summaryrefslogtreecommitdiff
path: root/boost/geometry/formulas/differential_quantities.hpp
diff options
context:
space:
mode:
Diffstat (limited to 'boost/geometry/formulas/differential_quantities.hpp')
-rw-r--r--boost/geometry/formulas/differential_quantities.hpp300
1 files changed, 300 insertions, 0 deletions
diff --git a/boost/geometry/formulas/differential_quantities.hpp b/boost/geometry/formulas/differential_quantities.hpp
new file mode 100644
index 0000000000..9a92f14e18
--- /dev/null
+++ b/boost/geometry/formulas/differential_quantities.hpp
@@ -0,0 +1,300 @@
+// Boost.Geometry
+
+// Copyright (c) 2016 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_INVERSE_DIFFERENTIAL_QUANTITIES_HPP
+#define BOOST_GEOMETRY_FORMULAS_INVERSE_DIFFERENTIAL_QUANTITIES_HPP
+
+
+#include <boost/geometry/util/condition.hpp>
+#include <boost/geometry/util/math.hpp>
+
+
+namespace boost { namespace geometry { namespace formula
+{
+
+/*!
+\brief The solution of a part of the inverse problem - differential quantities.
+\author See
+- Charles F.F Karney, Algorithms for geodesics, 2011
+https://arxiv.org/pdf/1109.4448.pdf
+*/
+template <
+ typename CT,
+ bool EnableReducedLength,
+ bool EnableGeodesicScale,
+ unsigned int Order = 2,
+ bool ApproxF = true
+>
+class differential_quantities
+{
+public:
+ static inline void apply(CT const& lon1, CT const& lat1,
+ CT const& lon2, CT const& lat2,
+ CT const& azimuth, CT const& reverse_azimuth,
+ CT const& b, CT const& f,
+ CT & reduced_length, CT & geodesic_scale)
+ {
+ CT const dlon = lon2 - lon1;
+ CT const sin_lat1 = sin(lat1);
+ CT const cos_lat1 = cos(lat1);
+ CT const sin_lat2 = sin(lat2);
+ CT const cos_lat2 = cos(lat2);
+
+ apply(dlon, sin_lat1, cos_lat1, sin_lat2, cos_lat2,
+ azimuth, reverse_azimuth,
+ b, f,
+ reduced_length, geodesic_scale);
+ }
+
+ static inline void apply(CT const& dlon,
+ CT const& sin_lat1, CT const& cos_lat1,
+ CT const& sin_lat2, CT const& cos_lat2,
+ CT const& azimuth, CT const& reverse_azimuth,
+ CT const& b, CT const& f,
+ CT & reduced_length, CT & geodesic_scale)
+ {
+ CT const c0 = 0;
+ CT const c1 = 1;
+ CT const one_minus_f = c1 - f;
+
+ CT const sin_bet1 = one_minus_f * sin_lat1;
+ CT const sin_bet2 = one_minus_f * sin_lat2;
+
+ // equator
+ if (math::equals(sin_bet1, c0) && math::equals(sin_bet2, c0))
+ {
+ CT const sig_12 = math::abs(dlon) / one_minus_f;
+ if (BOOST_GEOMETRY_CONDITION(EnableReducedLength))
+ {
+ CT m12 = sin(sig_12) * b;
+ reduced_length = m12;
+ }
+
+ if (BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
+ {
+ CT M12 = cos(sig_12);
+ geodesic_scale = M12;
+ }
+ }
+ else
+ {
+ CT const c2 = 2;
+ CT const e2 = f * (c2 - f);
+ CT const ep2 = e2 / math::sqr(one_minus_f);
+
+ CT const cos_bet1 = cos_lat1;
+ CT const cos_bet2 = cos_lat2;
+
+ CT const sin_alp1 = sin(azimuth);
+ CT const cos_alp1 = cos(azimuth);
+ //CT const sin_alp2 = sin(reverse_azimuth);
+ CT const cos_alp2 = cos(reverse_azimuth);
+
+ CT sin_sig1 = sin_bet1;
+ CT cos_sig1 = cos_alp1 * cos_bet1;
+ CT sin_sig2 = sin_bet2;
+ CT cos_sig2 = cos_alp2 * cos_bet2;
+
+ normalize(sin_sig1, cos_sig1);
+ normalize(sin_sig2, cos_sig2);
+
+ CT const sin_alp0 = sin_alp1 * cos_bet1;
+ CT const cos_alp0_sqr = c1 - math::sqr(sin_alp0);
+
+ CT const J12 = BOOST_GEOMETRY_CONDITION(ApproxF) ?
+ J12_f(sin_sig1, cos_sig1, sin_sig2, cos_sig2, cos_alp0_sqr, f) :
+ J12_ep_sqr(sin_sig1, cos_sig1, sin_sig2, cos_sig2, cos_alp0_sqr, ep2) ;
+
+ CT const dn1 = math::sqrt(c1 + e2 * math::sqr(sin_lat1));
+ CT const dn2 = math::sqrt(c1 + e2 * math::sqr(sin_lat2));
+
+ if (BOOST_GEOMETRY_CONDITION(EnableReducedLength))
+ {
+ CT const m12_b = dn2 * (cos_sig1 * sin_sig2)
+ - dn1 * (sin_sig1 * cos_sig2)
+ - cos_sig1 * cos_sig2 * J12;
+ CT const m12 = m12_b * b;
+
+ reduced_length = m12;
+ }
+
+ if (BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
+ {
+ CT const cos_sig12 = cos_sig1 * cos_sig2 + sin_sig1 * sin_sig2;
+ CT const t = ep2 * (cos_bet1 - cos_bet2) * (cos_bet1 + cos_bet2) / (dn1 + dn2);
+ CT const M12 = cos_sig12 + (t * sin_sig2 - cos_sig2 * J12) * sin_sig1 / dn1;
+
+ geodesic_scale = M12;
+ }
+ }
+ }
+
+private:
+ /*! Approximation of J12, expanded into taylor series in f
+ Maxima script:
+ ep2: f * (2-f) / ((1-f)^2);
+ k2: ca02 * ep2;
+ assume(f < 1);
+ assume(sig > 0);
+ I1(sig):= integrate(sqrt(1 + k2 * sin(s)^2), s, 0, sig);
+ I2(sig):= integrate(1/sqrt(1 + k2 * sin(s)^2), s, 0, sig);
+ J(sig):= I1(sig) - I2(sig);
+ S: taylor(J(sig), f, 0, 3);
+ S1: factor( 2*integrate(sin(s)^2,s,0,sig)*ca02*f );
+ S2: factor( ((integrate(-6*ca02^2*sin(s)^4+6*ca02*sin(s)^2,s,0,sig)+integrate(-2*ca02^2*sin(s)^4+6*ca02*sin(s)^2,s,0,sig))*f^2)/4 );
+ S3: factor( ((integrate(30*ca02^3*sin(s)^6-54*ca02^2*sin(s)^4+24*ca02*sin(s)^2,s,0,sig)+integrate(6*ca02^3*sin(s)^6-18*ca02^2*sin(s)^4+24*ca02*sin(s)^2,s,0,sig))*f^3)/12 );
+ */
+ static inline CT J12_f(CT const& sin_sig1, CT const& cos_sig1,
+ CT const& sin_sig2, CT const& cos_sig2,
+ CT const& cos_alp0_sqr, CT const& f)
+ {
+ if (Order == 0)
+ {
+ return 0;
+ }
+
+ CT const c2 = 2;
+
+ CT const sig_12 = atan2(cos_sig1 * sin_sig2 - sin_sig1 * cos_sig2,
+ cos_sig1 * cos_sig2 + sin_sig1 * sin_sig2);
+ CT const sin_2sig1 = c2 * cos_sig1 * sin_sig1; // sin(2sig1)
+ CT const sin_2sig2 = c2 * cos_sig2 * sin_sig2; // sin(2sig2)
+ CT const sin_2sig_12 = sin_2sig2 - sin_2sig1;
+ CT const L1 = sig_12 - sin_2sig_12 / c2;
+
+ if (Order == 1)
+ {
+ return cos_alp0_sqr * f * L1;
+ }
+
+ CT const sin_4sig1 = c2 * sin_2sig1 * (math::sqr(cos_sig1) - math::sqr(sin_sig1)); // sin(4sig1)
+ CT const sin_4sig2 = c2 * sin_2sig2 * (math::sqr(cos_sig2) - math::sqr(sin_sig2)); // sin(4sig2)
+ CT const sin_4sig_12 = sin_4sig2 - sin_4sig1;
+
+ CT const c8 = 8;
+ CT const c12 = 12;
+ CT const c16 = 16;
+ CT const c24 = 24;
+
+ CT const L2 = -( cos_alp0_sqr * sin_4sig_12
+ + (-c8 * cos_alp0_sqr + c12) * sin_2sig_12
+ + (c12 * cos_alp0_sqr - c24) * sig_12)
+ / c16;
+
+ if (Order == 2)
+ {
+ return cos_alp0_sqr * f * (L1 + f * L2);
+ }
+
+ CT const c4 = 4;
+ CT const c9 = 9;
+ CT const c48 = 48;
+ CT const c60 = 60;
+ CT const c64 = 64;
+ CT const c96 = 96;
+ CT const c128 = 128;
+ CT const c144 = 144;
+
+ CT const cos_alp0_quad = math::sqr(cos_alp0_sqr);
+ CT const sin3_2sig1 = math::sqr(sin_2sig1) * sin_2sig1;
+ CT const sin3_2sig2 = math::sqr(sin_2sig2) * sin_2sig2;
+ CT const sin3_2sig_12 = sin3_2sig2 - sin3_2sig1;
+
+ CT const A = (c9 * cos_alp0_quad - c12 * cos_alp0_sqr) * sin_4sig_12;
+ CT const B = c4 * cos_alp0_quad * sin3_2sig_12;
+ CT const C = (-c48 * cos_alp0_quad + c96 * cos_alp0_sqr - c64) * sin_2sig_12;
+ CT const D = (c60 * cos_alp0_quad - c144 * cos_alp0_sqr + c128) * sig_12;
+
+ CT const L3 = (A + B + C + D) / c64;
+
+ // Order 3 and higher
+ return cos_alp0_sqr * f * (L1 + f * (L2 + f * L3));
+ }
+
+ /*! Approximation of J12, expanded into taylor series in e'^2
+ Maxima script:
+ k2: ca02 * ep2;
+ assume(sig > 0);
+ I1(sig):= integrate(sqrt(1 + k2 * sin(s)^2), s, 0, sig);
+ I2(sig):= integrate(1/sqrt(1 + k2 * sin(s)^2), s, 0, sig);
+ J(sig):= I1(sig) - I2(sig);
+ S: taylor(J(sig), ep2, 0, 3);
+ S1: factor( integrate(sin(s)^2,s,0,sig)*ca02*ep2 );
+ S2: factor( (integrate(sin(s)^4,s,0,sig)*ca02^2*ep2^2)/2 );
+ S3: factor( (3*integrate(sin(s)^6,s,0,sig)*ca02^3*ep2^3)/8 );
+ */
+ static inline CT J12_ep_sqr(CT const& sin_sig1, CT const& cos_sig1,
+ CT const& sin_sig2, CT const& cos_sig2,
+ CT const& cos_alp0_sqr, CT const& ep_sqr)
+ {
+ if (Order == 0)
+ {
+ return 0;
+ }
+
+ CT const c2 = 2;
+ CT const c4 = 4;
+
+ CT const c2a0ep2 = cos_alp0_sqr * ep_sqr;
+
+ CT const sig_12 = atan2(cos_sig1 * sin_sig2 - sin_sig1 * cos_sig2,
+ cos_sig1 * cos_sig2 + sin_sig1 * sin_sig2); // sig2 - sig1
+ CT const sin_2sig1 = c2 * cos_sig1 * sin_sig1; // sin(2sig1)
+ CT const sin_2sig2 = c2 * cos_sig2 * sin_sig2; // sin(2sig2)
+ CT const sin_2sig_12 = sin_2sig2 - sin_2sig1;
+
+ CT const L1 = (c2 * sig_12 - sin_2sig_12) / c4;
+
+ if (Order == 1)
+ {
+ return c2a0ep2 * L1;
+ }
+
+ CT const c8 = 8;
+ CT const c64 = 64;
+
+ CT const sin_4sig1 = c2 * sin_2sig1 * (math::sqr(cos_sig1) - math::sqr(sin_sig1)); // sin(4sig1)
+ CT const sin_4sig2 = c2 * sin_2sig2 * (math::sqr(cos_sig2) - math::sqr(sin_sig2)); // sin(4sig2)
+ CT const sin_4sig_12 = sin_4sig2 - sin_4sig1;
+
+ CT const L2 = (sin_4sig_12 - c8 * sin_2sig_12 + 12 * sig_12) / c64;
+
+ if (Order == 2)
+ {
+ return c2a0ep2 * (L1 + c2a0ep2 * L2);
+ }
+
+ CT const sin3_2sig1 = math::sqr(sin_2sig1) * sin_2sig1;
+ CT const sin3_2sig2 = math::sqr(sin_2sig2) * sin_2sig2;
+ CT const sin3_2sig_12 = sin3_2sig2 - sin3_2sig1;
+
+ CT const c9 = 9;
+ CT const c48 = 48;
+ CT const c60 = 60;
+ CT const c512 = 512;
+
+ CT const L3 = (c9 * sin_4sig_12 + c4 * sin3_2sig_12 - c48 * sin_2sig_12 + c60 * sig_12) / c512;
+
+ // Order 3 and higher
+ return c2a0ep2 * (L1 + c2a0ep2 * (L2 + c2a0ep2 * L3));
+ }
+
+ static inline void normalize(CT & x, CT & y)
+ {
+ CT const len = math::sqrt(math::sqr(x) + math::sqr(y));
+ x /= len;
+ y /= len;
+ }
+};
+
+}}} // namespace boost::geometry::formula
+
+
+#endif // BOOST_GEOMETRY_FORMULAS_INVERSE_DIFFERENTIAL_QUANTITIES_HPP
generated by cgit v1.2.3 (git 2.25.1) at 2024-04-20 07:51:33 +0000