GeographicLib 1.52
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GeodesicLineExact.cpp
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1/**
2 * \file GeodesicLineExact.cpp
3 * \brief Implementation for GeographicLib::GeodesicLineExact class
4 *
5 * Copyright (c) Charles Karney (2012-2020) <charles@karney.com> and licensed
6 * under the MIT/X11 License. For more information, see
7 * https://geographiclib.sourceforge.io/
8 *
9 * This is a reformulation of the geodesic problem. The notation is as
10 * follows:
11 * - at a general point (no suffix or 1 or 2 as suffix)
12 * - phi = latitude
13 * - beta = latitude on auxiliary sphere
14 * - omega = longitude on auxiliary sphere
15 * - lambda = longitude
16 * - alpha = azimuth of great circle
17 * - sigma = arc length along great circle
18 * - s = distance
19 * - tau = scaled distance (= sigma at multiples of pi/2)
20 * - at northwards equator crossing
21 * - beta = phi = 0
22 * - omega = lambda = 0
23 * - alpha = alpha0
24 * - sigma = s = 0
25 * - a 12 suffix means a difference, e.g., s12 = s2 - s1.
26 * - s and c prefixes mean sin and cos
27 **********************************************************************/
28
30
31namespace GeographicLib {
32
33 using namespace std;
34
35 void GeodesicLineExact::LineInit(const GeodesicExact& g,
36 real lat1, real lon1,
37 real azi1, real salp1, real calp1,
38 unsigned caps) {
39 tiny_ = g.tiny_;
40 _lat1 = Math::LatFix(lat1);
41 _lon1 = lon1;
42 _azi1 = azi1;
43 _salp1 = salp1;
44 _calp1 = calp1;
45 _a = g._a;
46 _f = g._f;
47 _b = g._b;
48 _c2 = g._c2;
49 _f1 = g._f1;
50 _e2 = g._e2;
51 // Always allow latitude and azimuth and unrolling of longitude
52 _caps = caps | LATITUDE | AZIMUTH | LONG_UNROLL;
53
54 real cbet1, sbet1;
55 Math::sincosd(Math::AngRound(_lat1), sbet1, cbet1); sbet1 *= _f1;
56 // Ensure cbet1 = +epsilon at poles
57 Math::norm(sbet1, cbet1); cbet1 = max(tiny_, cbet1);
58 _dn1 = (_f >= 0 ? sqrt(1 + g._ep2 * Math::sq(sbet1)) :
59 sqrt(1 - _e2 * Math::sq(cbet1)) / _f1);
60
61 // Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
62 _salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|]
63 // Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
64 // is slightly better (consider the case salp1 = 0).
65 _calp0 = hypot(_calp1, _salp1 * sbet1);
66 // Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
67 // sig = 0 is nearest northward crossing of equator.
68 // With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
69 // With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
70 // With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
71 // Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
72 // With alp0 in (0, pi/2], quadrants for sig and omg coincide.
73 // No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
74 // With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
75 _ssig1 = sbet1; _somg1 = _salp0 * sbet1;
76 _csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1;
77 // Without normalization we have schi1 = somg1.
78 _cchi1 = _f1 * _dn1 * _comg1;
79 Math::norm(_ssig1, _csig1); // sig1 in (-pi, pi]
80 // Math::norm(_somg1, _comg1); -- don't need to normalize!
81 // Math::norm(_schi1, _cchi1); -- don't need to normalize!
82
83 _k2 = Math::sq(_calp0) * g._ep2;
84 _E.Reset(-_k2, -g._ep2, 1 + _k2, 1 + g._ep2);
85
86 if (_caps & CAP_E) {
87 _E0 = _E.E() / (Math::pi() / 2);
88 _E1 = _E.deltaE(_ssig1, _csig1, _dn1);
89 real s = sin(_E1), c = cos(_E1);
90 // tau1 = sig1 + B11
91 _stau1 = _ssig1 * c + _csig1 * s;
92 _ctau1 = _csig1 * c - _ssig1 * s;
93 // Not necessary because Einv inverts E
94 // _E1 = -_E.deltaEinv(_stau1, _ctau1);
95 }
96
97 if (_caps & CAP_D) {
98 _D0 = _E.D() / (Math::pi() / 2);
99 _D1 = _E.deltaD(_ssig1, _csig1, _dn1);
100 }
101
102 if (_caps & CAP_H) {
103 _H0 = _E.H() / (Math::pi() / 2);
104 _H1 = _E.deltaH(_ssig1, _csig1, _dn1);
105 }
106
107 if (_caps & CAP_C4) {
108 real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2);
109 g.C4f(eps, _C4a);
110 // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
111 _A4 = Math::sq(_a) * _calp0 * _salp0 * _e2;
112 _B41 = GeodesicExact::CosSeries(_ssig1, _csig1, _C4a, nC4_);
113 }
114
115 _a13 = _s13 = Math::NaN();
116 }
117
119 real lat1, real lon1, real azi1,
120 unsigned caps) {
121 azi1 = Math::AngNormalize(azi1);
122 real salp1, calp1;
123 // Guard against underflow in salp0. Also -0 is converted to +0.
124 Math::sincosd(Math::AngRound(azi1), salp1, calp1);
125 LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
126 }
127
129 real lat1, real lon1,
130 real azi1, real salp1, real calp1,
131 unsigned caps,
132 bool arcmode, real s13_a13) {
133 LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
134 GenSetDistance(arcmode, s13_a13);
135 }
136
137 Math::real GeodesicLineExact::GenPosition(bool arcmode, real s12_a12,
138 unsigned outmask,
139 real& lat2, real& lon2, real& azi2,
140 real& s12, real& m12,
141 real& M12, real& M21,
142 real& S12) const {
143 outmask &= _caps & OUT_MASK;
144 if (!( Init() && (arcmode || (_caps & (OUT_MASK & DISTANCE_IN))) ))
145 // Uninitialized or impossible distance calculation requested
146 return Math::NaN();
147
148 // Avoid warning about uninitialized B12.
149 real sig12, ssig12, csig12, E2 = 0, AB1 = 0;
150 if (arcmode) {
151 // Interpret s12_a12 as spherical arc length
152 sig12 = s12_a12 * Math::degree();
153 real s12a = abs(s12_a12);
154 s12a -= 180 * floor(s12a / 180);
155 ssig12 = s12a == 0 ? 0 : sin(sig12);
156 csig12 = s12a == 90 ? 0 : cos(sig12);
157 } else {
158 // Interpret s12_a12 as distance
159 real
160 tau12 = s12_a12 / (_b * _E0),
161 s = sin(tau12),
162 c = cos(tau12);
163 // tau2 = tau1 + tau12
164 E2 = - _E.deltaEinv(_stau1 * c + _ctau1 * s, _ctau1 * c - _stau1 * s);
165 sig12 = tau12 - (E2 - _E1);
166 ssig12 = sin(sig12);
167 csig12 = cos(sig12);
168 }
169
170 real ssig2, csig2, sbet2, cbet2, salp2, calp2;
171 // sig2 = sig1 + sig12
172 ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
173 csig2 = _csig1 * csig12 - _ssig1 * ssig12;
174 real dn2 = _E.Delta(ssig2, csig2);
175 if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
176 if (arcmode) {
177 E2 = _E.deltaE(ssig2, csig2, dn2);
178 }
179 AB1 = _E0 * (E2 - _E1);
180 }
181 // sin(bet2) = cos(alp0) * sin(sig2)
182 sbet2 = _calp0 * ssig2;
183 // Alt: cbet2 = hypot(csig2, salp0 * ssig2);
184 cbet2 = hypot(_salp0, _calp0 * csig2);
185 if (cbet2 == 0)
186 // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
187 cbet2 = csig2 = tiny_;
188 // tan(alp0) = cos(sig2)*tan(alp2)
189 salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
190
191 if (outmask & DISTANCE)
192 s12 = arcmode ? _b * (_E0 * sig12 + AB1) : s12_a12;
193
194 if (outmask & LONGITUDE) {
195 real somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize
196 E = copysign(real(1), _salp0); // east-going?
197 // Without normalization we have schi2 = somg2.
198 real cchi2 = _f1 * dn2 * comg2;
199 real chi12 = outmask & LONG_UNROLL
200 ? E * (sig12
201 - (atan2( ssig2, csig2) - atan2( _ssig1, _csig1))
202 + (atan2(E * somg2, cchi2) - atan2(E * _somg1, _cchi1)))
203 : atan2(somg2 * _cchi1 - cchi2 * _somg1,
204 cchi2 * _cchi1 + somg2 * _somg1);
205 real lam12 = chi12 -
206 _e2/_f1 * _salp0 * _H0 *
207 (sig12 + (_E.deltaH(ssig2, csig2, dn2) - _H1));
208 real lon12 = lam12 / Math::degree();
209 lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 :
211 Math::AngNormalize(lon12));
212 }
213
214 if (outmask & LATITUDE)
215 lat2 = Math::atan2d(sbet2, _f1 * cbet2);
216
217 if (outmask & AZIMUTH)
218 azi2 = Math::atan2d(salp2, calp2);
219
220 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
221 real J12 = _k2 * _D0 * (sig12 + (_E.deltaD(ssig2, csig2, dn2) - _D1));
222 if (outmask & REDUCEDLENGTH)
223 // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
224 // accurate cancellation in the case of coincident points.
225 m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
226 - _csig1 * csig2 * J12);
227 if (outmask & GEODESICSCALE) {
228 real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
229 M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
230 M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2;
231 }
232 }
233
234 if (outmask & AREA) {
235 real
236 B42 = GeodesicExact::CosSeries(ssig2, csig2, _C4a, nC4_);
237 real salp12, calp12;
238 if (_calp0 == 0 || _salp0 == 0) {
239 // alp12 = alp2 - alp1, used in atan2 so no need to normalize
240 salp12 = salp2 * _calp1 - calp2 * _salp1;
241 calp12 = calp2 * _calp1 + salp2 * _salp1;
242 // We used to include here some patch up code that purported to deal
243 // with nearly meridional geodesics properly. However, this turned out
244 // to be wrong once _salp1 = -0 was allowed (via
245 // GeodesicExact::InverseLine). In fact, the calculation of {s,c}alp12
246 // was already correct (following the IEEE rules for handling signed
247 // zeros). So the patch up code was unnecessary (as well as
248 // dangerous).
249 } else {
250 // tan(alp) = tan(alp0) * sec(sig)
251 // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
252 // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
253 // If csig12 > 0, write
254 // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
255 // else
256 // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
257 // No need to normalize
258 salp12 = _calp0 * _salp0 *
259 (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
260 ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
261 calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2;
262 }
263 S12 = _c2 * atan2(salp12, calp12) + _A4 * (B42 - _B41);
264 }
265
266 return arcmode ? s12_a12 : sig12 / Math::degree();
267 }
268
270 _s13 = s13;
271 real t;
272 // This will set _a13 to NaN if the GeodesicLineExact doesn't have the
273 // DISTANCE_IN capability.
274 _a13 = GenPosition(false, _s13, 0u, t, t, t, t, t, t, t, t);
275 }
276
278 _a13 = a13;
279 // In case the GeodesicLineExact doesn't have the DISTANCE capability.
280 _s13 = Math::NaN();
281 real t;
282 GenPosition(true, _a13, DISTANCE, t, t, t, _s13, t, t, t, t);
283 }
284
285 void GeodesicLineExact::GenSetDistance(bool arcmode, real s13_a13) {
286 arcmode ? SetArc(s13_a13) : SetDistance(s13_a13);
287 }
288
289} // namespace GeographicLib
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
Header for GeographicLib::GeodesicLineExact class.
Math::real deltaE(real sn, real cn, real dn) const
void Reset(real k2=0, real alpha2=0)
Math::real Delta(real sn, real cn) const
Math::real deltaD(real sn, real cn, real dn) const
Math::real deltaH(real sn, real cn, real dn) const
Math::real deltaEinv(real stau, real ctau) const
Exact geodesic calculations.
Math::real GenPosition(bool arcmode, real s12_a12, unsigned outmask, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21, real &S12) const
void GenSetDistance(bool arcmode, real s13_a13)
static T AngNormalize(T x)
Definition: Math.hpp:420
static T degree()
Definition: Math.hpp:159
static T LatFix(T x)
Definition: Math.hpp:433
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:126
static T atan2d(T y, T x)
Definition: Math.cpp:180
static void norm(T &x, T &y)
Definition: Math.hpp:355
static T AngRound(T x)
Definition: Math.cpp:117
static T sq(T x)
Definition: Math.hpp:171
static T pi()
Definition: Math.hpp:149
static T NaN()
Definition: Math.cpp:260
Namespace for GeographicLib.
Definition: Accumulator.cpp:12