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// Boost.Geometry
// Copyright (c) 2016-2017 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_THOMAS_DIRECT_HPP
#define BOOST_GEOMETRY_FORMULAS_THOMAS_DIRECT_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/formulas/differential_quantities.hpp>
#include <boost/geometry/formulas/flattening.hpp>
#include <boost/geometry/formulas/result_direct.hpp>
namespace boost { namespace geometry { namespace formula
{
/*!
\brief The solution of the direct problem of geodesics on latlong coordinates,
Forsyth-Andoyer-Lambert type approximation with second order terms.
\author See
- Technical Report: PAUL D. THOMAS, MATHEMATICAL MODELS FOR NAVIGATION SYSTEMS, 1965
http://www.dtic.mil/docs/citations/AD0627893
- Technical Report: PAUL D. THOMAS, SPHEROIDAL GEODESICS, REFERENCE SYSTEMS, AND LOCAL GEOMETRY, 1970
http://www.dtic.mil/docs/citations/AD0703541
*/
template <
typename CT,
bool EnableCoordinates = true,
bool EnableReverseAzimuth = false,
bool EnableReducedLength = false,
bool EnableGeodesicScale = false
>
class thomas_direct
{
static const bool CalcQuantities = EnableReducedLength || EnableGeodesicScale;
static const bool CalcCoordinates = EnableCoordinates || CalcQuantities;
static const bool CalcRevAzimuth = EnableReverseAzimuth || CalcCoordinates || CalcQuantities;
public:
typedef result_direct<CT> result_type;
template <typename T, typename Dist, typename Azi, typename Spheroid>
static inline result_type apply(T const& lo1,
T const& la1,
Dist const& distance,
Azi const& azimuth12,
Spheroid const& spheroid)
{
result_type result;
CT const lon1 = lo1;
CT const lat1 = la1;
if ( math::equals(distance, Dist(0)) || distance < Dist(0) )
{
result.lon2 = lon1;
result.lat2 = lat1;
return result;
}
CT const c0 = 0;
CT const c1 = 1;
CT const c2 = 2;
CT const c4 = 4;
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 pi = math::pi<CT>();
CT const pi_half = pi / c2;
// keep azimuth small - experiments show low accuracy
// if the azimuth is closer to (+-)180 deg.
CT azi12_alt = azimuth12;
CT lat1_alt = lat1;
bool alter_result = vflip_if_south(lat1, azimuth12, lat1_alt, azi12_alt);
CT const theta1 = math::equals(lat1_alt, pi_half) ? lat1_alt :
math::equals(lat1_alt, -pi_half) ? lat1_alt :
atan(one_minus_f * tan(lat1_alt));
CT const sin_theta1 = sin(theta1);
CT const cos_theta1 = cos(theta1);
CT const sin_a12 = sin(azi12_alt);
CT const cos_a12 = cos(azi12_alt);
CT const M = cos_theta1 * sin_a12; // cos_theta0
CT const theta0 = acos(M);
CT const sin_theta0 = sin(theta0);
CT const N = cos_theta1 * cos_a12;
CT const C1 = f * M; // lower-case c1 in the technical report
CT const C2 = f * (c1 - math::sqr(M)) / c4; // lower-case c2 in the technical report
CT const D = (c1 - C2) * (c1 - C2 - C1 * M);
CT const P = C2 * (c1 + C1 * M / c2) / D;
// special case for equator:
// sin_theta0 = 0 <=> lat1 = 0 ^ |azimuth12| = pi/2
// NOTE: in this case it doesn't matter what's the value of cos_sigma1 because
// theta1=0, theta0=0, M=1|-1, C2=0 so X=0 and Y=0 so d_sigma=d
// cos_a12=0 so N=0, therefore
// lat2=0, azi21=pi/2|-pi/2
// d_eta = atan2(sin_d_sigma, cos_d_sigma)
// H = C1 * d_sigma
CT const cos_sigma1 = math::equals(sin_theta0, c0)
? c1
: normalized1_1(sin_theta1 / sin_theta0);
CT const sigma1 = acos(cos_sigma1);
CT const d = distance / (a * D);
CT const u = 2 * (sigma1 - d);
CT const cos_d = cos(d);
CT const sin_d = sin(d);
CT const cos_u = cos(u);
CT const sin_u = sin(u);
CT const W = c1 - c2 * P * cos_u;
CT const V = cos_u * cos_d - sin_u * sin_d;
CT const X = math::sqr(C2) * sin_d * cos_d * (2 * math::sqr(V) - c1);
CT const Y = c2 * P * V * W * sin_d;
CT const d_sigma = d + X - Y;
CT const sin_d_sigma = sin(d_sigma);
CT const cos_d_sigma = cos(d_sigma);
if (BOOST_GEOMETRY_CONDITION(CalcRevAzimuth))
{
result.reverse_azimuth = atan2(M, N * cos_d_sigma - sin_theta1 * sin_d_sigma);
if (alter_result)
{
vflip_rev_azi(result.reverse_azimuth, azimuth12);
}
}
if (BOOST_GEOMETRY_CONDITION(CalcCoordinates))
{
CT const S_sigma = c2 * sigma1 - d_sigma;
CT const cos_S_sigma = cos(S_sigma);
CT const d_eta = atan2(sin_d_sigma * sin_a12, cos_theta1 * cos_d_sigma - sin_theta1 * sin_d_sigma * cos_a12);
CT const H = C1 * (c1 - C2) * d_sigma - C1 * C2 * sin_d_sigma * cos_S_sigma;
CT const d_lambda = d_eta - H;
result.lon2 = lon1 + d_lambda;
if (! math::equals(M, c0))
{
CT const sin_a21 = sin(result.reverse_azimuth);
CT const tan_theta2 = (sin_theta1 * cos_d_sigma + N * sin_d_sigma) * sin_a21 / M;
result.lat2 = atan(tan_theta2 / one_minus_f);
}
else
{
CT const sigma2 = S_sigma - sigma1;
//theta2 = asin(cos(sigma2)) <=> sin_theta0 = 1
// NOTE: cos(sigma2) defines the sign of tan_theta2
CT const tan_theta2 = cos(sigma2) / math::abs(sin(sigma2));
result.lat2 = atan(tan_theta2 / one_minus_f);
}
if (alter_result)
{
result.lat2 = -result.lat2;
}
}
if (BOOST_GEOMETRY_CONDITION(CalcQuantities))
{
typedef differential_quantities<CT, EnableReducedLength, EnableGeodesicScale, 2> quantities;
quantities::apply(lon1, lat1, result.lon2, result.lat2,
azimuth12, result.reverse_azimuth,
b, f,
result.reduced_length, result.geodesic_scale);
}
return result;
}
private:
static inline bool vflip_if_south(CT const& lat1, CT const& azi12, CT & lat1_alt, CT & azi12_alt)
{
CT const c2 = 2;
CT const pi = math::pi<CT>();
CT const pi_half = pi / c2;
if (azi12 > pi_half)
{
azi12_alt = pi - azi12;
lat1_alt = -lat1;
return true;
}
else if (azi12 < -pi_half)
{
azi12_alt = -pi - azi12;
lat1_alt = -lat1;
return true;
}
return false;
}
static inline void vflip_rev_azi(CT & rev_azi, CT const& azimuth12)
{
CT const c0 = 0;
CT const pi = math::pi<CT>();
if (rev_azi == c0)
{
rev_azi = azimuth12 >= 0 ? pi : -pi;
}
else if (rev_azi > c0)
{
rev_azi = pi - rev_azi;
}
else
{
rev_azi = -pi - rev_azi;
}
}
static inline CT normalized1_1(CT const& value)
{
CT const c1 = 1;
return value > c1 ? c1 :
value < -c1 ? -c1 :
value;
}
};
}}} // namespace boost::geometry::formula
#endif // BOOST_GEOMETRY_FORMULAS_THOMAS_DIRECT_HPP
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