3 // Copyright (c) 2016-2019 Oracle and/or its affiliates.
5 // Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle
7 // Use, modification and distribution is subject to the Boost Software License,
8 // Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
9 // http://www.boost.org/LICENSE_1_0.txt)
11 #ifndef BOOST_GEOMETRY_FORMULAS_INVERSE_DIFFERENTIAL_QUANTITIES_HPP
12 #define BOOST_GEOMETRY_FORMULAS_INVERSE_DIFFERENTIAL_QUANTITIES_HPP
14 #include <boost/geometry/core/assert.hpp>
16 #include <boost/geometry/util/condition.hpp>
17 #include <boost/geometry/util/math.hpp>
20 namespace boost { namespace geometry { namespace formula
24 \brief The solution of a part of the inverse problem - differential quantities.
26 - Charles F.F Karney, Algorithms for geodesics, 2011
27 https://arxiv.org/pdf/1109.4448.pdf
31 bool EnableReducedLength,
32 bool EnableGeodesicScale,
33 unsigned int Order = 2,
36 class differential_quantities
39 static inline void apply(CT const& lon1, CT const& lat1,
40 CT const& lon2, CT const& lat2,
41 CT const& azimuth, CT const& reverse_azimuth,
42 CT const& b, CT const& f,
43 CT & reduced_length, CT & geodesic_scale)
45 CT const dlon = lon2 - lon1;
46 CT const sin_lat1 = sin(lat1);
47 CT const cos_lat1 = cos(lat1);
48 CT const sin_lat2 = sin(lat2);
49 CT const cos_lat2 = cos(lat2);
51 apply(dlon, sin_lat1, cos_lat1, sin_lat2, cos_lat2,
52 azimuth, reverse_azimuth,
54 reduced_length, geodesic_scale);
57 static inline void apply(CT const& dlon,
58 CT const& sin_lat1, CT const& cos_lat1,
59 CT const& sin_lat2, CT const& cos_lat2,
60 CT const& azimuth, CT const& reverse_azimuth,
61 CT const& b, CT const& f,
62 CT & reduced_length, CT & geodesic_scale)
66 CT const one_minus_f = c1 - f;
68 CT sin_bet1 = one_minus_f * sin_lat1;
69 CT sin_bet2 = one_minus_f * sin_lat2;
72 if (math::equals(sin_bet1, c0) && math::equals(sin_bet2, c0))
74 CT const sig_12 = dlon / one_minus_f;
75 if (BOOST_GEOMETRY_CONDITION(EnableReducedLength))
77 BOOST_GEOMETRY_ASSERT((-math::pi<CT>() <= azimuth && azimuth <= math::pi<CT>()));
79 int azi_sign = math::sign(azimuth) >= 0 ? 1 : -1; // for antipodal
80 CT m12 = azi_sign * sin(sig_12) * b;
84 if (BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
93 CT const e2 = f * (c2 - f);
94 CT const ep2 = e2 / math::sqr(one_minus_f);
96 CT const sin_alp1 = sin(azimuth);
97 CT const cos_alp1 = cos(azimuth);
98 //CT const sin_alp2 = sin(reverse_azimuth);
99 CT const cos_alp2 = cos(reverse_azimuth);
101 CT cos_bet1 = cos_lat1;
102 CT cos_bet2 = cos_lat2;
104 normalize(sin_bet1, cos_bet1);
105 normalize(sin_bet2, cos_bet2);
107 CT sin_sig1 = sin_bet1;
108 CT cos_sig1 = cos_alp1 * cos_bet1;
109 CT sin_sig2 = sin_bet2;
110 CT cos_sig2 = cos_alp2 * cos_bet2;
112 normalize(sin_sig1, cos_sig1);
113 normalize(sin_sig2, cos_sig2);
115 CT const sin_alp0 = sin_alp1 * cos_bet1;
116 CT const cos_alp0_sqr = c1 - math::sqr(sin_alp0);
118 CT const J12 = BOOST_GEOMETRY_CONDITION(ApproxF) ?
119 J12_f(sin_sig1, cos_sig1, sin_sig2, cos_sig2, cos_alp0_sqr, f) :
120 J12_ep_sqr(sin_sig1, cos_sig1, sin_sig2, cos_sig2, cos_alp0_sqr, ep2) ;
122 CT const dn1 = math::sqrt(c1 + ep2 * math::sqr(sin_bet1));
123 CT const dn2 = math::sqrt(c1 + ep2 * math::sqr(sin_bet2));
125 if (BOOST_GEOMETRY_CONDITION(EnableReducedLength))
127 CT const m12_b = dn2 * (cos_sig1 * sin_sig2)
128 - dn1 * (sin_sig1 * cos_sig2)
129 - cos_sig1 * cos_sig2 * J12;
130 CT const m12 = m12_b * b;
132 reduced_length = m12;
135 if (BOOST_GEOMETRY_CONDITION(EnableGeodesicScale))
137 CT const cos_sig12 = cos_sig1 * cos_sig2 + sin_sig1 * sin_sig2;
138 CT const t = ep2 * (cos_bet1 - cos_bet2) * (cos_bet1 + cos_bet2) / (dn1 + dn2);
139 CT const M12 = cos_sig12 + (t * sin_sig2 - cos_sig2 * J12) * sin_sig1 / dn1;
141 geodesic_scale = M12;
147 /*! Approximation of J12, expanded into taylor series in f
149 ep2: f * (2-f) / ((1-f)^2);
153 I1(sig):= integrate(sqrt(1 + k2 * sin(s)^2), s, 0, sig);
154 I2(sig):= integrate(1/sqrt(1 + k2 * sin(s)^2), s, 0, sig);
155 J(sig):= I1(sig) - I2(sig);
156 S: taylor(J(sig), f, 0, 3);
157 S1: factor( 2*integrate(sin(s)^2,s,0,sig)*ca02*f );
158 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 );
159 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 );
161 static inline CT J12_f(CT const& sin_sig1, CT const& cos_sig1,
162 CT const& sin_sig2, CT const& cos_sig2,
163 CT const& cos_alp0_sqr, CT const& f)
172 CT const sig_12 = atan2(cos_sig1 * sin_sig2 - sin_sig1 * cos_sig2,
173 cos_sig1 * cos_sig2 + sin_sig1 * sin_sig2);
174 CT const sin_2sig1 = c2 * cos_sig1 * sin_sig1; // sin(2sig1)
175 CT const sin_2sig2 = c2 * cos_sig2 * sin_sig2; // sin(2sig2)
176 CT const sin_2sig_12 = sin_2sig2 - sin_2sig1;
177 CT const L1 = sig_12 - sin_2sig_12 / c2;
181 return cos_alp0_sqr * f * L1;
184 CT const sin_4sig1 = c2 * sin_2sig1 * (math::sqr(cos_sig1) - math::sqr(sin_sig1)); // sin(4sig1)
185 CT const sin_4sig2 = c2 * sin_2sig2 * (math::sqr(cos_sig2) - math::sqr(sin_sig2)); // sin(4sig2)
186 CT const sin_4sig_12 = sin_4sig2 - sin_4sig1;
193 CT const L2 = -( cos_alp0_sqr * sin_4sig_12
194 + (-c8 * cos_alp0_sqr + c12) * sin_2sig_12
195 + (c12 * cos_alp0_sqr - c24) * sig_12)
200 return cos_alp0_sqr * f * (L1 + f * L2);
212 CT const cos_alp0_quad = math::sqr(cos_alp0_sqr);
213 CT const sin3_2sig1 = math::sqr(sin_2sig1) * sin_2sig1;
214 CT const sin3_2sig2 = math::sqr(sin_2sig2) * sin_2sig2;
215 CT const sin3_2sig_12 = sin3_2sig2 - sin3_2sig1;
217 CT const A = (c9 * cos_alp0_quad - c12 * cos_alp0_sqr) * sin_4sig_12;
218 CT const B = c4 * cos_alp0_quad * sin3_2sig_12;
219 CT const C = (-c48 * cos_alp0_quad + c96 * cos_alp0_sqr - c64) * sin_2sig_12;
220 CT const D = (c60 * cos_alp0_quad - c144 * cos_alp0_sqr + c128) * sig_12;
222 CT const L3 = (A + B + C + D) / c64;
224 // Order 3 and higher
225 return cos_alp0_sqr * f * (L1 + f * (L2 + f * L3));
228 /*! Approximation of J12, expanded into taylor series in e'^2
232 I1(sig):= integrate(sqrt(1 + k2 * sin(s)^2), s, 0, sig);
233 I2(sig):= integrate(1/sqrt(1 + k2 * sin(s)^2), s, 0, sig);
234 J(sig):= I1(sig) - I2(sig);
235 S: taylor(J(sig), ep2, 0, 3);
236 S1: factor( integrate(sin(s)^2,s,0,sig)*ca02*ep2 );
237 S2: factor( (integrate(sin(s)^4,s,0,sig)*ca02^2*ep2^2)/2 );
238 S3: factor( (3*integrate(sin(s)^6,s,0,sig)*ca02^3*ep2^3)/8 );
240 static inline CT J12_ep_sqr(CT const& sin_sig1, CT const& cos_sig1,
241 CT const& sin_sig2, CT const& cos_sig2,
242 CT const& cos_alp0_sqr, CT const& ep_sqr)
252 CT const c2a0ep2 = cos_alp0_sqr * ep_sqr;
254 CT const sig_12 = atan2(cos_sig1 * sin_sig2 - sin_sig1 * cos_sig2,
255 cos_sig1 * cos_sig2 + sin_sig1 * sin_sig2); // sig2 - sig1
256 CT const sin_2sig1 = c2 * cos_sig1 * sin_sig1; // sin(2sig1)
257 CT const sin_2sig2 = c2 * cos_sig2 * sin_sig2; // sin(2sig2)
258 CT const sin_2sig_12 = sin_2sig2 - sin_2sig1;
260 CT const L1 = (c2 * sig_12 - sin_2sig_12) / c4;
270 CT const sin_4sig1 = c2 * sin_2sig1 * (math::sqr(cos_sig1) - math::sqr(sin_sig1)); // sin(4sig1)
271 CT const sin_4sig2 = c2 * sin_2sig2 * (math::sqr(cos_sig2) - math::sqr(sin_sig2)); // sin(4sig2)
272 CT const sin_4sig_12 = sin_4sig2 - sin_4sig1;
274 CT const L2 = (sin_4sig_12 - c8 * sin_2sig_12 + 12 * sig_12) / c64;
278 return c2a0ep2 * (L1 + c2a0ep2 * L2);
281 CT const sin3_2sig1 = math::sqr(sin_2sig1) * sin_2sig1;
282 CT const sin3_2sig2 = math::sqr(sin_2sig2) * sin_2sig2;
283 CT const sin3_2sig_12 = sin3_2sig2 - sin3_2sig1;
290 CT const L3 = (c9 * sin_4sig_12 + c4 * sin3_2sig_12 - c48 * sin_2sig_12 + c60 * sig_12) / c512;
292 // Order 3 and higher
293 return c2a0ep2 * (L1 + c2a0ep2 * (L2 + c2a0ep2 * L3));
296 static inline void normalize(CT & x, CT & y)
298 CT const len = math::sqrt(math::sqr(x) + math::sqr(y));
304 }}} // namespace boost::geometry::formula
307 #endif // BOOST_GEOMETRY_FORMULAS_INVERSE_DIFFERENTIAL_QUANTITIES_HPP