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// Boost.Geometry
// Copyright (c) 2017 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_ELLIPTIC_ARC_LENGTH_HPP
#define BOOST_GEOMETRY_FORMULAS_ELLIPTIC_ARC_LENGTH_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>
namespace boost { namespace geometry { namespace formula
{
/*!
\brief Compute the arc length of an ellipse.
*/
template <typename CT, unsigned int Order = 1>
class elliptic_arc_length
{
public :
struct result
{
result()
: distance(0)
, meridian(false)
{}
CT distance;
bool meridian;
};
template <typename T>
static bool meridian_not_crossing_pole(T lat1, T lat2, CT diff)
{
CT half_pi = math::pi<CT>()/CT(2);
return math::equals(diff, CT(0)) ||
(math::equals(lat2, half_pi) && math::equals(lat1, -half_pi));
}
static bool meridian_crossing_pole(CT diff)
{
return math::equals(math::abs(diff), math::pi<CT>());
}
template <typename T, typename Spheroid>
static CT meridian_not_crossing_pole_dist(T lat1, T lat2, Spheroid const& spheroid)
{
return math::abs(apply(lat2, spheroid) - apply(lat1, spheroid));
}
template <typename T, typename Spheroid>
static CT meridian_crossing_pole_dist(T lat1, T lat2, Spheroid const& spheroid)
{
CT c0 = 0;
CT half_pi = math::pi<CT>()/CT(2);
CT lat_sign = 1;
if (lat1+lat2 < c0)
{
lat_sign = CT(-1);
}
return math::abs(lat_sign * CT(2) * apply(half_pi, spheroid)
- apply(lat1, spheroid) - apply(lat2, spheroid));
}
template <typename T, typename Spheroid>
static result apply(T lon1, T lat1, T lon2, T lat2, Spheroid const& spheroid)
{
result res;
CT diff = geometry::math::longitude_distance_signed<geometry::radian>(lon1, lon2);
if (lat1 > lat2)
{
std::swap(lat1, lat2);
}
if ( meridian_not_crossing_pole(lat1, lat2, diff) )
{
res.distance = meridian_not_crossing_pole_dist(lat1, lat2, spheroid);
res.meridian = true;
}
else if ( meridian_crossing_pole(diff) )
{
res.distance = meridian_crossing_pole_dist(lat1, lat2, spheroid);
res.meridian = true;
}
return res;
}
// Distance computation on meridians using series approximations
// to elliptic integrals. Formula to compute distance from lattitude 0 to lat
// https://en.wikipedia.org/wiki/Meridian_arc
// latitudes are assumed to be in radians and in [-pi/2,pi/2]
template <typename T, typename Spheroid>
static CT apply(T lat, Spheroid const& spheroid)
{
CT const a = get_radius<0>(spheroid);
CT const f = formula::flattening<CT>(spheroid);
CT n = f / (CT(2) - f);
CT M = a/(1+n);
CT C0 = 1;
if (Order == 0)
{
return M * C0 * lat;
}
CT C2 = -1.5 * n;
if (Order == 1)
{
return M * (C0 * lat + C2 * sin(2*lat));
}
CT n2 = n * n;
C0 += .25 * n2;
CT C4 = 0.9375 * n2;
if (Order == 2)
{
return M * (C0 * lat + C2 * sin(2*lat) + C4 * sin(4*lat));
}
CT n3 = n2 * n;
C2 += 0.1875 * n3;
CT C6 = -0.729166667 * n3;
if (Order == 3)
{
return M * (C0 * lat + C2 * sin(2*lat) + C4 * sin(4*lat)
+ C6 * sin(6*lat));
}
CT n4 = n2 * n2;
C4 -= 0.234375 * n4;
CT C8 = 0.615234375 * n4;
if (Order == 4)
{
return M * (C0 * lat + C2 * sin(2*lat) + C4 * sin(4*lat)
+ C6 * sin(6*lat) + C8 * sin(8*lat));
}
CT n5 = n4 * n;
C6 += 0.227864583 * n5;
CT C10 = -0.54140625 * n5;
// Order 5 or higher
return M * (C0 * lat + C2 * sin(2*lat) + C4 * sin(4*lat)
+ C6 * sin(6*lat) + C8 * sin(8*lat) + C10 * sin(10*lat));
}
// Iterative method to elliptic arc length based on
// http://www.codeguru.com/cpp/cpp/algorithms/article.php/c5115/
// Geographic-Distance-and-Azimuth-Calculations.htm
// latitudes are assumed to be in radians and in [-pi/2,pi/2]
template <typename T1, typename T2, typename Spheroid>
CT interative_method(T1 lat1,
T2 lat2,
Spheroid const& spheroid)
{
CT result = 0;
CT const zero = 0;
CT const one = 1;
CT const c1 = 2;
CT const c2 = 0.5;
CT const c3 = 4000;
CT const a = get_radius<0>(spheroid);
CT const f = formula::flattening<CT>(spheroid);
// how many steps to use
CT lat1_deg = lat1 * geometry::math::r2d<CT>();
CT lat2_deg = lat2 * geometry::math::r2d<CT>();
int steps = c1 + (c2 + (lat2_deg > lat1_deg) ? CT(lat2_deg - lat1_deg)
: CT(lat1_deg - lat2_deg));
steps = (steps > c3) ? c3 : steps;
//std::cout << "Steps=" << steps << std::endl;
CT snLat1 = sin(lat1);
CT snLat2 = sin(lat2);
CT twoF = 2 * f - f * f;
// limits of integration
CT x1 = a * cos(lat1) /
sqrt(1 - twoF * snLat1 * snLat1);
CT x2 = a * cos(lat2) /
sqrt(1 - twoF * snLat2 * snLat2);
CT dx = (x2 - x1) / (steps - one);
CT x, y1, y2, dy, dydx;
CT adx = (dx < zero) ? -dx : dx; // absolute value of dx
CT a2 = a * a;
CT oneF = 1 - f;
// now loop through each step adding up all the little
// hypotenuses
for (int i = 0; i < (steps - 1); i++){
x = x1 + dx * i;
dydx = ((a * oneF * sqrt((one - ((x+dx)*(x+dx))/a2))) -
(a * oneF * sqrt((one - (x*x)/a2)))) / dx;
result += adx * sqrt(one + dydx*dydx);
}
return result;
}
};
}}} // namespace boost::geometry::formula
#endif // BOOST_GEOMETRY_FORMULAS_ELLIPTIC_ARC_LENGTH_HPP
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