos_sigma2, dn2,
cos_beta1, cos_beta2, s12x,
m12x, dummy, result.geodesic_scale,
M21, coeffs_C1);
if (sigma12 < c1 || m12x >= c0)
{
if (sigma12 < c3 * tiny)
{
sigma12 = m12x = s12x = c0;
}
m12x *= b;
s12x *= b;
}
else
{
// m12 < 0, i.e., prolate and too close to anti-podal.
meridian = false;
}
}
CT omega12;
if (!meridian && sin_beta1 == c0 &&
(f <= c0 || lon12_error >= f * c180))
{
// Points lie on the equator.
cos_alpha1 = cos_alpha2 = c0;
sin_alpha1 = sin_alpha2 = c1;
s12x = a * lam12;
sigma12 = omega12 = lam12 / one_minus_f;
m12x = b * sin(sigma12);
if (BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
{
result.geodesic_scale = cos(sigma12);
}
}
else if (!meridian)
{
// If point1 and point2 belong within a hemisphere bounded by a
// meridian and geodesic is neither meridional nor equatorial.
// Find the starting point for Newton's method.
CT dnm = c1;
sigma12 = newton_start(sin_beta1, cos_beta1, dn1,
sin_beta2, cos_beta2, dn2,
lam12, sin_lam12, cos_lam12,
sin_alpha1, cos_alpha1,
sin_alpha2, cos_alpha2,
dnm, coeffs_C1, ep2,
tol1, tol2, etol2,
n, f);
if (sigma12 >= c0)
{
// Short lines case (newton_start sets sin_alpha2, cos_alpha2, dnm).
s12x = sigma12 * b * dnm;
m12x = math::sqr(dnm) * b * sin(sigma12 / dnm);
if (BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
{
result.geodesic_scale = cos(sigma12 / dnm);
}
// Convert to radians.
omega12 = lam12 / (one_minus_f * dnm);
}
else
{
// Apply the Newton's method.
CT sin_sigma1 = c0, cos_sigma1 = c0;
CT sin_sigma2 = c0, cos_sigma2 = c0;
CT eps = c0, diff_omega12 = c0;
// Bracketing range.
CT sin_alpha1a = tiny, cos_alpha1a = c1;
CT sin_alpha1b = tiny, cos_alpha1b = -c1;
size_t iteration = 0;
size_t max_iterations = 20 + std::numeric_limits<size_t>::digits + 10;
for (bool tripn = false, tripb = false;
iteration < max_iterations;
++iteration)
{
CT dv = c0;
CT v = lambda12(sin_beta1, cos_beta1, dn1,
sin_beta2, cos_beta2, dn2,
sin_alpha1, cos_alpha1,
sin_lam12, cos_lam12,
sin_alpha2, cos_alpha2,
sigma12,
sin_sigma1, cos_sigma1,
sin_sigma2, cos_sigma2,
eps, diff_omega12,
iteration < max_iterations,
dv, f, n, ep2, tiny, coeffs_C1);
// Reversed test to allow escape with NaNs.
if (tripb || !(std::abs(v) >= (tripn ? c8 : c1) * tol0))
break;
// Update bracketing values.
if (v > c0 && (iteration > max_iterations ||
cos_alpha1 / sin_alpha1 > cos_alpha1b / sin_alpha1b))
{
sin_alpha1b = sin_alpha1;
cos_alpha1b = cos_alpha1;
}
else if (v < c0 && (iteration > max_iterations ||
cos_alpha1 / sin_alpha1 < cos_alpha1a / sin_alpha1a))
{
sin_alpha1a = sin_alpha1;
cos_alpha1a = cos_alpha1;
}
if (iteration < max_iterations && dv > c0)
{
CT diff_alpha1 = -v / dv;
CT sin_diff_alpha1 = sin(diff_alpha1);
CT cos_diff_alpha1 = cos(diff_alpha1);
CT nsin_alpha1 = sin_alpha1 * cos_diff_alpha1 +
cos_alpha1 * sin_diff_alpha1;
if (nsin_alpha1 > c0 && std::abs(diff_alpha1) < pi)
{
cos_alpha1 = cos_alpha1 * cos_diff_alpha1 - sin_alpha1 * sin_diff_alpha1;
sin_alpha1 = nsin_alpha1;
math::normalize_unit_vector<CT>(sin_alpha1, cos_alpha1);
// In some regimes we don't get quadratic convergence because
// slope -> 0. So use convergence conditions based on epsilon
// instead of sqrt(epsilon).
tripn = std::abs(v) <= c16 * tol0;
continue;
}
}
// Either dv was not positive or updated value was outside legal
// range. Use the midpoint of the bracket as the next estimate.
// This mechanism is not needed for the WGS84 ellipsoid, but it does
// catch problems with more eeccentric ellipsoids. Its efficacy is
// such for the WGS84 test set with the starting guess set to alp1 =
// 90deg:
// the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
// WGS84 and random input: mean = 4.74, sd = 0.99
sin_alpha1 = (sin_alpha1a + sin_alpha1b) / c2;
cos_alpha1 = (cos_alpha1a + cos_alpha1b) / c2;
math::normalize_unit_vector<CT>(sin_alpha1, cos_alpha1);
tripn = false;
tripb = (std::abs(sin_alpha1a - sin_alpha1) + (cos_alpha1a - cos_alpha1) < tol_bisection ||
std::abs(sin_alpha1 - sin_alpha1b) + (cos_alpha1 - cos_alpha1b) < tol_bisection);
}
CT dummy;
se::coeffs_C1<SeriesOrder, CT> const coeffs_C1_eps(eps);
// Ensure that the reduced length and geodesic scale are computed in
// a "canonical" way, with the I2 integral.
meridian_length(eps, ep2, sigma12, sin_sigma1, cos_sigma1, dn1,
sin_sigma2, cos_sigma2, dn2,
cos_beta1, cos_beta2, s12x,
m12x, dummy, result.geodesic_scale,
M21, coeffs_C1_eps);
m12x *= b;
s12x *= b;
}
}
if (swap_point < 0)
{
swap(sin_alpha1, sin_alpha2);
swap(cos_alpha1, cos_alpha2);
swap(result.geodesic_scale, M21);
}
sin_alpha1 *= swap_point * lon12_sign;
cos_alpha1 *= swap_point * lat_sign;
sin_alpha2 *= swap_point * lon12_sign;
cos_alpha2 *= swap_point * lat_sign;
if (BOOST_GEOMETRY_CONDITION(EnableReducedLength))
{
result.reduced_length = m12x;
}
if (BOOST_GEOMETRY_CONDITION(CalcAzimuths))
{
if (BOOST_GEOMETRY_CONDITION(CalcFwdAzimuth))
{
result.azimuth = atan2(sin_alpha1, cos_alpha1);
}
if (BOOST_GEOMETRY_CONDITION(CalcRevAzimuth))
{
result.reverse_azimuth = atan2(sin_alpha2, cos_alpha2);
}
}
if (BOOST_GEOMETRY_CONDITION(EnableDistance))
{
result.distance = s12x;
}
return result;
}
template <typename CoeffsC1>
static inline void meridian_length(CT const& epsilon, CT const& ep2, CT const& sigma12,
CT const& sin_sigma1, CT const& cos_sigma1, CT const& dn1,
CT const& sin_sigma2, CT const& cos_sigma2, CT const& dn2,
CT const& cos_beta1, CT const& cos_beta2,
CT& s12x, CT& m12x, CT& m0,
CT& M12, CT& M21,
CoeffsC1 const& coeffs_C1)
{
static CT const c1 = 1;
CT A12x = 0, J12 = 0;
CT expansion_A1, expansion_A2;
// Evaluate the coefficients for C2.
se::coeffs_C2<SeriesOrder, CT> coeffs_C2(epsilon);
if (BOOST_GEOMETRY_CONDITION(EnableDistance) ||
BOOST_GEOMETRY_CONDITION(EnableReducedLength) ||
BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
{
// Find the coefficients for A1 by computing the
// series expansion using Horner scehme.
expansion_A1 = se::evaluate_A1<SeriesOrder>(epsilon);
if (BOOST_GEOMETRY_CONDITION(EnableReducedLength) ||
BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
{
// Find the coefficients for A2 by computing the
// series expansion using Horner scehme.
expansion_A2 = se::evaluate_A2<SeriesOrder>(epsilon);
A12x = expansion_A1 - expansion_A2;
expansion_A2 += c1;
}
expansion_A1 += c1;
}
if (BOOST_GEOMETRY_CONDITION(EnableDistance))
{
CT B1 = se::sin_cos_series(sin_sigma2, cos_sigma2, coeffs_C1)
- se::sin_cos_series(sin_sigma1, cos_sigma1, coeffs_C1);
s12x = expansion_A1 * (sigma12 + B1);
if (BOOST_GEOMETRY_CONDITION(EnableReducedLength) ||
BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
{
CT B2 = se::sin_cos_series(sin_sigma2, cos_sigma2, coeffs_C2)
- se::sin_cos_series(sin_sigma1, cos_sigma1, coeffs_C2);
J12 = A12x * sigma12 + (expansion_A1 * B1 - expansion_A2 * B2);
}
}
else if (BOOST_GEOMETRY_CONDITION(EnableReducedLength) ||
BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
{
for (size_t i = 1; i <= SeriesOrder; ++i)
{
coeffs_C2[i] = expansion_A1 * coeffs_C1[i] -
expansion_A2 * coeffs_C2[i];
}
J12 = A12x * sigma12 +
(se::sin_cos_series(sin_sigma2, cos_sigma2, coeffs_C2)
- se::sin_cos_series(sin_sigma1, cos_sigma1, coeffs_C2));
}
if (BOOST_GEOMETRY_CONDITION(EnableReducedLength))
{
m0 = A12x;
m12x = dn2 * (cos_sigma1 * sin_sigma2) -
dn1 * (sin_sigma1 * cos_sigma2) -
cos_sigma1 * cos_sigma2 * J12;
}
if (BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
{
CT cos_sigma12 = cos_sigma1 * cos_sigma2 + sin_sigma1 * sin_sigma2;
CT t = ep2 * (cos_beta1 - cos_beta2) *
(cos_beta1 + cos_beta2) / (dn1 + dn2);
M12 = cos_sigma12 + (t * sin_sigma2 - cos_sigma2 * J12) * sin_sigma1 / dn1;
M21 = cos_sigma12 - (t * sin_sigma1 - cos_sigma1 * J12) * sin_sigma2 / dn2;
}
}
/*
Return a starting point for Newton's method in sin_alpha1 and
cos_alpha1 (function value is -1). If Newton's method
doesn't need to be used, return also sin_alpha2 and
cos_alpha2 and function value is sig12.
*/
template <typename CoeffsC1>
static inline CT newton_start(CT const& sin_beta1, CT const& cos_beta1, CT const& dn1,
CT const& sin_beta2, CT const& cos_beta2, CT dn2,
CT const& lam12, CT const& sin_lam12, CT const& cos_lam12,
CT& sin_alpha1, CT& cos_alpha1,
CT& sin_alpha2, CT& cos_alpha2,
CT& dnm, CoeffsC1 const& coeffs_C1, CT const& ep2,
CT const& tol1, CT const& tol2, CT const& etol2, CT const& n,
CT const& f)
{
static CT const c0 = 0;
static CT const c0_01 = 0.01;
static CT const c0_1 = 0.1;
static CT const c0_5 = 0.5;
static CT const c1 = 1;
static CT const c2 = 2;
static CT const c6 = 6;
static CT const c1000 = 1000;
static CT const pi = math::pi<CT>();
CT const one_minus_f = c1 - f;
CT const x_thresh = c1000 * tol2;
// Return a starting point for Newton's method in sin_alpha1
// and cos_alpha1 (function value is -1). If Newton's method
// doesn't need to be used, return also sin_alpha2 and
// cos_alpha2 and function value is sig12.
CT sig12 = -c1;
// bet12 = bet2 - bet1 in [0, pi); beta12a = bet2 + bet1 in (-pi, 0]
CT sin_beta12 = sin_beta2 * cos_beta1 - cos_beta2 * sin_beta1;
CT cos_beta12 = cos_beta2 * cos_beta1 + sin_beta2 * sin_beta1;
CT sin_beta12a = sin_beta2 * cos_beta1 + cos_beta2 * sin_beta1;
bool shortline = cos_beta12 >= c0 && sin_beta12 < c0_5 &&
cos_beta2 * lam12 < c0_5;
CT sin_omega12, cos_omega12;
if (shortline)
{
CT sin_beta_m2 = math::sqr(sin_beta1 + sin_beta2);
sin_beta_m2 /= sin_beta_m2 + math::sqr(cos_beta1 + cos_beta2);
dnm = math::sqrt(c1 + ep2 * sin_beta_m2);
CT omega12 = lam12 / (one_minus_f * dnm);
sin_omega12 = sin(omega12);
cos_omega12 = cos(omega12);
}
else
{
sin_omega12 = sin_lam12;
cos_omega12 = cos_lam12;
}
sin_alpha1 = cos_beta2 * sin_omega12;
cos_alpha1 = cos_omega12 >= c0 ?
sin_beta12 + cos_beta2 * sin_beta1 * math::sqr(sin_omega12) / (c1 + cos_omega12) :
sin_beta12a - cos_beta2 * sin_beta1 * math::sqr(sin_omega12) / (c1 - cos_omega12);
CT sin_sigma12 = boost::math::hypot(sin_alpha1, cos_alpha1);
CT cos_sigma12 = sin_beta1 * sin_beta2 + cos_beta1 * cos_beta2 * cos_omega12;
if (shortline && sin_sigma12 < etol2)
{
sin_alpha2 = cos_beta1 * sin_omega12;
cos_alpha2 = sin_beta12 - cos_beta1 * sin_beta2 *
(cos_omega12 >= c0 ? math::sqr(sin_omega12) /
(c1 + cos_omega12) : c1 - cos_omega12);
math::normalize_unit_vector<CT>(sin_alpha2, cos_alpha2);
// Set return value.
sig12 = atan2(sin_sigma12, cos_sigma12);
}
// Skip astroid calculation if too eccentric.
else if (std::abs(n) > c0_1 ||
cos_sigma12 >= c0 ||
sin_sigma12 >= c6 * std::abs(n) * pi *
math::sqr(cos_beta1))
{
// Nothing to do, zeroth order spherical approximation will do.
}
else
{
// Scale lam12 and bet2 to x, y coordinate system where antipodal
// point is at origin and singular point is at y = 0, x = -1.
CT lambda_scale, beta_scale;
CT y;
volatile CT x;
CT lam12x = atan2(-sin_lam12, -cos_lam12);
if (f >= c0)
{
CT k2 = math::sqr(sin_beta1) * ep2;
CT eps = k2 / (c2 * (c1 + sqrt(c1 + k2)) + k2);
se::coeffs_A3<SeriesOrder, CT> const coeffs_A3(n);
CT const A3 = math::horner_evaluate(eps, coeffs_A3.begin(), coeffs_A3.end());
lambda_scale = f * cos_beta1 * A3 * pi;
beta_scale = lambda_scale * cos_beta1;
x = lam12x / lambda_scale;
y = sin_beta12a / beta_scale;
}
else
{
CT cos_beta12a = cos_beta2 * cos_beta1 - sin_beta2 * sin_beta1;
CT beta12a = atan2(sin_beta12a, cos_beta12a);
CT m12b = c0;
CT m0 = c1;
CT dummy;
meridian_length(n, ep2, pi + beta12a,
sin_beta1, -cos_beta1, dn1,
sin_beta2, cos_beta2, dn2,
cos_beta1, cos_beta2, dummy,
m12b, m0, dummy, dummy, coeffs_C1);
x = -c1 + m12b / (cos_beta1 * cos_beta2 * m0 * pi);
beta_scale = x < -c0_01
? sin_beta12a / x
: -f * math::sqr(cos_beta1) * pi;
lambda_scale = beta_scale / cos_beta1;
y = lam12x / lambda_scale;
}
if (y > -tol1 && x > -c1 - x_thresh)
{
// Strip near cut.
if (f >= c0)
{
sin_alpha1 = (std::min)(c1, -CT(x));
cos_alpha1 = - math::sqrt(c1 - math::sqr(sin_alpha1));
}
else
{
cos_alpha1 = (std::max)(CT(x > -tol1 ? c0 : -c1), CT(x));
sin_alpha1 = math::sqrt(c1 - math::sqr(cos_alpha1));
}
}
else
{
// Solve the astroid problem.
CT k = astroid(CT(x), y);
CT omega12a = lambda_scale * (f >= c0 ? -x * k /
(c1 + k) : -y * (c1 + k) / k);
sin_omega12 = sin(omega12a);
cos_omega12 = -cos(omega12a);
// Update spherical estimate of alpha1 using omgega12 instead of lam12.
sin_alpha1 = cos_beta2 * sin_omega12;
cos_alpha1 = sin_beta12a - cos_beta2 * sin_beta1 *
math::sqr(sin_omega12) / (c1 - cos_omega12);
}
}
// Sanity check on starting guess. Backwards check allows NaN through.
if (!(sin_alpha1 <= c0))
{
math::normalize_unit_vector<CT>(sin_alpha1, cos_alpha1);
}
else
{
sin_alpha1 = c1;
cos_alpha1 = c0;
}
return sig12;
}
/*
Solve the astroid problem using the equation:
κ4 + 2κ3 + (1 − x2 − y 2 )κ2 − 2y 2 κ − y 2 = 0.
For details, please refer to Eq. (65) in,
Geodesics on an ellipsoid of revolution, Charles F.F Karney,
https://arxiv.org/abs/1102.1215
*/
static inline CT astroid(CT const& x, CT const& y)
{
static CT const c0 = 0;
static CT const c1 = 1;
static CT const c2 = 2;
static CT const c3 = 3;
static CT const c4 = 4;
static CT const c6 = 6;
CT k;
CT p = math::sqr(x);
CT q = math::sqr(y);
CT r = (p + q - c1) / c6;
if (!(q == c0 && r <= c0))
{
// Avoid possible division by zero when r = 0 by multiplying
// equations for s and t by r^3 and r, respectively.
CT S = p * q / c4;
CT r2 = math::sqr(r);
CT r3 = r * r2;
// The discriminant of the quadratic equation for T3. This is
// zero on the evolute curve p^(1/3)+q^(1/3) = 1.
CT discriminant = S * (S + c2 * r3);
CT u = r;
if (discriminant >= c0)
{
CT T3 = S + r3;
// Pick the sign on the sqrt to maximize abs(T3). This minimizes
// loss of precision due to cancellation. The result is unchanged
// because of the way the T is used in definition of u.
T3 += T3 < c0 ? -std::sqrt(discriminant) : std::sqrt(discriminant);
CT T = std::cbrt(T3);
// T can be zero; but then r2 / T -> 0.
u += T + (T != c0 ? r2 / T : c0);
}
else
{
CT ang = std::atan2(std::sqrt(-discriminant), -(S + r3));
// There are three possible cube roots. We choose the root which avoids
// cancellation. Note that discriminant < 0 implies that r < 0.
u += c2 * r * cos(ang / c3);
}
CT v = std::sqrt(math::sqr(u) + q);
// Avoid loss of accuracy when u < 0.
CT uv = u < c0 ? q / (v - u) : u + v;
CT w = (uv - q) / (c2 * v);
// Rearrange expression for k to avoid loss of accuracy due to
// subtraction. Division by 0 not possible because uv > 0, w >= 0.
k = uv / (std::sqrt(uv + math::sqr(w)) + w);
}
else // q == 0 && r <= 0
{
// y = 0 with |x| <= 1. Handle this case directly.
// For y small, positive root is k = abs(y)/sqrt(1-x^2).
k = c0;
}
return k;
}
template <typename CoeffsC1>
static inline CT lambda12(CT const& sin_beta1, CT const& cos_beta1, CT const& dn1,
CT const& sin_beta2, CT const& cos_beta2, CT const& dn2,
CT const& sin_alpha1, CT cos_alpha1,
CT const& sin_lam120, CT const& cos_lam120,
CT& sin_alpha2, CT& cos_alpha2,
CT& sigma12,
CT& sin_sigma1, CT& cos_sigma1,
CT& sin_sigma2, CT& cos_sigma2,
CT& eps, CT& diff_omega12,
bool diffp, CT& diff_lam12,
CT const& f, CT const& n, CT const& ep2, CT const& tiny,
CoeffsC1 const& coeffs_C1)
{
static CT const c0 = 0;
static CT const c1 = 1;
static CT const c2 = 2;
CT const one_minus_f = c1 - f;
if (sin_beta1 == c0 && cos_alpha1 == c0)
{
// Break degeneracy of equatorial line.
cos_alpha1 = -tiny;
}
CT sin_alpha0 = sin_alpha1 * cos_beta1;
CT cos_alpha0 = boost::math::hypot(cos_alpha1, sin_alpha1 * sin_beta1);
CT sin_omega1, cos_omega1;
CT sin_omega2, cos_omega2;
CT sin_omega12, cos_omega12;
CT lam12;
sin_sigma1 = sin_beta1;
sin_omega1 = sin_alpha0 * sin_beta1;
cos_sigma1 = cos_omega1 = cos_alpha1 * cos_beta1;
math::normalize_unit_vector<CT>(sin_sigma1, cos_sigma1);
// Enforce symmetries in the case abs(beta2) = -beta1.
// Otherwise, this can yield singularities in the Newton iteration.
// sin(alpha2) * cos(beta2) = sin(alpha0).
sin_alpha2 = cos_beta2 != cos_beta1 ?
sin_alpha0 / cos_beta2 : sin_alpha1;
cos_alpha2 = cos_beta2 != cos_beta1 || std::abs(sin_beta2) != -sin_beta1 ?
sqrt(math::sqr(cos_alpha1 * cos_beta1) +
(cos_beta1 < -sin_beta1 ?
(cos_beta2 - cos_beta1) * (cos_beta1 + cos_beta2) :
(sin_beta1 - sin_beta2) * (sin_beta1 + sin_beta2))) / cos_beta2 :
std::abs(cos_alpha1);
sin_sigma2 = sin_beta2;
sin_omega2 = sin_alpha0 * sin_beta2;
cos_sigma2 = cos_omega2 =
(cos_alpha2 * cos_beta2);
// Break degeneracy of equatorial line.
math::normalize_unit_vector<CT>(sin_sigma2, cos_sigma2);
// sig12 = sig2 - sig1, limit to [0, pi].
sigma12 = atan2((std::max)(c0, cos_sigma1 * sin_sigma2 - sin_sigma1 * cos_sigma2),
cos_sigma1 * cos_sigma2 + sin_sigma1 * sin_sigma2);
// omg12 = omg2 - omg1, limit to [0, pi].
sin_omega12 = (std::max)(c0, cos_omega1 * sin_omega2 - sin_omega1 * cos_omega2);
cos_omega12 = cos_omega1 * cos_omega2 + sin_omega1 * sin_omega2;
// eta = omg12 - lam120.
CT eta = atan2(sin_omega12 * cos_lam120 - cos_omega12 * sin_lam120,
cos_omega12 * cos_lam120 + sin_omega12 * sin_lam120);
CT B312;
CT k2 = math::sqr(cos_alpha0) * ep2;
eps = k2 / (c2 * (c1 + std::sqrt(c1 + k2)) + k2);
se::coeffs_C3<SeriesOrder, CT> const coeffs_C3(n, eps);
B312 = se::sin_cos_series(sin_sigma2, cos_sigma2, coeffs_C3)
- se::sin_cos_series(sin_sigma1, cos_sigma1, coeffs_C3);
se::coeffs_A3<SeriesOrder, CT> const coeffs_A3(n);
CT const A3 = math::horner_evaluate(eps, coeffs_A3.begin(), coeffs_A3.end());
diff_omega12 = -f * A3 * sin_alpha0 * (sigma12 + B312);
lam12 = eta + diff_omega12;
if (diffp)
{
if (cos_alpha2 == c0)
{
diff_lam12 = - c2 * one_minus_f * dn1 / sin_beta1;
}
else
{
CT dummy;
meridian_length(eps, ep2, sigma12, sin_sigma1, cos_sigma1, dn1,
sin_sigma2, cos_sigma2, dn2,
cos_beta1, cos_beta2, dummy,
diff_lam12, dummy, dummy,
dummy, coeffs_C1);
diff_lam12 *= one_minus_f / (cos_alpha2 * cos_beta2);
}
}
return lam12;
}
};
} // namespace detail
/*!
\brief The solution of the inverse problem of geodesics on latlong coordinates,
after Karney (2011).
\author See
- Charles F.F Karney, Algorithms for geodesics, 2011
https://arxiv.org/pdf/1109.4448.pdf
*/
template <
typename CT,
bool EnableDistance,
bool EnableAzimuth,
bool EnableReverseAzimuth = false,
bool EnableReducedLength = false,
bool EnableGeodesicScale = false
>
struct karney_inverse
: detail::karney_inverse
<
CT,
EnableDistance,
EnableAzimuth,
EnableReverseAzimuth,
EnableReducedLength,
EnableGeodesicScale
>
{};
}}} // namespace boost::geometry::formula
#endif // BOOST_GEOMETRY_FORMULAS_KARNEY_INVERSE_HPP
Reference in New Issue Copy Permalink